Theorem iscrngo2 38057

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 38056	. 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
2		relrngo 37956	. . . . 5 ⊢ Rel RingOps
3		1st2nd 7977	. . . . 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 2821	. . . . 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 5881	. . . . . . . 8 ⊢ ran 𝐺 = ran (1st ‘𝑅)
9	6, 8	eqtri 2756	. . . . . . 7 ⊢ 𝑋 = ran (1st ‘𝑅)
10	9	raleqi 3291	. . . . . 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 7365	. . . . . . . . 9 ⊢ (𝑥𝐻𝑦) = (𝑥(2nd ‘𝑅)𝑦)
13	11	oveqi 7365	. . . . . . . . 9 ⊢ (𝑦𝐻𝑥) = (𝑦(2nd ‘𝑅)𝑥)
14	12, 13	eqeq12i 2751	. . . . . . . 8 ⊢ ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
15	9, 14	raleqbii 3311	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
16	15	ralbii 3079	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ran (1st ‘𝑅)(𝑥(2nd ‘𝑅)𝑦) = (𝑦(2nd ‘𝑅)𝑥))
17		fvex 6841	. . . . . . 7 ⊢ (1st ‘𝑅) ∈ V
18		fvex 6841	. . . . . . 7 ⊢ (2nd ‘𝑅) ∈ V
19		iscom2 38055	. . . . . . 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 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437  ⟨cop 4581  ran crn 5620  Rel wrel 5624  ‘cfv 6486  (class class class)co 7352  1^st c1st 7925  2^nd c2nd 7926  RingOpscrngo 37954  Com2ccm2 38049  CRingOpsccring 38053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-1st 7927  df-2nd 7928  df-rngo 37955  df-com2 38050  df-crngo 38054

This theorem is referenced by:  crngocom  38061  crngohomfo  38066