rhmpreimaidl - Mathbox for Thierry Arnoux

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Theorem rhmpreimaidl 33178

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 6081	. . . 4 ⊢ (^◡𝐹 “ 𝐽) ⊆ dom 𝐹
2		eqid 2725	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2725	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
4	2, 3	rhmf 20423	. . . 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 479	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
7	4	adantr 479	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
8	7	ffund 6721	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → Fun 𝐹)
9		rhmrcl1 20414	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	adantr 479	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → 𝑅 ∈ Ring)
11		eqid 2725	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
12	2, 11	ring0cl 20202	. . . . . 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 2828	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑅) ∈ dom 𝐹)
16		rhmghm 20422	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
17		ghmmhm 19179	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹 ∈ (𝑅 MndHom 𝑆))
18		eqid 2725	. . . . . . . 8 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
19	11, 18	mhm0 18745	. . . . . . 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 479	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
22		rhmrcl2 20415	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
23		eqid 2725	. . . . . . 7 ⊢ (LIdeal‘𝑆) = (LIdeal‘𝑆)
24	23, 18	lidl0cl 21115	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑆) ∈ 𝐽)
25	22, 24	sylan 578	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (0_g‘𝑆) ∈ 𝐽)
26	21, 25	eqeltrd 2825	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹‘(0_g‘𝑅)) ∈ 𝐽)
27		fvimacnv 7055	. . . . 5 ⊢ ((Fun 𝐹 ∧ (0_g‘𝑅) ∈ dom 𝐹) → ((𝐹‘(0_g‘𝑅)) ∈ 𝐽 ↔ (0_g‘𝑅) ∈ (^◡𝐹 “ 𝐽)))
28	27	biimpa 475	. . . 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 4332	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ≠ ∅)
31	7	ffnd 6718	. . . . . . 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 3973	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
37	36	adantr 479	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
38		eqid 2725	. . . . . . . . 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 1368	. . . . . . 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 479	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) → (^◡𝐹 “ 𝐽) ⊆ (Base‘𝑅))
42	41	sselda 3973	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑏 ∈ (Base‘𝑅))
43		eqid 2725	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	2, 43	ringacl 20213	. . . . . . 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 1368	. . . . . 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 2725	. . . . . . . . 9 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
48	2, 43, 47	ghmlin 19174	. . . . . . . 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 1368	. . . . . . 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 483	. . . . . . . . 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 2725	. . . . . . . . . . 11 ⊢ (._r‘𝑆) = (._r‘𝑆)
55	2, 38, 54	rhmmul 20424	. . . . . . . . . 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 1368	. . . . . . . . 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 7087	. . . . . . . . . . 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 7060	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑎 ∈ (^◡𝐹 “ 𝐽) ↔ (𝑎 ∈ (Base‘𝑅) ∧ (𝐹‘𝑎) ∈ 𝐽)))
61	60	simplbda 498	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑎) ∈ 𝐽)
62	32, 59, 61	syl2anc 582	. . . . . . . . . 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 21120	. . . . . . . . . 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 2825	. . . . . . . 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 483	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → 𝑏 ∈ (^◡𝐹 “ 𝐽))
67		elpreima 7060	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → (𝑏 ∈ (^◡𝐹 “ 𝐽) ↔ (𝑏 ∈ (Base‘𝑅) ∧ (𝐹‘𝑏) ∈ 𝐽)))
68	67	simplbda 498	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
69	32, 66, 68	syl2anc 582	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 ∈ (^◡𝐹 “ 𝐽)) ∧ 𝑏 ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ 𝐽)
70	23, 47	lidlacl 21116	. . . . . . . 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 2825	. . . . . 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 7061	. . . . 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 465	. . . 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 3191	. . 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 3136	. 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 21110	. 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 1340	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ⊆ wss 3941 ∅c0 4319 ^◡ccnv 5672 dom cdm 5673 “ cima 5676 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 Basecbs 17174 +_gcplusg 17227 ._rcmulr 17228 0_gc0g 17415 MndHom cmhm 18732 GrpHom cghm 19166 Ringcrg 20172 RingHom crh 20407 LIdealclidl 21101

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-ghm 19167 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-rhm 20410 df-subrg 20507 df-lmod 20744 df-lss 20815 df-sra 21057 df-rgmod 21058 df-lidl 21103

This theorem is referenced by: kerlidl 33179 rhmpreimaprmidl 33212 ply1annidl 33426

W3C validator