Theorem rhmpreimaidl 21598

Description: The preimage of an ideal by a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)

Hypothesis

Ref	Expression
rhmpreimaidl.i	⊢ 𝐼 = (LIdeal‘𝑅)

Assertion

Ref	Expression
rhmpreimaidl	⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝐼)

Proof of Theorem rhmpreimaidl

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

Step	Hyp	Ref	Expression
1		cnvimass 6210	. . . 4 ⊢ (◡𝐹 “ 𝐽) ⊆ dom 𝐹
2		eqid 2763	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2763	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	2, 3	rhmf 20742	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
5	1, 4	fssdm 6748	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
6	5	adantr 486	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
7	4	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
8	7	ffund 6733	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → Fun 𝐹)
9		rhmrcl1 20734	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	adantr 486	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝑅 ∈ Ring)
11		eqid 2763	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
12	2, 11	ring0cl 20523	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
13	10, 12	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑅) ∈ (Base‘𝑅))
14	7	fdmd 6739	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → dom 𝐹 = (Base‘𝑅))
15	13, 14	eleqtrrd 2867	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑅) ∈ dom 𝐹)
16		rhmghm 20741	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
17		ghmmhm 19466	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹 ∈ (𝑅 MndHom 𝑆))
18		eqid 2763	. . . . . . . 8 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	11, 18	mhm0 19015	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
20	16, 17, 19	3syl 19	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
21	20	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
22		rhmrcl2 20735	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
23		eqid 2763	. . . . . . 7 ⊢ (LIdeal‘𝑆) = (LIdeal‘𝑆)
24	23, 18	lidl0cl 21526	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑆) ∈ 𝐽)
25	22, 24	sylan 592	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑆) ∈ 𝐽)
26	21, 25	eqeltrd 2864	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0g‘𝑅)) ∈ 𝐽)
27		fvimacnv 7072	. . . . 5 ⊢ ((Fun 𝐹 ∧ (0g‘𝑅) ∈ dom 𝐹) → ((𝐹‘(0g‘𝑅)) ∈ 𝐽 ↔ (0g‘𝑅) ∈ (◡𝐹 “ 𝐽)))
28	27	biimpa 482	. . . 4 ⊢ (((Fun 𝐹 ∧ (0g‘𝑅) ∈ dom 𝐹) ∧ (𝐹‘(0g‘𝑅)) ∈ 𝐽) → (0g‘𝑅) ∈ (◡𝐹 “ 𝐽))
29	8, 15, 26, 28	syl21anc 851	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑅) ∈ (◡𝐹 “ 𝐽))
30	29	ne0d 4291	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ ∅)
31	7	ffnd 6729	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹 Fn (Base‘𝑅))
32	31	ad3antrrr 743	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐹 Fn (Base‘𝑅))
33	10	ad3antrrr 743	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑅 ∈ Ring)
34		simpllr 788	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑥 ∈ (Base‘𝑅))
35	5	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
36	35	sselda 3913	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
37	36	adantr 486	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
38		eqid 2763	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
39	2, 38	ringcl 20504	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑎) ∈ (Base‘𝑅))
40	33, 34, 37, 39	syl3anc 1399	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝑥(.r‘𝑅)𝑎) ∈ (Base‘𝑅))
41	35	adantr 486	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
42	41	sselda 3913	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑏 ∈ (Base‘𝑅))
43		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
44	2, 43	ringacl 20534	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑥(.r‘𝑅)𝑎) ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ (Base‘𝑅))
45	33, 40, 42, 44	syl3anc 1399	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ (Base‘𝑅))
46	16	ad4antr 745	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
47		eqid 2763	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
48	2, 43, 47	ghmlin 19461	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑥(.r‘𝑅)𝑎) ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏)) = ((𝐹‘(𝑥(.r‘𝑅)𝑎))(+g‘𝑆)(𝐹‘𝑏)))
49	46, 40, 42, 48	syl3anc 1399	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏)) = ((𝐹‘(𝑥(.r‘𝑅)𝑎))(+g‘𝑆)(𝐹‘𝑏)))
50		simp-4l 795	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
51	50, 22	syl 18	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑆 ∈ Ring)
52		simpr 490	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
53	52	ad3antrrr 743	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐽 ∈ (LIdeal‘𝑆))
54		eqid 2763	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
55	2, 38, 54	rhmmul 20747	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑎)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑎)))
56	50, 34, 37, 55	syl3anc 1399	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘(𝑥(.r‘𝑅)𝑎)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑎)))
57	7	ffvelcdmda 7105	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
58	57	ad2antrr 739	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
59		simplr 781	. . . . . . . . . . 11 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (◡𝐹 “ 𝐽))
60		elpreima 7077	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑎 ∈ (◡𝐹 “ 𝐽) ↔ (𝑎 ∈ (Base‘𝑅) ∧ (𝐹‘𝑎) ∈ 𝐽)))
61	60	simplbda 505	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑎) ∈ 𝐽)
62	32, 59, 61	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑎) ∈ 𝐽)
63	23, 3, 54	lidlmcl 21531	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ ((𝐹‘𝑥) ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝐽)) → ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑎)) ∈ 𝐽)
64	51, 53, 58, 62, 63	syl22anc 852	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑎)) ∈ 𝐽)
65	56, 64	eqeltrd 2864	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘(𝑥(.r‘𝑅)𝑎)) ∈ 𝐽)
66		simpr 490	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑏 ∈ (◡𝐹 “ 𝐽))
67		elpreima 7077	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑏 ∈ (◡𝐹 “ 𝐽) ↔ (𝑏 ∈ (Base‘𝑅) ∧ (𝐹‘𝑏) ∈ 𝐽)))
68	67	simplbda 505	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
69	32, 66, 68	syl2anc 596	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
70	23, 47	lidlacl 21527	. . . . . . . 8 ⊢ (((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ ((𝐹‘(𝑥(.r‘𝑅)𝑎)) ∈ 𝐽 ∧ (𝐹‘𝑏) ∈ 𝐽)) → ((𝐹‘(𝑥(.r‘𝑅)𝑎))(+g‘𝑆)(𝐹‘𝑏)) ∈ 𝐽)
71	51, 53, 65, 69, 70	syl22anc 852	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → ((𝐹‘(𝑥(.r‘𝑅)𝑎))(+g‘𝑆)(𝐹‘𝑏)) ∈ 𝐽)
72	49, 71	eqeltrd 2864	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏)) ∈ 𝐽)
73	32, 45, 72	elpreimad 7079	. . . . 5 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽))
74	73	anasss 472	. . . 4 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑎 ∈ (◡𝐹 “ 𝐽) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽))) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽))
75	74	ralrimivva 3209	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑎 ∈ (◡𝐹 “ 𝐽)∀𝑏 ∈ (◡𝐹 “ 𝐽)((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽))
76	75	ralrimiva 3158	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑎 ∈ (◡𝐹 “ 𝐽)∀𝑏 ∈ (◡𝐹 “ 𝐽)((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽))
77		rhmpreimaidl.i	. . 3 ⊢ 𝐼 = (LIdeal‘𝑅)
78	77, 2, 43, 38	islidl 21521	. 2 ⊢ ((◡𝐹 “ 𝐽) ∈ 𝐼 ↔ ((◡𝐹 “ 𝐽) ⊆ (Base‘𝑅) ∧ (◡𝐹 “ 𝐽) ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑎 ∈ (◡𝐹 “ 𝐽)∀𝑏 ∈ (◡𝐹 “ 𝐽)((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)))
79	6, 30, 76, 78	syl3anbrc 1362	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝐼)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  ^◡ccnv 5654  dom cdm 5655   “ cima 5658  Fun wfun 6549   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  ._rcmulr 17455  0_gc0g 17636   MndHom cmhm 19002   GrpHom cghm 19453  Ringcrg 20486   RingHom crh 20726  LIdealclidl 21511

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-rep 5235  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-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-sca 17470  df-vsca 17471  df-ip 17472  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-grp 19173  df-minusg 19174  df-sbg 19175  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-rhm 20729  df-subrg 20849  df-lmod 21164  df-lss 21234  df-sra 21475  df-rgmod 21476  df-lidl 21513

This theorem is used by:  kerlidl  21599  rhmpreimaprmidl  21662  ply1annidl  34401