imacrhmcl - Mathbox for Steven Nguyen

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Theorem imacrhmcl 41648

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 20506	. . . 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 20476	. . 3 ⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝐶 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ Ring)
8	5	ressbasss2 17194	. . . . . 6 ⊢ (Base‘𝐶) ⊆ (𝐹 “ 𝑆)
9	8	sseli 3973	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶) → 𝑥 ∈ (𝐹 “ 𝑆))
10	8	sseli 3973	. . . . 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 2726	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
13		eqid 2726	. . . . . . . . . 10 ⊢ (Base‘𝑁) = (Base‘𝑁)
14	12, 13	rhmf 20387	. . . . . . . . 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 6715	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
17		fvelima 6951	. . . . . . 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 714	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥)
20		fvelima 6951	. . . . . . . . 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 713	. . . . . . 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 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑀 ∈ CRing)
26	12	subrgss 20474	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
27	2, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑀))
28	27	ad3antrrr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑆 ⊆ (Base‘𝑀))
29		simplrl 774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ 𝑆)
30	28, 29	sseldd 3978	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ (Base‘𝑀))
31		simprl 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ 𝑆)
32	28, 31	sseldd 3978	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ (Base‘𝑀))
33		eqid 2726	. . . . . . . . . . 11 ⊢ (._r‘𝑀) = (._r‘𝑀)
34	12, 33	crngcom 20156	. . . . . . . . . 10 ⊢ ((𝑀 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(._r‘𝑀)𝑏) = (𝑏(._r‘𝑀)𝑎))
35	25, 30, 32, 34	syl3anc 1368	. . . . . . . . 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 6889	. . . . . . . 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 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝐹 ∈ (𝑀 RingHom 𝑁))
38		eqid 2726	. . . . . . . . . 10 ⊢ (._r‘𝑁) = (._r‘𝑁)
39	12, 33, 38	rhmmul 20388	. . . . . . . . 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 1368	. . . . . . . 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 20388	. . . . . . . . 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 1368	. . . . . . . 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 2774	. . . . . . 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 7903	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (𝐹 “ 𝑆) ∈ V)
45	5, 38	ressmulr 17261	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑆) ∈ V → (._r‘𝑁) = (._r‘𝐶))
46	1, 44, 45	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (._r‘𝑁) = (._r‘𝐶))
47	46	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (._r‘𝑁) = (._r‘𝐶))
48		simplrr 775	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘𝑎) = 𝑥)
49		simprr 770	. . . . . . . 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 7426	. . . . . . 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 7426	. . . . . . 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 2774	. . . . . 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 3155	. . . . 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 3155	. . . 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 3194	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(._r‘𝐶)𝑦) = (𝑦(._r‘𝐶)𝑥))
57		eqid 2726	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
58		eqid 2726	. . 3 ⊢ (._r‘𝐶) = (._r‘𝐶)
59	57, 58	iscrng2 20157	. 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)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ⊆ wss 3943 “ cima 5672 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 ↾_s cress 17182 ._rcmulr 17207 Ringcrg 20138 CRingccrg 20139 RingHom crh 20371 SubRingcsubrg 20469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-subg 19050 df-ghm 19139 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-rhm 20374 df-subrng 20446 df-subrg 20471

This theorem is referenced by: riccrng1 41655

W3C validator