Theorem imacrhmcl 42806

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 20539	. . . 4 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
4	1, 2, 3	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
5		imacrhmcl.c	. . . 4 ⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆))
6	5	subrgring 20509	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝐶 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ Ring)
8	5	ressbasss2 17170	. . . . . 6 ⊢ (Base‘𝐶) ⊆ (𝐹 “ 𝑆)
9	8	sseli 3928	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶) → 𝑥 ∈ (𝐹 “ 𝑆))
10	8	sseli 3928	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝐶) → 𝑦 ∈ (𝐹 “ 𝑆))
11	9, 10	anim12i 614	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆)))
12		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
13		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝑁) = (Base‘𝑁)
14	12, 13	rhmf 20422	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
15	1, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
16	15	ffund 6665	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
17		fvelima 6898	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
18	16, 17	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
19	18	adantrr 718	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
20		fvelima 6898	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
21	16, 20	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
22	21	adantrl 717	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
23	22	adantr 480	. . . . . 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 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑀 ∈ CRing)
26	12	subrgss 20507	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
27	2, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑀))
28	27	ad3antrrr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑆 ⊆ (Base‘𝑀))
29		simplrl 777	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ 𝑆)
30	28, 29	sseldd 3933	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ (Base‘𝑀))
31		simprl 771	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ 𝑆)
32	28, 31	sseldd 3933	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ (Base‘𝑀))
33		eqid 2735	. . . . . . . . . . 11 ⊢ (.r‘𝑀) = (.r‘𝑀)
34	12, 33	crngcom 20188	. . . . . . . . . 10 ⊢ ((𝑀 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(.r‘𝑀)𝑏) = (𝑏(.r‘𝑀)𝑎))
35	25, 30, 32, 34	syl3anc 1374	. . . . . . . . 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 6837	. . . . . . . 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 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝐹 ∈ (𝑀 RingHom 𝑁))
38		eqid 2735	. . . . . . . . . 10 ⊢ (.r‘𝑁) = (.r‘𝑁)
39	12, 33, 38	rhmmul 20423	. . . . . . . . 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 1374	. . . . . . . 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 20423	. . . . . . . . 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 1374	. . . . . . . 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 2778	. . . . . . 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 7855	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (𝐹 “ 𝑆) ∈ V)
45	5, 38	ressmulr 17229	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑆) ∈ V → (.r‘𝑁) = (.r‘𝐶))
46	1, 44, 45	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (.r‘𝑁) = (.r‘𝐶))
47	46	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (.r‘𝑁) = (.r‘𝐶))
48		simplrr 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘𝑎) = 𝑥)
49		simprr 773	. . . . . . . 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 7379	. . . . . . 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 7379	. . . . . . 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 2778	. . . . . 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 3142	. . . . 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 3142	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
55	11, 54	sylan2 594	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
56	55	ralrimivva 3178	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
57		eqid 2735	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
58		eqid 2735	. . 3 ⊢ (.r‘𝐶) = (.r‘𝐶)
59	57, 58	iscrng2 20189	. 2 ⊢ (𝐶 ∈ CRing ↔ (𝐶 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)))
60	7, 56, 59	sylanbrc 584	1 ⊢ (𝜑 → 𝐶 ∈ CRing)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃wrex 3059  Vcvv 3439   ⊆ wss 3900   “ cima 5626  Fun wfun 6485  ⟶wf 6487  ‘cfv 6491  (class class class)co 7358  Basecbs 17138   ↾_s cress 17159  ._rcmulr 17180  Ringcrg 20170  CRingccrg 20171   RingHom crh 20407  SubRingcsubrg 20504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-0g 17363  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-subg 19055  df-ghm 19144  df-cmn 19713  df-abl 19714  df-mgp 20078  df-rng 20090  df-ur 20119  df-ring 20172  df-cring 20173  df-rhm 20410  df-subrng 20481  df-subrg 20505

This theorem is referenced by:  riccrng1  42813