Theorem imacrhmcl 43772

Description: The image of a commutative ring homomorphism is a commutative ring. (Contributed by SN, 10-Jan-2025.)

Hypotheses

Ref	Expression
imacrhmcl.c	⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆))
imacrhmcl.h	⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))
imacrhmcl.m	⊢ (𝜑 → 𝑀 ∈ CRing)
imacrhmcl.s	⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀))

Assertion

Ref	Expression
imacrhmcl	⊢ (𝜑 → 𝐶 ∈ CRing)

Proof of Theorem imacrhmcl

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

Step	Hyp	Ref	Expression
1		imacrhmcl.h	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))
2		imacrhmcl.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀))
3		rhmima 20883	. . . 4 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
4	1, 2, 3	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
5		imacrhmcl.c	. . . 4 ⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆))
6	5	subrgring 20853	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝐶 ∈ Ring)
7	4, 6	syl 18	. 2 ⊢ (𝜑 → 𝐶 ∈ Ring)
8	5	ressbasss2 17445	. . . . . 6 ⊢ (Base‘𝐶) ⊆ (𝐹 “ 𝑆)
9	8	sseli 3909	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶) → 𝑥 ∈ (𝐹 “ 𝑆))
10	8	sseli 3909	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝐶) → 𝑦 ∈ (𝐹 “ 𝑆))
11	9, 10	anim12i 625	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆)))
12		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
13		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝑁) = (Base‘𝑁)
14	12, 13	rhmf 20742	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
15	1, 14	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
16	15	ffund 6733	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
17		fvelima 6970	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
18	16, 17	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
19	18	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
20		fvelima 6970	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
21	16, 20	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
22	21	adantrl 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
23	22	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
24		imacrhmcl.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ CRing)
25	24	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑀 ∈ CRing)
26	12	subrgss 20851	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
27	2, 26	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑀))
28	27	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑆 ⊆ (Base‘𝑀))
29		simplrl 789	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ 𝑆)
30	28, 29	sseldd 3914	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ (Base‘𝑀))
31		simprl 783	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ 𝑆)
32	28, 31	sseldd 3914	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ (Base‘𝑀))
33		eqid 2763	. . . . . . . . . . 11 ⊢ (.r‘𝑀) = (.r‘𝑀)
34	12, 33	crngcom 20505	. . . . . . . . . 10 ⊢ ((𝑀 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(.r‘𝑀)𝑏) = (𝑏(.r‘𝑀)𝑎))
35	25, 30, 32, 34	syl3anc 1399	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝑎(.r‘𝑀)𝑏) = (𝑏(.r‘𝑀)𝑎))
36	35	fveq2d 6909	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘(𝑎(.r‘𝑀)𝑏)) = (𝐹‘(𝑏(.r‘𝑀)𝑎)))
37	1	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝐹 ∈ (𝑀 RingHom 𝑁))
38		eqid 2763	. . . . . . . . . 10 ⊢ (.r‘𝑁) = (.r‘𝑁)
39	12, 33, 38	rhmmul 20747	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝐹‘(𝑎(.r‘𝑀)𝑏)) = ((𝐹‘𝑎)(.r‘𝑁)(𝐹‘𝑏)))
40	37, 30, 32, 39	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘(𝑎(.r‘𝑀)𝑏)) = ((𝐹‘𝑎)(.r‘𝑁)(𝐹‘𝑏)))
41	12, 33, 38	rhmmul 20747	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑏 ∈ (Base‘𝑀) ∧ 𝑎 ∈ (Base‘𝑀)) → (𝐹‘(𝑏(.r‘𝑀)𝑎)) = ((𝐹‘𝑏)(.r‘𝑁)(𝐹‘𝑎)))
42	37, 32, 30, 41	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘(𝑏(.r‘𝑀)𝑎)) = ((𝐹‘𝑏)(.r‘𝑁)(𝐹‘𝑎)))
43	36, 40, 42	3eqtr3d 2806	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → ((𝐹‘𝑎)(.r‘𝑁)(𝐹‘𝑏)) = ((𝐹‘𝑏)(.r‘𝑁)(𝐹‘𝑎)))
44		imaexg 7946	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (𝐹 “ 𝑆) ∈ V)
45	5, 38	ressmulr 17504	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑆) ∈ V → (.r‘𝑁) = (.r‘𝐶))
46	1, 44, 45	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → (.r‘𝑁) = (.r‘𝐶))
47	46	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (.r‘𝑁) = (.r‘𝐶))
48		simplrr 790	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘𝑎) = 𝑥)
49		simprr 785	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘𝑏) = 𝑦)
50	47, 48, 49	oveq123d 7461	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → ((𝐹‘𝑎)(.r‘𝑁)(𝐹‘𝑏)) = (𝑥(.r‘𝐶)𝑦))
51	47, 49, 48	oveq123d 7461	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → ((𝐹‘𝑏)(.r‘𝑁)(𝐹‘𝑎)) = (𝑦(.r‘𝐶)𝑥))
52	43, 50, 51	3eqtr3d 2806	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
53	23, 52	rexlimddv 3173	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
54	19, 53	rexlimddv 3173	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
55	11, 54	sylan2 605	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
56	55	ralrimivva 3209	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
57		eqid 2763	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
58		eqid 2763	. . 3 ⊢ (.r‘𝐶) = (.r‘𝐶)
59	57, 58	iscrng2 20506	. 2 ⊢ (𝐶 ∈ CRing ↔ (𝐶 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)))
60	7, 56, 59	sylanbrc 595	1 ⊢ (𝜑 → 𝐶 ∈ CRing)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   “ cima 5658  Fun wfun 6549  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  Basecbs 17413   ↾_s cress 17434  ._rcmulr 17455  Ringcrg 20486  CRingccrg 20487   RingHom crh 20726  SubRingcsubrg 20848

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-submnd 19005  df-grp 19173  df-minusg 19174  df-subg 19359  df-ghm 19454  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-ring 20488  df-cring 20489  df-rhm 20729  df-subrng 20825  df-subrg 20849

This theorem is used by:  riccrng1  43773