Theorem crngrhmfo 20735

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 20717	. . 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 7112	. . . . . . . 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 7112	. . . . . . . 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 3234	. . . . . 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 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑆) = (Base‘𝑆)
14	12, 13	rhmf 20724	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
15	14	fdmd 6725	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
16	15	eleq2d 2846	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑎 ∈ dom 𝐹 ↔ 𝑎 ∈ (Base‘𝑅)))
17	15	eleq2d 2846	. . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
25	12, 24	crngcom 20487	. . . . . . . . . . 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 6894	. . . . . . . . 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 2760	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
32	12, 24, 31	rhmmul 20729	. . . . . . . . . 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 20729	. . . . . . . . . 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 2803	. . . . . . . 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 7433	. . . . . . . . 9 ⊢ ((𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐹‘𝑏)) → (𝑥(.r‘𝑆)𝑦) = ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)))
41		oveq12 7433	. . . . . . . . . 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 2776	. . . . . . . 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 3218	. . . 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 3203	. 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 20488	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  dom cdm 5651  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  ._rcmulr 17438  Ringcrg 20468  CRingccrg 20469   RingHom crh 20708

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-er 8724  df-map 8856  df-en 8981  df-dom 8982  df-sdom 8983  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-plusg 17450  df-0g 17621  df-mhm 18987  df-ghm 19437  df-cmn 20005  df-mgp 20370  df-ur 20417  df-ring 20470  df-cring 20471  df-rhm 20711

This theorem is used by: (None)