Theorem crngocom 38577

Description: The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)

Hypotheses

Ref	Expression
crngocom.1	⊢ 𝐺 = (1st ‘𝑅)
crngocom.2	⊢ 𝐻 = (2nd ‘𝑅)
crngocom.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
crngocom	⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))

Proof of Theorem crngocom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngocom.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		crngocom.2	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
3		crngocom.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	iscrngo2 38573	. . . 4 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
5	4	simprbi 502	. . 3 ⊢ (𝑅 ∈ CRingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
6		oveq1 7420	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
7		oveq2 7421	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴))
8	6, 7	eqeq12d 2785	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴)))
9		oveq2 7421	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
10		oveq1 7420	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴))
11	9, 10	eqeq12d 2785	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
12	8, 11	rspc2v 3601	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
13	5, 12	mpan9 515	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
14	13	3impb 1130	1 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ran crn 5665  ‘cfv 6539  (class class class)co 7413  1^st c1st 7986  2^nd c2nd 7987  RingOpscrngo 38470  CRingOpsccring 38569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-iota 6495  df-fun 6541  df-fv 6547  df-ov 7416  df-1st 7988  df-2nd 7989  df-rngo 38471  df-com2 38566  df-crngo 38570

This theorem is referenced by:  crngm23  38578  crngohomfo  38582  isidlc  38591  dmncan2  38653