Theorem imacrhmcl 42509

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 20520	. . . 4 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
5		imacrhmcl.c	. . . 4 ⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆))
6	5	subrgring 20490	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝐶 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ Ring)
8	5	ressbasss2 17218	. . . . . 6 ⊢ (Base‘𝐶) ⊆ (𝐹 “ 𝑆)
9	8	sseli 3945	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶) → 𝑥 ∈ (𝐹 “ 𝑆))
10	8	sseli 3945	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝐶) → 𝑦 ∈ (𝐹 “ 𝑆))
11	9, 10	anim12i 613	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆)))
12		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
13		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑁) = (Base‘𝑁)
14	12, 13	rhmf 20401	. . . . . . . . 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 6695	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
17		fvelima 6929	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
18	16, 17	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
19	18	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
20		fvelima 6929	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
21	16, 20	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
22	21	adantrl 716	. . . . . . 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 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑀 ∈ CRing)
26	12	subrgss 20488	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
27	2, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑀))
28	27	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑆 ⊆ (Base‘𝑀))
29		simplrl 776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ 𝑆)
30	28, 29	sseldd 3950	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ (Base‘𝑀))
31		simprl 770	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ 𝑆)
32	28, 31	sseldd 3950	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ (Base‘𝑀))
33		eqid 2730	. . . . . . . . . . 11 ⊢ (.r‘𝑀) = (.r‘𝑀)
34	12, 33	crngcom 20167	. . . . . . . . . 10 ⊢ ((𝑀 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(.r‘𝑀)𝑏) = (𝑏(.r‘𝑀)𝑎))
35	25, 30, 32, 34	syl3anc 1373	. . . . . . . . 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 6865	. . . . . . . 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 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝐹 ∈ (𝑀 RingHom 𝑁))
38		eqid 2730	. . . . . . . . . 10 ⊢ (.r‘𝑁) = (.r‘𝑁)
39	12, 33, 38	rhmmul 20402	. . . . . . . . 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 1373	. . . . . . . 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 20402	. . . . . . . . 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 1373	. . . . . . . 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 2773	. . . . . . 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 7892	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (𝐹 “ 𝑆) ∈ V)
45	5, 38	ressmulr 17277	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑆) ∈ V → (.r‘𝑁) = (.r‘𝐶))
46	1, 44, 45	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (.r‘𝑁) = (.r‘𝐶))
47	46	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (.r‘𝑁) = (.r‘𝐶))
48		simplrr 777	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘𝑎) = 𝑥)
49		simprr 772	. . . . . . . 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 7411	. . . . . . 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 7411	. . . . . . 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 2773	. . . . . 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 3141	. . . . 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 3141	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
55	11, 54	sylan2 593	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
56	55	ralrimivva 3181	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
57		eqid 2730	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
58		eqid 2730	. . 3 ⊢ (.r‘𝐶) = (.r‘𝐶)
59	57, 58	iscrng2 20168	. 2 ⊢ (𝐶 ∈ CRing ↔ (𝐶 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)))
60	7, 56, 59	sylanbrc 583	1 ⊢ (𝜑 → 𝐶 ∈ CRing)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917   “ cima 5644  Fun wfun 6508  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186   ↾_s cress 17207  ._rcmulr 17228  Ringcrg 20149  CRingccrg 20150   RingHom crh 20385  SubRingcsubrg 20485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-subg 19062  df-ghm 19152  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-cring 20152  df-rhm 20388  df-subrng 20462  df-subrg 20486

This theorem is referenced by:  riccrng1  42516