Theorem crngocom 38936

Description: Obsolete theorem, use crngcom 20478 instead. The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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 38932	. . . 4 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
5	4	simprbi 503	. . 3 ⊢ (𝑅 ∈ CRingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
6		oveq1 7427	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
7		oveq2 7428	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴))
8	6, 7	eqeq12d 2776	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴)))
9		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
10		oveq1 7427	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴))
11	9, 10	eqeq12d 2776	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
12	8, 11	rspc2v 3586	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)))
13	5, 12	mpan9 516	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
14	13	3impb 1132	1 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ran crn 5651  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  RingOpscrngo 38829  CRingOpsccring 38928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-1st 8001  df-2nd 8002  df-rngo 38830  df-com2 38925  df-crngo 38929

This theorem is used by:  crngm23  38937  crngohomfo  38941  isidlc  38950  dmncan2  39012