Theorem crngohomfo 38146

Description: The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)

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 768	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ RingOps)
2		foelrn 7050	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑦 ∈ 𝑌) → ∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤))
3	2	ex 412	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → (𝑦 ∈ 𝑌 → ∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤)))
4		foelrn 7050	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥))
5	4	ex 412	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → (𝑧 ∈ 𝑌 → ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥)))
6	3, 5	anim12d 609	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤) ∧ ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥))))
7		reeanv 3206	. . . . . 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 252	. . . . 5 ⊢ (𝐹:𝑋–onto→𝑌 → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → ∃𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝑋 (𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥))))
9	8	ad2antll 729	. . . 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 2734	. . . . . . . . . . . . . 14 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
12		crngohomfo.2	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran 𝐺
13	10, 11, 12	crngocom 38141	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
14	13	3expb 1120	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRingOps ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
15	14	3ad2antl1 1186	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
16	15	fveq2d 6836	. . . . . . . . . 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 38140	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
18		eqid 2734	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆)
19	10, 12, 11, 18	rngohommul 38110	. . . . . . . . . . 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 1414	. . . . . . . . . 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 38110	. . . . . . . . . . . 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 650	. . . . . . . . . . 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 1414	. . . . . . . . . 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 2777	. . . . . . . . 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 7365	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)))
26		oveq12 7365	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝐹‘𝑥) ∧ 𝑦 = (𝐹‘𝑤)) → (𝑧(2nd ‘𝑆)𝑦) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
27	26	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑧(2nd ‘𝑆)𝑦) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
28	25, 27	eqeq12d 2750	. . . . . . . . 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 247	. . . . . . . 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 412	. . . . . . 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 1118	. . . . . 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 717	. . . . 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 3190	. . . 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 47	. . 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 3175	. 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 38137	. 2 ⊢ (𝑆 ∈ CRingOps ↔ (𝑆 ∈ RingOps ∧ ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑌 (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦)))
39	1, 35, 38	sylanbrc 583	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ CRingOps)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  ran crn 5623  –onto→wfo 6488  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  RingOpscrngo 38034   RingOpsHom crngohom 38100  CRingOpsccring 38133

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 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-rngo 38035  df-rngohom 38103  df-com2 38130  df-crngo 38134

This theorem is referenced by: (None)