Theorem rhmpreimaidl 21395

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 6084	. . . 4 ⊢ (◡𝐹 “ 𝐽) ⊆ dom 𝐹
2		eqid 2761	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2761	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	2, 3	rhmf 20565	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
5	1, 4	fssdm 6725	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
6	5	adantr 485	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
7	4	adantr 485	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
8	7	ffund 6710	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → Fun 𝐹)
9		rhmrcl1 20557	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	adantr 485	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝑅 ∈ Ring)
11		eqid 2761	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
12	2, 11	ring0cl 20349	. . . . . 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 6716	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → dom 𝐹 = (Base‘𝑅))
15	13, 14	eleqtrrd 2864	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑅) ∈ dom 𝐹)
16		rhmghm 20564	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
17		ghmmhm 19295	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹 ∈ (𝑅 MndHom 𝑆))
18		eqid 2761	. . . . . . . 8 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	11, 18	mhm0 18851	. . . . . . 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 485	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆))
22		rhmrcl2 20558	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
23		eqid 2761	. . . . . . 7 ⊢ (LIdeal‘𝑆) = (LIdeal‘𝑆)
24	23, 18	lidl0cl 21324	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑆) ∈ 𝐽)
25	22, 24	sylan 591	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑆) ∈ 𝐽)
26	21, 25	eqeltrd 2861	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0g‘𝑅)) ∈ 𝐽)
27		fvimacnv 7048	. . . . 5 ⊢ ((Fun 𝐹 ∧ (0g‘𝑅) ∈ dom 𝐹) → ((𝐹‘(0g‘𝑅)) ∈ 𝐽 ↔ (0g‘𝑅) ∈ (◡𝐹 “ 𝐽)))
28	27	biimpa 481	. . . 4 ⊢ (((Fun 𝐹 ∧ (0g‘𝑅) ∈ dom 𝐹) ∧ (𝐹‘(0g‘𝑅)) ∈ 𝐽) → (0g‘𝑅) ∈ (◡𝐹 “ 𝐽))
29	8, 15, 26, 28	syl21anc 850	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0g‘𝑅) ∈ (◡𝐹 “ 𝐽))
30	29	ne0d 4294	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ ∅)
31	7	ffnd 6706	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹 Fn (Base‘𝑅))
32	31	ad3antrrr 742	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐹 Fn (Base‘𝑅))
33	10	ad3antrrr 742	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑅 ∈ Ring)
34		simpllr 787	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑥 ∈ (Base‘𝑅))
35	5	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
36	35	sselda 3936	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
37	36	adantr 485	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
38		eqid 2761	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
39	2, 38	ringcl 20331	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑎) ∈ (Base‘𝑅))
40	33, 34, 37, 39	syl3anc 1396	. . . . . . 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 485	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → (◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
42	41	sselda 3936	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑏 ∈ (Base‘𝑅))
43		eqid 2761	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
44	2, 43	ringacl 20360	. . . . . . 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 1396	. . . . . 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 744	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
47		eqid 2761	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
48	2, 43, 47	ghmlin 19290	. . . . . . . 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 1396	. . . . . . 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 794	. . . . . . . . 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 489	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
53	52	ad3antrrr 742	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝐽 ∈ (LIdeal‘𝑆))
54		eqid 2761	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
55	2, 38, 54	rhmmul 20567	. . . . . . . . . 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 1396	. . . . . . . . 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 7079	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
58	57	ad2antrr 738	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
59		simplr 780	. . . . . . . . . . 11 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (◡𝐹 “ 𝐽))
60		elpreima 7053	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑎 ∈ (◡𝐹 “ 𝐽) ↔ (𝑎 ∈ (Base‘𝑅) ∧ (𝐹‘𝑎) ∈ 𝐽)))
61	60	simplbda 504	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑎) ∈ 𝐽)
62	32, 59, 61	syl2anc 595	. . . . . . . . . 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 21329	. . . . . . . . . 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 851	. . . . . . . . 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 2861	. . . . . . . 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 489	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → 𝑏 ∈ (◡𝐹 “ 𝐽))
67		elpreima 7053	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑏 ∈ (◡𝐹 “ 𝐽) ↔ (𝑏 ∈ (Base‘𝑅) ∧ (𝐹‘𝑏) ∈ 𝐽)))
68	67	simplbda 504	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
69	32, 66, 68	syl2anc 595	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
70	23, 47	lidlacl 21325	. . . . . . . 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 851	. . . . . . 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 2861	. . . . . 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 7054	. . . . 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 471	. . . 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 3206	. . 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 3155	. 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 21319	. 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 1360	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077   ⊆ wss 3904  ∅c0 4285  ^◡ccnv 5660  dom cdm 5661   “ cima 5664  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  +_gcplusg 17309  ._rcmulr 17310  0_gc0g 17491   MndHom cmhm 18838   GrpHom cghm 19282  Ringcrg 20314   RingHom crh 20550  LIdealclidl 21309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-sca 17325  df-vsca 17326  df-ip 17327  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-mhm 18840  df-grp 19002  df-minusg 19003  df-sbg 19004  df-subg 19188  df-ghm 19283  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-rhm 20553  df-subrg 20654  df-lmod 20962  df-lss 21032  df-sra 21273  df-rgmod 21274  df-lidl 21311

This theorem is referenced by:  kerlidl  21396  rhmpreimaprmidl  21458  ply1annidl  34058