rhmimaidl - Mathbox for Thierry Arnoux

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Theorem rhmimaidl 33507

Description: The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.)

**Hypotheses**
Ref	Expression
rhmimaidl.b	⊢ 𝐵 = (Base‘𝑆)
rhmimaidl.t	⊢ 𝑇 = (LIdeal‘𝑅)
rhmimaidl.u	⊢ 𝑈 = (LIdeal‘𝑆)

**Assertion**
Ref	Expression
rhmimaidl	⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)

Proof of Theorem rhmimaidl Dummy variables 𝑎 𝑏 𝑖 𝑗 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		rhmimaidl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
3	1, 2	rhmf 20455	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐵)
4		fimass 6682	. . . . 5 ⊢ (𝐹:(Base‘𝑅)⟶𝐵 → (𝐹 “ 𝐼) ⊆ 𝐵)
5	3, 4	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ 𝐼) ⊆ 𝐵)
6	5	ad2antrr 727	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ⊆ 𝐵)
7	3	ffnd 6663	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
8	7	ad2antrr 727	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐹 Fn (Base‘𝑅))
9		rhmrcl1 20447	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	ad2antrr 727	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝑅 ∈ Ring)
11		eqid 2737	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
12	1, 11	ring0cl 20239	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
13	10, 12	syl 17	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ (Base‘𝑅))
14		simpr 484	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐼 ∈ 𝑇)
15		rhmimaidl.t	. . . . . . 7 ⊢ 𝑇 = (LIdeal‘𝑅)
16	15, 11	lidl0cl 21210	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ 𝐼)
17	10, 14, 16	syl2anc 585	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ 𝐼)
18	8, 13, 17	fnfvimad 7182	. . . 4 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹‘(0_g‘𝑅)) ∈ (𝐹 “ 𝐼))
19	18	ne0d 4283	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ≠ ∅)
20		rhmghm 20454	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
21	20	ad4antr 733	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
22	9	ad4antr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
23		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑧 ∈ (Base‘𝑅))
24	1, 15	lidlss 21202	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐼 ∈ 𝑇 → 𝐼 ⊆ (Base‘𝑅))
25	24	ad4antlr 734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐼 ⊆ (Base‘𝑅))
26		simplr 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ 𝐼)
27	25, 26	sseldd 3923	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ (Base‘𝑅))
28		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._r‘𝑅) = (._r‘𝑅)
29	1, 28	ringcl 20222	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅))
30	22, 23, 27, 29	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅))
31		simpllr 776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ 𝐼)
32	25, 31	sseldd 3923	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ (Base‘𝑅))
33		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
34		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
35	1, 33, 34	ghmlin 19187	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅) ∧ 𝑗 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = ((𝐹‘(𝑧(._r‘𝑅)𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
36	21, 30, 32, 35	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = ((𝐹‘(𝑧(._r‘𝑅)𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
37		simp-4l 783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
38		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._r‘𝑆) = (._r‘𝑆)
39	1, 28, 38	rhmmul 20456	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝐹‘(𝑧(._r‘𝑅)𝑖)) = ((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖)))
40	37, 23, 27, 39	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘(𝑧(._r‘𝑅)𝑖)) = ((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖)))
41	40	oveq1d 7375	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → ((𝐹‘(𝑧(._r‘𝑅)𝑖))(+_g‘𝑆)(𝐹‘𝑗)) = (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
42	36, 41	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
43	42	adantl4r 756	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
44	43	adantl3r 751	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
45	44	adantl3r 751	. . . . . . . . . . . . . 14 ⊢ ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
46	45	adantl3r 751	. . . . . . . . . . . . 13 ⊢ (((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
47	46	adantllr 720	. . . . . . . . . . . 12 ⊢ ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
48	47	ad4ant13 752	. . . . . . . . . . 11 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)))
49		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → (𝐹‘𝑧) = 𝑥)
50		simpllr 776	. . . . . . . . . . . . 13 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → (𝐹‘𝑖) = 𝑎)
51	49, 50	oveq12d 7378	. . . . . . . . . . . 12 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → ((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖)) = (𝑥(._r‘𝑆)𝑎))
52		simp-5r 786	. . . . . . . . . . . 12 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → (𝐹‘𝑗) = 𝑏)
53	51, 52	oveq12d 7378	. . . . . . . . . . 11 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → (((𝐹‘𝑧)(._r‘𝑆)(𝐹‘𝑖))(+_g‘𝑆)(𝐹‘𝑗)) = ((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏))
54	48, 53	eqtrd 2772	. . . . . . . . . 10 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) = ((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏))
55	8	ad9antr 743	. . . . . . . . . . 11 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → 𝐹 Fn (Base‘𝑅))
56	14, 24	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐼 ⊆ (Base‘𝑅))
57	56	ad9antr 743	. . . . . . . . . . . 12 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → 𝐼 ⊆ (Base‘𝑅))
58	14	ad9antr 743	. . . . . . . . . . . . 13 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → 𝐼 ∈ 𝑇)
59		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → 𝑧 ∈ (Base‘𝑅))
60		simp-4r 784	. . . . . . . . . . . . 13 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → 𝑖 ∈ 𝐼)
61		simp-6r 788	. . . . . . . . . . . . 13 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → 𝑗 ∈ 𝐼)
62	15, 1, 33, 28	islidl 21205	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ 𝑇 ↔ (𝐼 ⊆ (Base‘𝑅) ∧ 𝐼 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘𝑅)∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼))
63	62	simp3bi 1148	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ 𝑇 → ∀𝑧 ∈ (Base‘𝑅)∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
64	63	r19.21bi 3230	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
65	64	r19.21bi 3230	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
66	65	r19.21bi 3230	. . . . . . . . . . . . 13 ⊢ ((((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
67	58, 59, 60, 61, 66	syl1111anc 841	. . . . . . . . . . . 12 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
68	57, 67	sseldd 3923	. . . . . . . . . . 11 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ (Base‘𝑅))
69	55, 68, 67	fnfvimad 7182	. . . . . . . . . 10 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → (𝐹‘((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗)) ∈ (𝐹 “ 𝐼))
70	54, 69	eqeltrrd 2838	. . . . . . . . 9 ⊢ ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹‘𝑧) = 𝑥) → ((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏) ∈ (𝐹 “ 𝐼))
71	3	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐹:(Base‘𝑅)⟶𝐵)
72	71	ffund 6666	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → Fun 𝐹)
73	72	ad7antr 739	. . . . . . . . . 10 ⊢ ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) → Fun 𝐹)
74	3	fdmd 6672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
75	74	imaeq2d 6019	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ dom 𝐹) = (𝐹 “ (Base‘𝑅)))
76		imadmrn 6029	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
77	75, 76	eqtr3di 2787	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ (Base‘𝑅)) = ran 𝐹)
78	77	eqeq1d 2739	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 “ (Base‘𝑅)) = 𝐵 ↔ ran 𝐹 = 𝐵))
79	78	biimpar 477	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝐹 “ (Base‘𝑅)) = 𝐵)
80	79	eleq2d 2823	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝑥 ∈ (𝐹 “ (Base‘𝑅)) ↔ 𝑥 ∈ 𝐵))
81	80	biimpar 477	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
82	81	adantlr 716	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
83	82	ad6antr 737	. . . . . . . . . 10 ⊢ ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
84		fvelima 6899	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ (Base‘𝑅))) → ∃𝑧 ∈ (Base‘𝑅)(𝐹‘𝑧) = 𝑥)
85	73, 83, 84	syl2anc 585	. . . . . . . . 9 ⊢ ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑧 ∈ (Base‘𝑅)(𝐹‘𝑧) = 𝑥)
86	70, 85	r19.29a 3146	. . . . . . . 8 ⊢ ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑖 ∈ 𝐼) ∧ (𝐹‘𝑖) = 𝑎) → ((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏) ∈ (𝐹 “ 𝐼))
87	72	ad5antr 735	. . . . . . . . 9 ⊢ ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) → Fun 𝐹)
88		simp-4r 784	. . . . . . . . 9 ⊢ ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) → 𝑎 ∈ (𝐹 “ 𝐼))
89		fvelima 6899	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑎 ∈ (𝐹 “ 𝐼)) → ∃𝑖 ∈ 𝐼 (𝐹‘𝑖) = 𝑎)
90	87, 88, 89	syl2anc 585	. . . . . . . 8 ⊢ ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) → ∃𝑖 ∈ 𝐼 (𝐹‘𝑖) = 𝑎)
91	86, 90	r19.29a 3146	. . . . . . 7 ⊢ ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) ∧ 𝑗 ∈ 𝐼) ∧ (𝐹‘𝑗) = 𝑏) → ((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏) ∈ (𝐹 “ 𝐼))
92	72	ad3antrrr 731	. . . . . . . 8 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → Fun 𝐹)
93		simpr 484	. . . . . . . 8 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → 𝑏 ∈ (𝐹 “ 𝐼))
94		fvelima 6899	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → ∃𝑗 ∈ 𝐼 (𝐹‘𝑗) = 𝑏)
95	92, 93, 94	syl2anc 585	. . . . . . 7 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → ∃𝑗 ∈ 𝐼 (𝐹‘𝑗) = 𝑏)
96	91, 95	r19.29a 3146	. . . . . 6 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → ((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏) ∈ (𝐹 “ 𝐼))
97	96	anasss 466	. . . . 5 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ (𝑎 ∈ (𝐹 “ 𝐼) ∧ 𝑏 ∈ (𝐹 “ 𝐼))) → ((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏) ∈ (𝐹 “ 𝐼))
98	97	ralrimivva 3181	. . . 4 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) → ∀𝑎 ∈ (𝐹 “ 𝐼)∀𝑏 ∈ (𝐹 “ 𝐼)((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏) ∈ (𝐹 “ 𝐼))
99	98	ralrimiva 3130	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ (𝐹 “ 𝐼)∀𝑏 ∈ (𝐹 “ 𝐼)((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏) ∈ (𝐹 “ 𝐼))
100		rhmimaidl.u	. . . 4 ⊢ 𝑈 = (LIdeal‘𝑆)
101	100, 2, 34, 38	islidl 21205	. . 3 ⊢ ((𝐹 “ 𝐼) ∈ 𝑈 ↔ ((𝐹 “ 𝐼) ⊆ 𝐵 ∧ (𝐹 “ 𝐼) ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ (𝐹 “ 𝐼)∀𝑏 ∈ (𝐹 “ 𝐼)((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏) ∈ (𝐹 “ 𝐼)))
102	6, 19, 99, 101	syl3anbrc 1345	. 2 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)
103	102	3impa 1110	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ⊆ wss 3890 ∅c0 4274 dom cdm 5624 ran crn 5625 “ cima 5627 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 +_gcplusg 17211 ._rcmulr 17212 0_gc0g 17393 GrpHom cghm 19178 Ringcrg 20205 RingHom crh 20440 LIdealclidl 21196

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

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 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-ghm 19179 df-mgp 20113 df-ur 20154 df-ring 20207 df-rhm 20443 df-subrg 20538 df-lmod 20848 df-lss 20918 df-sra 21160 df-rgmod 21161 df-lidl 21198

This theorem is referenced by: rhmpreimacnlem 34044 rhmpreimacn 34045

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