rhmpreimaidl - Mathbox for Thierry Arnoux

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Theorem rhmpreimaidl 32525

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 6077	. . . 4 ⊢ (^◡𝐹 “ 𝐽) ⊆ dom 𝐹
2		eqid 2732	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2732	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	2, 3	rhmf 20255	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
5	1, 4	fssdm 6734	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (^◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
6	5	adantr 481	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
7	4	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
8	7	ffund 6718	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → Fun 𝐹)
9		rhmrcl1 20247	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	adantr 481	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝑅 ∈ Ring)
11		eqid 2732	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
12	2, 11	ring0cl 20077	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
13	10, 12	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑅) ∈ (Base‘𝑅))
14	7	fdmd 6725	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → dom 𝐹 = (Base‘𝑅))
15	13, 14	eleqtrrd 2836	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑅) ∈ dom 𝐹)
16		rhmghm 20254	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
17		ghmmhm 19096	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹 ∈ (𝑅 MndHom 𝑆))
18		eqid 2732	. . . . . . . 8 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
19	11, 18	mhm0 18676	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
20	16, 17, 19	3syl 18	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
21	20	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
22		rhmrcl2 20248	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
23		eqid 2732	. . . . . . 7 ⊢ (LIdeal‘𝑆) = (LIdeal‘𝑆)
24	23, 18	lidl0cl 20827	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑆) ∈ 𝐽)
25	22, 24	sylan 580	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑆) ∈ 𝐽)
26	21, 25	eqeltrd 2833	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0_g‘𝑅)) ∈ 𝐽)
27		fvimacnv 7051	. . . . 5 ⊢ ((Fun 𝐹 ∧ (0_g‘𝑅) ∈ dom 𝐹) → ((𝐹‘(0_g‘𝑅)) ∈ 𝐽 ↔ (0_g‘𝑅) ∈ (^◡𝐹 “ 𝐽)))
28	27	biimpa 477	. . . 4 ⊢ (((Fun 𝐹 ∧ (0_g‘𝑅) ∈ dom 𝐹) ∧ (𝐹‘(0_g‘𝑅)) ∈ 𝐽) → (0_g‘𝑅) ∈ (^◡𝐹 “ 𝐽))
29	8, 15, 26, 28	syl21anc 836	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑅) ∈ (^◡𝐹 “ 𝐽))
30	29	ne0d 4334	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ≠ ∅)
31	7	ffnd 6715	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹 Fn (Base‘𝑅))
32	31	ad3antrrr 728	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝐹 Fn (Base‘𝑅))
33	10	ad3antrrr 728	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑅 ∈ Ring)
34		simpllr 774	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑥 ∈ (Base‘𝑅))
35	5	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (^◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
36	35	sselda 3981	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
37	36	adantr 481	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
38		eqid 2732	. . . . . . . . 9 ⊢ (._r‘𝑅) = (._r‘𝑅)
39	2, 38	ringcl 20066	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑥(._r‘𝑅)𝑎) ∈ (Base‘𝑅))
40	33, 34, 37, 39	syl3anc 1371	. . . . . . 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 481	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) → (^◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
42	41	sselda 3981	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑏 ∈ (Base‘𝑅))
43		eqid 2732	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	2, 43	ringacl 20088	. . . . . . 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 1371	. . . . . 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 730	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
47		eqid 2732	. . . . . . . . 9 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
48	2, 43, 47	ghmlin 19091	. . . . . . . 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 1371	. . . . . . 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 781	. . . . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
53	52	ad3antrrr 728	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝐽 ∈ (LIdeal‘𝑆))
54		eqid 2732	. . . . . . . . . . 11 ⊢ (._r‘𝑆) = (._r‘𝑆)
55	2, 38, 54	rhmmul 20256	. . . . . . . . . 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 1371	. . . . . . . . 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 7083	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
58	57	ad2antrr 724	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
59		simplr 767	. . . . . . . . . . 11 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑎 ∈ (^◡𝐹 “ 𝐽))
60		elpreima 7056	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑎 ∈ (^◡𝐹 “ 𝐽) ↔ (𝑎 ∈ (Base‘𝑅) ∧ (𝐹‘𝑎) ∈ 𝐽)))
61	60	simplbda 500	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑎) ∈ 𝐽)
62	32, 59, 61	syl2anc 584	. . . . . . . . . 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 20832	. . . . . . . . . 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 837	. . . . . . . . 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 2833	. . . . . . . 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 485	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑏 ∈ (^◡𝐹 “ 𝐽))
67		elpreima 7056	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑏 ∈ (^◡𝐹 “ 𝐽) ↔ (𝑏 ∈ (Base‘𝑅) ∧ (𝐹‘𝑏) ∈ 𝐽)))
68	67	simplbda 500	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
69	32, 66, 68	syl2anc 584	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
70	23, 47	lidlacl 20828	. . . . . . . 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 837	. . . . . . 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 2833	. . . . . 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 7057	. . . . 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 467	. . . 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 3200	. . 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 3146	. 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 20826	. 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 1343	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3947 ∅c0 4321 ^◡ccnv 5674 dom cdm 5675 “ cima 5678 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 0_gc0g 17381 MndHom cmhm 18665 GrpHom cghm 19083 Ringcrg 20049 RingHom crh 20240 LIdealclidl 20775

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-ghm 19084 df-mgp 19982 df-ur 19999 df-ring 20051 df-rnghom 20243 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-lidl 20779

This theorem is referenced by: kerlidl 32526 rhmpreimaprmidl 32558 ply1annidl 32751

W3C validator