Theorem crngrhmfo 20604

Description: The image of a surjective homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.) (Revised by AV, 19-Jul-2026.)

Hypothesis

Ref	Expression
crngrhmfo.b	⊢ 𝐵 = (Base‘𝑆)

Assertion

Ref	Expression
crngrhmfo	⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → 𝑆 ∈ CRing)

Proof of Theorem crngrhmfo

Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmrcl2 20586	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
2	1	3ad2ant2 1152	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → 𝑆 ∈ Ring)
3		foelrn 7106	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑎 ∈ dom 𝐹 𝑥 = (𝐹‘𝑎))
4	3	ex 418	. . . . . . 7 ⊢ (𝐹:dom 𝐹–onto→𝐵 → (𝑥 ∈ 𝐵 → ∃𝑎 ∈ dom 𝐹 𝑥 = (𝐹‘𝑎)))
5		foelrn 7106	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑏 ∈ dom 𝐹 𝑦 = (𝐹‘𝑏))
6	5	ex 418	. . . . . . 7 ⊢ (𝐹:dom 𝐹–onto→𝐵 → (𝑦 ∈ 𝐵 → ∃𝑏 ∈ dom 𝐹 𝑦 = (𝐹‘𝑏)))
7	4, 6	anim12d 621	. . . . . 6 ⊢ (𝐹:dom 𝐹–onto→𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ dom 𝐹 𝑥 = (𝐹‘𝑎) ∧ ∃𝑏 ∈ dom 𝐹 𝑦 = (𝐹‘𝑏))))
8		reeanv 3239	. . . . . 6 ⊢ (∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹(𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) ↔ (∃𝑎 ∈ dom 𝐹 𝑥 = (𝐹‘𝑎) ∧ ∃𝑏 ∈ dom 𝐹 𝑦 = (𝐹‘𝑏)))
9	7, 8	imbitrrdi 255	. . . . 5 ⊢ (𝐹:dom 𝐹–onto→𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹(𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏))))
10	9	3ad2ant3 1153	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹(𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏))))
11		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → 𝑅 ∈ CRing)
12		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑆) = (Base‘𝑆)
14	12, 13	rhmf 20593	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
15	14	fdmd 6720	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
16	15	eleq2d 2851	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑎 ∈ dom 𝐹 ↔ 𝑎 ∈ (Base‘𝑅)))
17	15	eleq2d 2851	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑏 ∈ dom 𝐹 ↔ 𝑏 ∈ (Base‘𝑅)))
18	16, 17	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹) ↔ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))))
19	18	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))))
20	19	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) → ((𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))))
21	20	imp 412	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)))
22		3anass 1111	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) ↔ (𝑅 ∈ CRing ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))))
23	11, 21, 22	sylanbrc 595	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝑅 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)))
24		eqid 2765	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
25	12, 24	crngcom 20357	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(.r‘𝑅)𝑏) = (𝑏(.r‘𝑅)𝑎))
26	23, 25	syl 18	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝑎(.r‘𝑅)𝑏) = (𝑏(.r‘𝑅)𝑎))
27	26	fveq2d 6889	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = (𝐹‘(𝑏(.r‘𝑅)𝑎)))
28		simplr 781	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
29		3anass 1111	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))))
30	28, 21, 29	sylanbrc 595	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)))
31		eqid 2765	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
32	12, 24, 31	rhmmul 20598	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)))
33	30, 32	syl 18	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)))
34	21	ancomd 467	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝑏 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)))
35		3anass 1111	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑏 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅))))
36	28, 34, 35	sylanbrc 595	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)))
37	12, 24, 31	rhmmul 20598	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → (𝐹‘(𝑏(.r‘𝑅)𝑎)) = ((𝐹‘𝑏)(.r‘𝑆)(𝐹‘𝑎)))
38	36, 37	syl 18	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝐹‘(𝑏(.r‘𝑅)𝑎)) = ((𝐹‘𝑏)(.r‘𝑆)(𝐹‘𝑎)))
39	27, 33, 38	3eqtr3d 2808	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)) = ((𝐹‘𝑏)(.r‘𝑆)(𝐹‘𝑎)))
40		oveq12 7425	. . . . . . . . 9 ⊢ ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑥(.r‘𝑆)𝑦) = ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)))
41		oveq12 7425	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑏) ∧ 𝑥 = (𝐹‘𝑎)) → (𝑦(.r‘𝑆)𝑥) = ((𝐹‘𝑏)(.r‘𝑆)(𝐹‘𝑎)))
42	41	ancoms 464	. . . . . . . . 9 ⊢ ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑦(.r‘𝑆)𝑥) = ((𝐹‘𝑏)(.r‘𝑆)(𝐹‘𝑎)))
43	40, 42	eqeq12d 2781	. . . . . . . 8 ⊢ ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → ((𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥) ↔ ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)) = ((𝐹‘𝑏)(.r‘𝑆)(𝐹‘𝑎))))
44	39, 43	syl5ibrcom 250	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥)))
45	44	ex 418	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) → ((𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹) → ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥))))
46	45	3adant3 1150	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → ((𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹) → ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥))))
47	46	rexlimdvv 3223	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → (∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹(𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥)))
48	10, 47	syld 48	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥)))
49	48	ralrimivv 3208	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥))
50		crngrhmfo.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
51	50, 31	iscrng2 20358	. 2 ⊢ (𝑆 ∈ CRing ↔ (𝑆 ∈ Ring ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥)))
52	2, 49, 51	sylanbrc 595	1 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → 𝑆 ∈ CRing)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  dom cdm 5663  –onto→wfo 6538  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  ._rcmulr 17329  Ringcrg 20339  CRingccrg 20340   RingHom crh 20577

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-sets 17242  df-slot 17260  df-ndx 17272  df-base 17288  df-plusg 17341  df-0g 17512  df-mhm 18865  df-ghm 19308  df-cmn 19876  df-mgp 20241  df-ur 20288  df-ring 20341  df-cring 20342  df-rhm 20580

This theorem is used by: (None)