rhmpreimaidl - Mathbox for Thierry Arnoux

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Theorem rhmpreimaidl 33057

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 6079	. . . 4 ⊢ (^◡𝐹 “ 𝐽) ⊆ dom 𝐹
2		eqid 2727	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2727	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	2, 3	rhmf 20406	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
5	1, 4	fssdm 6736	. . 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 6720	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → Fun 𝐹)
9		rhmrcl1 20397	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝑅 ∈ Ring)
11		eqid 2727	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
12	2, 11	ring0cl 20185	. . . . . 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 6727	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → dom 𝐹 = (Base‘𝑅))
15	13, 14	eleqtrrd 2831	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑅) ∈ dom 𝐹)
16		rhmghm 20405	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
17		ghmmhm 19164	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹 ∈ (𝑅 MndHom 𝑆))
18		eqid 2727	. . . . . . . 8 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
19	11, 18	mhm0 18736	. . . . . . 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 480	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
22		rhmrcl2 20398	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
23		eqid 2727	. . . . . . 7 ⊢ (LIdeal‘𝑆) = (LIdeal‘𝑆)
24	23, 18	lidl0cl 21098	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑆) ∈ 𝐽)
25	22, 24	sylan 579	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑆) ∈ 𝐽)
26	21, 25	eqeltrd 2828	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0_g‘𝑅)) ∈ 𝐽)
27		fvimacnv 7056	. . . . 5 ⊢ ((Fun 𝐹 ∧ (0_g‘𝑅) ∈ dom 𝐹) → ((𝐹‘(0_g‘𝑅)) ∈ 𝐽 ↔ (0_g‘𝑅) ∈ (^◡𝐹 “ 𝐽)))
28	27	biimpa 476	. . . 4 ⊢ (((Fun 𝐹 ∧ (0_g‘𝑅) ∈ dom 𝐹) ∧ (𝐹‘(0_g‘𝑅)) ∈ 𝐽) → (0_g‘𝑅) ∈ (^◡𝐹 “ 𝐽))
29	8, 15, 26, 28	syl21anc 837	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑅) ∈ (^◡𝐹 “ 𝐽))
30	29	ne0d 4331	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ≠ ∅)
31	7	ffnd 6717	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹 Fn (Base‘𝑅))
32	31	ad3antrrr 729	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝐹 Fn (Base‘𝑅))
33	10	ad3antrrr 729	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑅 ∈ Ring)
34		simpllr 775	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑥 ∈ (Base‘𝑅))
35	5	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (^◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
36	35	sselda 3978	. . . . . . . . 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 2727	. . . . . . . . 9 ⊢ (._r‘𝑅) = (._r‘𝑅)
39	2, 38	ringcl 20174	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ (Base‘𝑅)) → (𝑥(._r‘𝑅)𝑎) ∈ (Base‘𝑅))
40	33, 34, 37, 39	syl3anc 1369	. . . . . . 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 3978	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑏 ∈ (Base‘𝑅))
43		eqid 2727	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	2, 43	ringacl 20196	. . . . . . 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 1369	. . . . . 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 731	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
47		eqid 2727	. . . . . . . . 9 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
48	2, 43, 47	ghmlin 19159	. . . . . . . 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 1369	. . . . . . 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 782	. . . . . . . . 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 729	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝐽 ∈ (LIdeal‘𝑆))
54		eqid 2727	. . . . . . . . . . 11 ⊢ (._r‘𝑆) = (._r‘𝑆)
55	2, 38, 54	rhmmul 20407	. . . . . . . . . 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 1369	. . . . . . . . 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 7088	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
58	57	ad2antrr 725	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑥) ∈ (Base‘𝑆))
59		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑎 ∈ (^◡𝐹 “ 𝐽))
60		elpreima 7061	. . . . . . . . . . . 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 583	. . . . . . . . . 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 21103	. . . . . . . . . 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 838	. . . . . . . . 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 2828	. . . . . . . 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 7061	. . . . . . . . . 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 583	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
70	23, 47	lidlacl 21099	. . . . . . . 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 838	. . . . . . 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 2828	. . . . . 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 7062	. . . . 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 3195	. . 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 3141	. 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 21093	. 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 1341	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ⊆ wss 3944 ∅c0 4318 ^◡ccnv 5671 dom cdm 5672 “ cima 5675 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 Basecbs 17165 +_gcplusg 17218 ._rcmulr 17219 0_gc0g 17406 MndHom cmhm 18723 GrpHom cghm 19151 Ringcrg 20157 RingHom crh 20390 LIdealclidl 21084

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-0g 17408 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19062 df-ghm 19152 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-rhm 20393 df-subrg 20490 df-lmod 20727 df-lss 20798 df-sra 21040 df-rgmod 21041 df-lidl 21086

This theorem is referenced by: kerlidl 33058 rhmpreimaprmidl 33090 ply1annidl 33296

W3C validator