Theorem iscrngo2 36529

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 36528	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2		relrngo 36428	. . . . 5 ⊢ Rel RingOps
3		1st2nd 7976	. . . . 5 ⊢ ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
4	2, 3	mpan 688	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
5		eleq1 2820	. . . . 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 5897	. . . . . . . 8 ⊢ ran 𝐺 = ran (1st ‘𝑅)
9	6, 8	eqtri 2759	. . . . . . 7 ⊢ 𝑋 = ran (1st ‘𝑅)
10	9	raleqi 3309	. . . . . 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 7375	. . . . . . . . 9 ⊢ (𝑥𝐻𝑦) = (𝑥(2nd ‘𝑅)𝑦)
13	11	oveqi 7375	. . . . . . . . 9 ⊢ (𝑦𝐻𝑥) = (𝑦(2nd ‘𝑅)𝑥)
14	12, 13	eqeq12i 2749	. . . . . . . 8 ⊢ ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
15	9, 14	raleqbii 3313	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
16	15	ralbii 3092	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
17		fvex 6860	. . . . . . 7 ⊢ (1st ‘𝑅) ∈ V
18		fvex 6860	. . . . . . 7 ⊢ (2nd ‘𝑅) ∈ V
19		iscom2 36527	. . . . . . 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 690	. . . . . 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 575	. 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  Vcvv 3446  ⟨cop 4597  ran crn 5639  Rel wrel 5643  ‘cfv 6501  (class class class)co 7362  1^st c1st 7924  2^nd c2nd 7925  RingOpscrngo 36426  Com2ccm2 36521  CRingOpsccring 36525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-1st 7926  df-2nd 7927  df-rngo 36427  df-com2 36522  df-crngo 36526

This theorem is referenced by:  crngocom  36533  crngohomfo  36538