Theorem crngocom 37172

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 37168	. . . 4 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
5	4	simprbi 495	. . 3 ⊢ (𝑅 ∈ CRingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
6		oveq1 7418	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
7		oveq2 7419	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴))
8	6, 7	eqeq12d 2746	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴)))
9		oveq2 7419	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
10		oveq1 7418	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴))
11	9, 10	eqeq12d 2746	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
12	8, 11	rspc2v 3621	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
13	5, 12	mpan9 505	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
14	13	3impb 1113	1 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ran crn 5676  ‘cfv 6542  (class class class)co 7411  1^st c1st 7975  2^nd c2nd 7976  RingOpscrngo 37065  CRingOpsccring 37164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-1st 7977  df-2nd 7978  df-rngo 37066  df-com2 37161  df-crngo 37165

This theorem is referenced by:  crngm23  37173  crngohomfo  37177  isidlc  37186  dmncan2  37248