Theorem iscrngo2 38021

Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)

Hypotheses

Ref	Expression
iscring2.1	⊢ 𝐺 = (1st ‘𝑅)
iscring2.2	⊢ 𝐻 = (2nd ‘𝑅)
iscring2.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
iscrngo2	⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem iscrngo2

Step	Hyp	Ref	Expression
1		iscrngo 38020	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2		relrngo 37920	. . . . 5 ⊢ Rel RingOps
3		1st2nd 8038	. . . . 5 ⊢ ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
4	2, 3	mpan 690	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
5		eleq1 2822	. . . . 5 ⊢ (𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ → (𝑅 ∈ Com2 ↔ ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ ∈ Com2))
6		iscring2.3	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
7		iscring2.1	. . . . . . . . 9 ⊢ 𝐺 = (1st ‘𝑅)
8	7	rneqi 5917	. . . . . . . 8 ⊢ ran 𝐺 = ran (1st ‘𝑅)
9	6, 8	eqtri 2758	. . . . . . 7 ⊢ 𝑋 = ran (1st ‘𝑅)
10	9	raleqi 3303	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥) ↔ ∀𝑥 ∈ ran (1st ‘𝑅)∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
11		iscring2.2	. . . . . . . . . 10 ⊢ 𝐻 = (2nd ‘𝑅)
12	11	oveqi 7418	. . . . . . . . 9 ⊢ (𝑥𝐻𝑦) = (𝑥(2nd ‘𝑅)𝑦)
13	11	oveqi 7418	. . . . . . . . 9 ⊢ (𝑦𝐻𝑥) = (𝑦(2nd ‘𝑅)𝑥)
14	12, 13	eqeq12i 2753	. . . . . . . 8 ⊢ ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
15	9, 14	raleqbii 3323	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
16	15	ralbii 3082	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
17		fvex 6889	. . . . . . 7 ⊢ (1st ‘𝑅) ∈ V
18		fvex 6889	. . . . . . 7 ⊢ (2nd ‘𝑅) ∈ V
19		iscom2 38019	. . . . . . 7 ⊢ (((1st ‘𝑅) ∈ V ∧ (2nd ‘𝑅) ∈ V) → (⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st ‘𝑅)∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥)))
20	17, 18, 19	mp2an 692	. . . . . 6 ⊢ (⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran (1st ‘𝑅)∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
21	10, 16, 20	3bitr4ri 304	. . . . 5 ⊢ (⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
22	5, 21	bitrdi 287	. . . 4 ⊢ (𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ → (𝑅 ∈ Com2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
23	4, 22	syl 17	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
24	23	pm5.32i 574	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
25	1, 24	bitri 275	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  Vcvv 3459  ⟨cop 4607  ran crn 5655  Rel wrel 5659  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  RingOpscrngo 37918  Com2ccm2 38013  CRingOpsccring 38017

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-1st 7988  df-2nd 7989  df-rngo 37919  df-com2 38014  df-crngo 38018

This theorem is referenced by:  crngocom  38025  crngohomfo  38030