Theorem iscrngo2 36082

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 36081	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2		relrngo 35981	. . . . 5 ⊢ Rel RingOps
3		1st2nd 7853	. . . . 5 ⊢ ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
4	2, 3	mpan 686	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
5		eleq1 2826	. . . . 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 5835	. . . . . . . 8 ⊢ ran 𝐺 = ran (1st ‘𝑅)
9	6, 8	eqtri 2766	. . . . . . 7 ⊢ 𝑋 = ran (1st ‘𝑅)
10	9	raleqi 3337	. . . . . 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 7268	. . . . . . . . 9 ⊢ (𝑥𝐻𝑦) = (𝑥(2nd ‘𝑅)𝑦)
13	11	oveqi 7268	. . . . . . . . 9 ⊢ (𝑦𝐻𝑥) = (𝑦(2nd ‘𝑅)𝑥)
14	12, 13	eqeq12i 2756	. . . . . . . 8 ⊢ ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
15	9, 14	raleqbii 3160	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
16	15	ralbii 3090	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
17		fvex 6769	. . . . . . 7 ⊢ (1st ‘𝑅) ∈ V
18		fvex 6769	. . . . . . 7 ⊢ (2nd ‘𝑅) ∈ V
19		iscom2 36080	. . . . . . 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 688	. . . . . 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 303	. . . . 5 ⊢ (⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ ∈ Com2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
22	5, 21	bitrdi 286	. . . 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 274	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  ⟨cop 4564  ran crn 5581  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  RingOpscrngo 35979  Com2ccm2 36074  CRingOpsccring 36078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-1st 7804  df-2nd 7805  df-rngo 35980  df-com2 36075  df-crngo 36079

This theorem is referenced by:  crngocom  36086  crngohomfo  36091