Theorem rhmpreimaidl 21236

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 6042	. . . 4 ⊢ (◡𝐹 “ 𝐽) ⊆ dom 𝐹
2		eqid 2737	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2737	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	2, 3	rhmf 20424	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
5	1, 4	fssdm 6682	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
6	5	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
7	4	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
8	7	ffund 6667	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → Fun 𝐹)
9		rhmrcl1 20416	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝑅 ∈ Ring)
11		eqid 2737	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
12	2, 11	ring0cl 20206	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
13	10, 12	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑅) ∈ (Base‘𝑅))
14	7	fdmd 6673	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → dom 𝐹 = (Base‘𝑅))
15	13, 14	eleqtrrd 2840	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑅) ∈ dom 𝐹)
16		rhmghm 20423	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
17		ghmmhm 19159	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹 ∈ (𝑅 MndHom 𝑆))
18		eqid 2737	. . . . . . . 8 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	11, 18	mhm0 18723	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
20	16, 17, 19	3syl 18	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
21	20	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
22		rhmrcl2 20417	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
23		eqid 2737	. . . . . . 7 ⊢ (LIdeal‘𝑆) = (LIdeal‘𝑆)
24	23, 18	lidl0cl 21179	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑆) ∈ 𝐽)
25	22, 24	sylan 581	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑆) ∈ 𝐽)
26	21, 25	eqeltrd 2837	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0g‘𝑅)) ∈ 𝐽)
27		fvimacnv 7000	. . . . 5 ⊢ ((Fun 𝐹 ∧ (0g‘𝑅) ∈ dom 𝐹) → ((𝐹‘(0g‘𝑅)) ∈ 𝐽 ↔ (0g‘𝑅) ∈ (◡𝐹 “ 𝐽)))
28	27	biimpa 476	. . . 4 ⊢ (((Fun 𝐹 ∧ (0g‘𝑅) ∈ dom 𝐹) ∧ (𝐹‘(0g‘𝑅)) ∈ 𝐽) → (0g‘𝑅) ∈ (◡𝐹 “ 𝐽))
29	8, 15, 26, 28	syl21anc 838	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑅) ∈ (◡𝐹 “ 𝐽))
30	29	ne0d 4295	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ ∅)
31	7	ffnd 6664	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹 Fn (Base‘𝑅))
32	31	ad3antrrr 731	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐹 Fn (Base‘𝑅))
33	10	ad3antrrr 731	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑅 ∈ Ring)
34		simpllr 776	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑥 ∈ (Base‘𝑅))
35	5	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
36	35	sselda 3934	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
37	36	adantr 480	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
38		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
39	2, 38	ringcl 20189	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑎) ∈ (Base‘𝑅))
40	33, 34, 37, 39	syl3anc 1374	. . . . . . 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 480	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
42	41	sselda 3934	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑏 ∈ (Base‘𝑅))
43		eqid 2737	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
44	2, 43	ringacl 20217	. . . . . . 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 1374	. . . . . 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 733	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
47		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
48	2, 43, 47	ghmlin 19154	. . . . . . . 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 1374	. . . . . . 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 783	. . . . . . . . 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 17	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑆 ∈ Ring)
52		simpr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
53	52	ad3antrrr 731	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐽 ∈ (LIdeal‘𝑆))
54		eqid 2737	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
55	2, 38, 54	rhmmul 20425	. . . . . . . . . 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 1374	. . . . . . . . 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 7031	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
58	57	ad2antrr 727	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
59		simplr 769	. . . . . . . . . . 11 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (◡𝐹 “ 𝐽))
60		elpreima 7005	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑎 ∈ (◡𝐹 “ 𝐽) ↔ (𝑎 ∈ (Base‘𝑅) ∧ (𝐹‘𝑎) ∈ 𝐽)))
61	60	simplbda 499	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑎) ∈ 𝐽)
62	32, 59, 61	syl2anc 585	. . . . . . . . . 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 21184	. . . . . . . . . 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 839	. . . . . . . . 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 2837	. . . . . . . 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 484	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑏 ∈ (◡𝐹 “ 𝐽))
67		elpreima 7005	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑏 ∈ (◡𝐹 “ 𝐽) ↔ (𝑏 ∈ (Base‘𝑅) ∧ (𝐹‘𝑏) ∈ 𝐽)))
68	67	simplbda 499	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
69	32, 66, 68	syl2anc 585	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
70	23, 47	lidlacl 21180	. . . . . . . 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 839	. . . . . . 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 2837	. . . . . 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 7006	. . . . 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 466	. . . 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 3180	. . 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 3129	. 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 21174	. 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 1345	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ⊆ wss 3902  ∅c0 4286  ^◡ccnv 5624  dom cdm 5625   “ cima 5628  Fun wfun 6487   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  +_gcplusg 17181  ._rcmulr 17182  0_gc0g 17363   MndHom cmhm 18710   GrpHom cghm 19145  Ringcrg 20172   RingHom crh 20409  LIdealclidl 21165

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 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-sca 17197  df-vsca 17198  df-ip 17199  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-mhm 18712  df-grp 18870  df-minusg 18871  df-sbg 18872  df-subg 19057  df-ghm 19146  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-rhm 20412  df-subrg 20507  df-lmod 20817  df-lss 20887  df-sra 21129  df-rgmod 21130  df-lidl 21167

This theorem is referenced by:  kerlidl  21237  rhmpreimaprmidl  33534  ply1annidl  33861