Theorem crngohomfo 39083

Description: Obsolete theorem, use crngrhmfo 20741 instead. The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
crngohomfo.1	⊢ 𝐺 = (1st ‘𝑅)
crngohomfo.2	⊢ 𝑋 = ran 𝐺
crngohomfo.3	⊢ 𝐽 = (1st ‘𝑆)
crngohomfo.4	⊢ 𝑌 = ran 𝐽

Assertion

Ref	Expression
crngohomfo	⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ CRingOps)

Proof of Theorem crngohomfo

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 781	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ RingOps)
2		foelrn 7118	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑦 ∈ 𝑌) → ∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤))
3	2	ex 418	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → (𝑦 ∈ 𝑌 → ∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤)))
4		foelrn 7118	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥))
5	4	ex 418	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → (𝑧 ∈ 𝑌 → ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥)))
6	3, 5	anim12d 621	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤) ∧ ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥))))
7		reeanv 3238	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝑋 (𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) ↔ (∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤) ∧ ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥)))
8	6, 7	imbitrrdi 255	. . . . 5 ⊢ (𝐹:𝑋–onto→𝑌 → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → ∃𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝑋 (𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥))))
9	8	ad2antll 742	. . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → ∃𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝑋 (𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥))))
10		crngohomfo.1	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (1st ‘𝑅)
11		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
12		crngohomfo.2	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran 𝐺
13	10, 11, 12	crngocom 39078	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
14	13	3expb 1138	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRingOps ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
15	14	3ad2antl1 1204	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
16	15	fveq2d 6900	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑤(2nd ‘𝑅)𝑥)) = (𝐹‘(𝑥(2nd ‘𝑅)𝑤)))
17		crngorngo 39077	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
18		eqid 2763	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆)
19	10, 12, 11, 18	rngohommul 39047	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑤(2nd ‘𝑅)𝑥)) = ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)))
20	17, 19	syl3anl1 1439	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑤(2nd ‘𝑅)𝑥)) = ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)))
21	10, 12, 11, 18	rngohommul 39047	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘(𝑥(2nd ‘𝑅)𝑤)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
22	21	ancom2s 663	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑥(2nd ‘𝑅)𝑤)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
23	17, 22	syl3anl1 1439	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑥(2nd ‘𝑅)𝑤)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
24	16, 20, 23	3eqtr3d 2806	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
25		oveq12 7439	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)))
26		oveq12 7439	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝐹‘𝑥) ∧ 𝑦 = (𝐹‘𝑤)) → (𝑧(2nd ‘𝑆)𝑦) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
27	26	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑧(2nd ‘𝑆)𝑦) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
28	25, 27	eqeq12d 2779	. . . . . . . . 9 ⊢ ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → ((𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦) ↔ ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤))))
29	24, 28	syl5ibrcom 250	. . . . . . . 8 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦)))
30	29	ex 418	. . . . . . 7 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦))))
31	30	3expa 1136	. . . . . 6 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦))))
32	31	adantrr 730	. . . . 5 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → ((𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦))))
33	32	rexlimdvv 3222	. . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → (∃𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝑋 (𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦)))
34	9, 33	syld 48	. . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦)))
35	34	ralrimivv 3207	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑌 (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦))
36		crngohomfo.3	. . 3 ⊢ 𝐽 = (1st ‘𝑆)
37		crngohomfo.4	. . 3 ⊢ 𝑌 = ran 𝐽
38	36, 18, 37	iscrngo2 39074	. 2 ⊢ (𝑆 ∈ CRingOps ↔ (𝑆 ∈ RingOps ∧ ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑌 (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦)))
39	1, 35, 38	sylanbrc 595	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ CRingOps)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3080  ∃wrex 3090  ran crn 5656  –onto→wfo 6545  ‘cfv 6547  (class class class)co 7430  1^st c1st 8009  2^nd c2nd 8010  RingOpscrngo 38971   RingOpsHom crngohom 39037  CRingOpsccring 39070

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

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 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fo 6553  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-map 8863  df-rngo 38972  df-rngohom 39040  df-com2 39067  df-crngo 39071

This theorem is used by: (None)