Theorem imacrhmcl 42469

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 20632	. . . 4 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
4	1, 2, 3	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁))
5		imacrhmcl.c	. . . 4 ⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆))
6	5	subrgring 20602	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝐶 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ Ring)
8	5	ressbasss2 17299	. . . . . 6 ⊢ (Base‘𝐶) ⊆ (𝐹 “ 𝑆)
9	8	sseli 4004	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶) → 𝑥 ∈ (𝐹 “ 𝑆))
10	8	sseli 4004	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝐶) → 𝑦 ∈ (𝐹 “ 𝑆))
11	9, 10	anim12i 612	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆)))
12		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
13		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘𝑁) = (Base‘𝑁)
14	12, 13	rhmf 20511	. . . . . . . . 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 6751	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
17		fvelima 6987	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
18	16, 17	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
19	18	adantrr 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
20		fvelima 6987	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
21	16, 20	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦)
22	21	adantrl 715	. . . . . . 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 729	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑀 ∈ CRing)
26	12	subrgss 20600	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
27	2, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑀))
28	27	ad3antrrr 729	. . . . . . . . . . 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 4009	. . . . . . . . . 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 4009	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ (Base‘𝑀))
33		eqid 2740	. . . . . . . . . . 11 ⊢ (.r‘𝑀) = (.r‘𝑀)
34	12, 33	crngcom 20278	. . . . . . . . . 10 ⊢ ((𝑀 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(.r‘𝑀)𝑏) = (𝑏(.r‘𝑀)𝑎))
35	25, 30, 32, 34	syl3anc 1371	. . . . . . . . 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 6924	. . . . . . . 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 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝐹 ∈ (𝑀 RingHom 𝑁))
38		eqid 2740	. . . . . . . . . 10 ⊢ (.r‘𝑁) = (.r‘𝑁)
39	12, 33, 38	rhmmul 20512	. . . . . . . . 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 1371	. . . . . . . 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 20512	. . . . . . . . 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 1371	. . . . . . . 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 2788	. . . . . . 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 7953	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (𝐹 “ 𝑆) ∈ V)
45	5, 38	ressmulr 17366	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑆) ∈ V → (.r‘𝑁) = (.r‘𝐶))
46	1, 44, 45	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (.r‘𝑁) = (.r‘𝐶))
47	46	ad3antrrr 729	. . . . . . . 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 7469	. . . . . . 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 7469	. . . . . . 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 2788	. . . . . 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 3167	. . . . 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 3167	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
55	11, 54	sylan2 592	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
56	55	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))
57		eqid 2740	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
58		eqid 2740	. . 3 ⊢ (.r‘𝐶) = (.r‘𝐶)
59	57, 58	iscrng2 20279	. 2 ⊢ (𝐶 ∈ CRing ↔ (𝐶 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)))
60	7, 56, 59	sylanbrc 582	1 ⊢ (𝜑 → 𝐶 ∈ CRing)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976   “ cima 5703  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   ↾_s cress 17287  ._rcmulr 17312  Ringcrg 20260  CRingccrg 20261   RingHom crh 20495  SubRingcsubrg 20595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-subg 19163  df-ghm 19253  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-rhm 20498  df-subrng 20572  df-subrg 20597

This theorem is referenced by:  riccrng1  42476