Theorem crngohomfo 38579

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 780	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ RingOps)
2		foelrn 7103	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑦 ∈ 𝑌) → ∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤))
3	2	ex 417	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → (𝑦 ∈ 𝑌 → ∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤)))
4		foelrn 7103	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥))
5	4	ex 417	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → (𝑧 ∈ 𝑌 → ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥)))
6	3, 5	anim12d 620	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 𝑦 = (𝐹‘𝑤) ∧ ∃𝑥 ∈ 𝑋 𝑧 = (𝐹‘𝑥))))
7		reeanv 3243	. . . . . 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 741	. . . 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 2769	. . . . . . . . . . . . . 14 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
12		crngohomfo.2	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran 𝐺
13	10, 11, 12	crngocom 38574	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
14	13	3expb 1136	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRingOps ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
15	14	3ad2antl1 1202	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑤(2nd ‘𝑅)𝑥) = (𝑥(2nd ‘𝑅)𝑤))
16	15	fveq2d 6886	. . . . . . . . . 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 38573	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
18		eqid 2769	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆)
19	10, 12, 11, 18	rngohommul 38543	. . . . . . . . . . 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 1437	. . . . . . . . . 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 38543	. . . . . . . . . . . 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 662	. . . . . . . . . . 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 1437	. . . . . . . . . 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 2812	. . . . . . . . 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 7420	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑦(2nd ‘𝑆)𝑧) = ((𝐹‘𝑤)(2nd ‘𝑆)(𝐹‘𝑥)))
26		oveq12 7420	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝐹‘𝑥) ∧ 𝑦 = (𝐹‘𝑤)) → (𝑧(2nd ‘𝑆)𝑦) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
27	26	ancoms 463	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑤) ∧ 𝑧 = (𝐹‘𝑥)) → (𝑧(2nd ‘𝑆)𝑦) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑤)))
28	25, 27	eqeq12d 2785	. . . . . . . . 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 417	. . . . . . 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 1134	. . . . . 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 729	. . . . 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 3227	. . . 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 3212	. 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 38570	. 2 ⊢ (𝑆 ∈ CRingOps ↔ (𝑆 ∈ RingOps ∧ ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑌 (𝑦(2nd ‘𝑆)𝑧) = (𝑧(2nd ‘𝑆)𝑦)))
39	1, 35, 38	sylanbrc 594	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ CRingOps)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ran crn 5663  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  2^nd c2nd 7985  RingOpscrngo 38467   RingOpsHom crngohom 38533  CRingOpsccring 38566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826  df-rngo 38468  df-rngohom 38536  df-com2 38563  df-crngo 38567

This theorem is referenced by: (None)