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Question 13532: A Power-of-Product Identity in a Qualified Commutative Ring

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0026-13532 · q-20261014-advance-20261028-r01-finite-commutative-ring-0057

Original question and givens

Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 4, 7, 0], [2, 3, 5, 4, 7, 6, 0, 1], [3, 5, 4, 6, 0, 7, 1, 2], [4, 6, 7, 0, 2, 1, 3, 5], [5, 4, 6, 7, 1, 0, 2, 3], [6, 7, 0, 1, 3, 2, 5, 4], [7, 0, 1, 2, 5, 3, 4, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 2, 0, 5, 6], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 6, 0, 5, 2], [0, 7, 6, 4, 3, 5, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Require multiplication to commute. a=7, b=5, nonnegative exponent=7. Give (a,b,exponent,ab,lhs=(ab)^exponent,rhs=a^exponent*b^exponent,equal?).

  • G001 — Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 4, 7, 0], [2, 3, 5, 4, 7, 6, 0, 1], [3, 5, 4, 6, 0, 7, 1, 2], [4, 6, 7, 0, 2, 1, 3, 5], [5, 4, 6, 7, 1, 0, 2, 3], [6, 7, 0, 1, 3, 2, 5, 4], [7, 0, 1, 2, 5, 3, 4, 6]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 2, 0, 5, 6], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 6, 0, 5, 2], [0, 7, 6, 4, 3, 5, 2, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Require multiplication to commute. a=7, b=5, nonnegative exponent=7. Give (a,b,exponent,ab,lhs=(ab)^exponent,rhs=a^exponent*b^exponent,equal?).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

same parent carrier Fin(8);addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 5, 6, 4, 7, 0], [2, 3, 5, 4, 7, 6, 0, 1], [3, 5, 4, 6, 0, 7, 1, 2], [4, 6, 7, 0, 2, 1, 3, 5], [5, 4, 6, 7, 1, 0, 2, 3], [6, 7, 0, 1, 3, 2, 5, 4], [7, 0, 1, 2, 5, 3, 4, 6]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 5, 6, 2, 0, 5, 6], [0, 3, 6, 1, 7, 5, 2, 4], [0, 4, 2, 7, 1, 5, 6, 3], [0, 5, 0, 5, 5, 0, 0, 5], [0, 6, 5, 2, 6, 0, 5, 2], [0, 7, 6, 4, 3, 5, 2, 1]]

After

both complete operation tables typed and closed

Conditions: none

Bind subsequent ideals or commutativity to this same parent ring.Preserve the complete original tables;carrier labels are not integer arithmetic values.

Step identifiers and dependencies

Rule: parent_ring_original_complete_tables · Uses: G001

S002 · implies

Before

every additive triple checked;first associativity failure=null;all additive pairs commute=Yes

After

additive zero candidates=[0];actual zero=0;two-sided additive inverses in label order=[0, 7, 6, 4, 3, 5, 2, 1]

Conditions: none

Audit additive associativity,commutativity,the two-sided zero and each inverse over the complete carrier.This qualified parent is the problem's premise,checked by complete finite conditions rather than samples.

Step identifiers and dependencies

Rule: parent_additive_abelian_group_complete_audit · Uses: G001, S001

S003 · implies

Before

every multiplicative triple checked;first associativity failure=null

After

multiplicative identity candidates=[1];actual one=1

Conditions: none

Multiplication here requires associativity and a two-sided one,not commutativity or inverses of every nonzero element.

Step identifiers and dependencies

Rule: parent_multiplicative_monoid_complete_audit · Uses: G001, S002

S004 · implies

Before

all 512 left and 512 right distributivity triples checked

After

first left failure=null;first right failure=null;zero=0;one=1;distinct=Yes

Conditions: none

Audit both distributive obligations completely and separately.Zero and one must be distinct actual elements.Unital alone implies neither a commutative ring,domain nor field.

Step identifiers and dependencies

Rule: parent_both_distributivities_and_distinct_identities · Uses: G001, S003

S005 · implies

Before

unordered parent elements=[0, 1];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S004

S006 · implies

Before

unordered parent elements=[0, 2];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S005

S007 · implies

Before

unordered parent elements=[0, 3];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S006

S008 · implies

Before

unordered parent elements=[0, 4];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S007

S009 · implies

Before

unordered parent elements=[0, 5];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S008

S010 · implies

Before

unordered parent elements=[0, 6];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S009

S011 · implies

Before

unordered parent elements=[0, 7];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S010

S012 · implies

Before

unordered parent elements=[1, 2];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S011

S013 · implies

Before

unordered parent elements=[1, 3];a*b=3;b*a=3

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S012

S014 · implies

Before

unordered parent elements=[1, 4];a*b=4;b*a=4

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S013

S015 · implies

Before

unordered parent elements=[1, 5];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S014

S016 · implies

Before

unordered parent elements=[1, 6];a*b=6;b*a=6

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S015

S017 · implies

Before

unordered parent elements=[1, 7];a*b=7;b*a=7

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S016

S018 · implies

Before

unordered parent elements=[2, 3];a*b=6;b*a=6

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S017

S019 · implies

Before

unordered parent elements=[2, 4];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S018

S020 · implies

Before

unordered parent elements=[2, 5];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S019

S021 · implies

Before

unordered parent elements=[2, 6];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S020

S022 · implies

Before

unordered parent elements=[2, 7];a*b=6;b*a=6

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S021

S023 · implies

Before

unordered parent elements=[3, 4];a*b=7;b*a=7

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S022

S024 · implies

Before

unordered parent elements=[3, 5];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S023

S025 · implies

Before

unordered parent elements=[3, 6];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S024

S026 · implies

Before

unordered parent elements=[3, 7];a*b=4;b*a=4

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S025

S027 · implies

Before

unordered parent elements=[4, 5];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S026

S028 · implies

Before

unordered parent elements=[4, 6];a*b=6;b*a=6

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S027

S029 · implies

Before

unordered parent elements=[4, 7];a*b=3;b*a=3

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S028

S030 · implies

Before

unordered parent elements=[5, 6];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S029

S031 · implies

Before

unordered parent elements=[5, 7];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S030

S032 · implies

Before

unordered parent elements=[6, 7];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S031

S033 · implies

Before

all multiplicative noncommuting records=[]

After

parent ring commutative=Yes;zero divisors not excluded

Conditions: none

This adds multiplicative commutativity only.It excludes neither zero divisors nor lack of multiplicative inverses.

Step identifiers and dependencies

Rule: commutative_ring_extra_obligation_not_field · Uses: G001, S032

S034 · implies

Before

a times b:operation=multiplication;left label=7;right label=5;table row=[0, 7, 6, 4, 3, 5, 2, 1]

After

result label=5

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S033

S035 · implies

Before

product power:base label=5;nonnegative exponent=7;actual one=1

After

initial power0=1

Conditions: none

Start every nonnegative power at the actual one,including exponent0;its label need not be1.

Step identifiers and dependencies

Rule: unital_ring_power_zero_boundary · Uses: G001, S034

S036 · implies

Before

product power step1:operation=multiplication;left label=1;right label=5;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=5

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S035

S037 · implies

Before

product power step2:operation=multiplication;left label=5;right label=5;table row=[0, 5, 0, 5, 5, 0, 0, 5]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S036

S038 · implies

Before

product power step3:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S037

S039 · implies

Before

product power step4:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S038

S040 · implies

Before

product power step5:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S039

S041 · implies

Before

product power step6:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S040

S042 · implies

Before

product power step7:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S041

S043 · implies

Before

a power:base label=7;nonnegative exponent=7;actual one=1

After

initial power0=1

Conditions: none

Start every nonnegative power at the actual one,including exponent0;its label need not be1.

Step identifiers and dependencies

Rule: unital_ring_power_zero_boundary · Uses: G001, S042

S044 · implies

Before

a power step1:operation=multiplication;left label=1;right label=7;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=7

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S043

S045 · implies

Before

a power step2:operation=multiplication;left label=7;right label=7;table row=[0, 7, 6, 4, 3, 5, 2, 1]

After

result label=1

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S044

S046 · implies

Before

a power step3:operation=multiplication;left label=1;right label=7;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=7

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S045

S047 · implies

Before

a power step4:operation=multiplication;left label=7;right label=7;table row=[0, 7, 6, 4, 3, 5, 2, 1]

After

result label=1

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S046

S048 · implies

Before

a power step5:operation=multiplication;left label=1;right label=7;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=7

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S047

S049 · implies

Before

a power step6:operation=multiplication;left label=7;right label=7;table row=[0, 7, 6, 4, 3, 5, 2, 1]

After

result label=1

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S048

S050 · implies

Before

a power step7:operation=multiplication;left label=1;right label=7;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=7

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S049

S051 · implies

Before

b power:base label=5;nonnegative exponent=7;actual one=1

After

initial power0=1

Conditions: none

Start every nonnegative power at the actual one,including exponent0;its label need not be1.

Step identifiers and dependencies

Rule: unital_ring_power_zero_boundary · Uses: G001, S050

S052 · implies

Before

b power step1:operation=multiplication;left label=1;right label=5;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=5

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S051

S053 · implies

Before

b power step2:operation=multiplication;left label=5;right label=5;table row=[0, 5, 0, 5, 5, 0, 0, 5]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S052

S054 · implies

Before

b power step3:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S053

S055 · implies

Before

b power step4:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S054

S056 · implies

Before

b power step5:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S055

S057 · implies

Before

b power step6:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S056

S058 · implies

Before

b power step7:operation=multiplication;left label=0;right label=5;table row=[0, 0, 0, 0, 0, 0, 0, 0]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S057

S059 · implies

Before

product of powers:operation=multiplication;left label=7;right label=0;table row=[0, 7, 6, 4, 3, 5, 2, 1]

After

result label=0

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S058

S060 · implies

Before

whole parent multiplication commutes;exponent=7;lhs=0;rhs=0

After

(a*b)^k=a^k*b^k under associativity and commutativity,including k0

Conditions: none

Associativity and whole-ring commutativity permit regrouping the finite factors.At k=0 both sides are one.This instance is not used to establish commutativity.

Step identifiers and dependencies

Rule: commutative_power_product_rearrangement · Uses: G001, S059

Reverse checks

  • S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(7, 5, 7, 5, 0, 0, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 7594f7afbfaa0724c691e18d0859803de16c98f0548dbb7f19afe6c06988b7d6
  • candidate record SHA-256: 7dffc25cb3a3672d7f2addb51cc5243ed5878890004bc43983e20dc62dda7338
  • semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798
  • question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON