Theorem crngrhmfo 20574

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 20556	. . 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 7102	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑎 ∈ dom 𝐹 𝑥 = (𝐹‘𝑎))
4	3	ex 417	. . . . . . 7 ⊢ (𝐹:dom 𝐹–onto→𝐵 → (𝑥 ∈ 𝐵 → ∃𝑎 ∈ dom 𝐹 𝑥 = (𝐹‘𝑎)))
5		foelrn 7102	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑏 ∈ dom 𝐹 𝑦 = (𝐹‘𝑏))
6	5	ex 417	. . . . . . 7 ⊢ (𝐹:dom 𝐹–onto→𝐵 → (𝑦 ∈ 𝐵 → ∃𝑏 ∈ dom 𝐹 𝑦 = (𝐹‘𝑏)))
7	4, 6	anim12d 620	. . . . . 6 ⊢ (𝐹:dom 𝐹–onto→𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ dom 𝐹 𝑥 = (𝐹‘𝑎) ∧ ∃𝑏 ∈ dom 𝐹 𝑦 = (𝐹‘𝑏))))
8		reeanv 3237	. . . . . 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 778	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → 𝑅 ∈ CRing)
12		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑆) = (Base‘𝑆)
14	12, 13	rhmf 20563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
15	14	fdmd 6716	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
16	15	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑎 ∈ dom 𝐹 ↔ 𝑎 ∈ (Base‘𝑅)))
17	15	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑏 ∈ dom 𝐹 ↔ 𝑏 ∈ (Base‘𝑅)))
18	16, 17	anbi12d 643	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) → ((𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))))
21	20	imp 411	. . . . . . . . . . . 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 594	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ (𝑎 ∈ dom 𝐹 ∧ 𝑏 ∈ dom 𝐹)) → (𝑅 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)))
24		eqid 2763	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
25	12, 24	crngcom 20328	. . . . . . . . . . 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 6885	. . . . . . . . 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 780	. . . . . . . . . . 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 594	. . . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
32	12, 24, 31	rhmmul 20568	. . . . . . . . . 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 466	. . . . . . . . . . 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 594	. . . . . . . . . 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 20568	. . . . . . . . . 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 2806	. . . . . . . 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 7419	. . . . . . . . 9 ⊢ ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑥(.r‘𝑆)𝑦) = ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)))
41		oveq12 7419	. . . . . . . . . 10 ⊢ ((𝑦 = (𝐹‘𝑏) ∧ 𝑥 = (𝐹‘𝑎)) → (𝑦(.r‘𝑆)𝑥) = ((𝐹‘𝑏)(.r‘𝑆)(𝐹‘𝑎)))
42	41	ancoms 463	. . . . . . . . 9 ⊢ ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑦(.r‘𝑆)𝑥) = ((𝐹‘𝑏)(.r‘𝑆)(𝐹‘𝑎)))
43	40, 42	eqeq12d 2779	. . . . . . . 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 417	. . . . . 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 3221	. . . 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 3206	. 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 20329	. 2 ⊢ (𝑆 ∈ CRing ↔ (𝑆 ∈ Ring ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥)))
52	2, 49, 51	sylanbrc 594	1 ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → 𝑆 ∈ CRing)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  dom cdm 5661  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  ._rcmulr 17306  Ringcrg 20310  CRingccrg 20311   RingHom crh 20547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-plusg 17318  df-0g 17489  df-mhm 18836  df-ghm 19279  df-cmn 19847  df-mgp 20212  df-ur 20259  df-ring 20312  df-cring 20313  df-rhm 20550

This theorem is referenced by: (None)