Theorem crngohomfo 36091

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

Hypotheses

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

Assertion

Ref	Expression
crngohomfo	⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 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 765	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ RingOps)
2		foelrn 6964	. . . . . . . 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 6964	. . . . . . . 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 608	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤) ∧ ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥))))
7		reeanv 3292	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝑋 (𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) ↔ (∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤) ∧ ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥)))
8	6, 7	syl6ibr 251	. . . . 5 ⊢ (𝐹:𝑋–onto→𝑌 → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → ∃𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝑋 (𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥))))
9	8	ad2antll 725	. . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → ∃𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝑋 (𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥))))
10		crnghomfo.1	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (1st ‘𝑅)
11		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
12		crnghomfo.2	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran 𝐺
13	10, 11, 12	crngocom 36086	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
14	13	3expb 1118	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRingOps ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
15	14	3ad2antl1 1183	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
16	15	fveq2d 6760	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑤(2nd ‘𝑅)𝑥)) = (𝐹‘(𝑥(2nd ‘𝑅)𝑤)))
17		crngorngo 36085	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
18		eqid 2738	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆)
19	10, 12, 11, 18	rngohommul 36055	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑤(2nd ‘𝑅)𝑥)) = ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)))
20	17, 19	syl3anl1 1410	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑤(2nd ‘𝑅)𝑥)) = ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)))
21	10, 12, 11, 18	rngohommul 36055	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘(𝑥(2nd ‘𝑅)𝑤)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
22	21	ancom2s 646	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑥(2nd ‘𝑅)𝑤)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
23	17, 22	syl3anl1 1410	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑥(2nd ‘𝑅)𝑤)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
24	16, 20, 23	3eqtr3d 2786	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
25		oveq12 7264	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)))
26		oveq12 7264	. . . . . . . . . . 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 2754	. . . . . . . . 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 246	. . . . . . . 8 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 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 RngHom 𝑆)) → ((𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦))))
31	30	3expa 1116	. . . . . 6 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦))))
32	31	adantrr 713	. . . . 5 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 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 3221	. . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 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 RngHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦)))
35	34	ralrimivv 3113	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑌 (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦))
36		crnghomfo.3	. . 3 ⊢ 𝐽 = (1st ‘𝑆)
37		crnghomfo.4	. . 3 ⊢ 𝑌 = ran 𝐽
38	36, 18, 37	iscrngo2 36082	. 2 ⊢ (𝑆 ∈ CRingOps ↔ (𝑆 ∈ RingOps ∧ ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑌 (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦)))
39	1, 35, 38	sylanbrc 582	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ CRingOps)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ran crn 5581  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  RingOpscrngo 35979   RngHom crnghom 36045  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-pow 5283  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-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  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-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-rngo 35980  df-rngohom 36048  df-com2 36075  df-crngo 36079

This theorem is referenced by: (None)