Theorem crngrhmfo 20638

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 20620	. . 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 7103	. . . . . . . 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 7103	. . . . . . . 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 3236	. . . . . 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 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑆) = (Base‘𝑆)
14	12, 13	rhmf 20627	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
15	14	fdmd 6717	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
16	15	eleq2d 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑎 ∈ dom 𝐹 ↔ 𝑎 ∈ (Base‘𝑅)))
17	15	eleq2d 2848	. . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
25	12, 24	crngcom 20391	. . . . . . . . . . 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 6886	. . . . . . . . 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 2762	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
32	12, 24, 31	rhmmul 20632	. . . . . . . . . 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 20632	. . . . . . . . . 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 2805	. . . . . . . 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 2778	. . . . . . . 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 3220	. . . 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 3205	. 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 20392	. 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 2145  ∀wral 3078  ∃wrex 3088  dom cdm 5659  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  ._rcmulr 17347  Ringcrg 20373  CRingccrg 20374   RingHom crh 20611

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-plusg 17359  df-0g 17530  df-mhm 18892  df-ghm 19342  df-cmn 19910  df-mgp 20275  df-ur 20322  df-ring 20375  df-cring 20376  df-rhm 20614

This theorem is used by: (None)