rhmimaidl - Mathbox for Thierry Arnoux

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Theorem rhmimaidl 33680

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 2769	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		rhmimaidl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
3	1, 2	rhmf 20562	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐵)
4		fimass 6724	. . . . 5 ⊢ (𝐹:(Base‘𝑅)⟶𝐵 → (𝐹 “ 𝐼) ⊆ 𝐵)
5	3, 4	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ 𝐼) ⊆ 𝐵)
6	5	ad2antrr 738	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ⊆ 𝐵)
7	3	ffnd 6704	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
8	7	ad2antrr 738	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐹 Fn (Base‘𝑅))
9		rhmrcl1 20554	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	ad2antrr 738	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝑅 ∈ Ring)
11		eqid 2769	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
12	1, 11	ring0cl 20346	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
13	10, 12	syl 18	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ (Base‘𝑅))
14		simpr 489	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐼 ∈ 𝑇)
15		rhmimaidl.t	. . . . . . 7 ⊢ 𝑇 = (LIdeal‘𝑅)
16	15, 11	lidl0cl 21319	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ 𝐼)
17	10, 14, 16	syl2anc 595	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ 𝐼)
18	8, 13, 17	fnfvimad 7230	. . . 4 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹‘(0_g‘𝑅)) ∈ (𝐹 “ 𝐼))
19	18	ne0d 4303	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ≠ ∅)
20		rhmghm 20561	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
21	20	ad4antr 744	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
22	9	ad4antr 744	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
23		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑧 ∈ (Base‘𝑅))
24	1, 15	lidlss 21310	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐼 ∈ 𝑇 → 𝐼 ⊆ (Base‘𝑅))
25	24	ad4antlr 745	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐼 ⊆ (Base‘𝑅))
26		simplr 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ 𝐼)
27	25, 26	sseldd 3946	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ (Base‘𝑅))
28		eqid 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._r‘𝑅) = (._r‘𝑅)
29	1, 28	ringcl 20328	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅))
30	22, 23, 27, 29	syl3anc 1396	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅))
31		simpllr 787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ 𝐼)
32	25, 31	sseldd 3946	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ (Base‘𝑅))
33		eqid 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
34		eqid 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
35	1, 33, 34	ghmlin 19287	. . . . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
38		eqid 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._r‘𝑆) = (._r‘𝑆)
39	1, 28, 38	rhmmul 20564	. . . . . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . . 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 731	. . . . . . . . . . . 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 763	. . . . . . . . . . 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 489	. . . . . . . . . . . . 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 787	. . . . . . . . . . . . 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 7426	. . . . . . . . . . . 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 797	. . . . . . . . . . . 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 7426	. . . . . . . . . . 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 2804	. . . . . . . . . 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 754	. . . . . . . . . . 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 18	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐼 ⊆ (Base‘𝑅))
57	56	ad9antr 754	. . . . . . . . . . . 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 754	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 795	. . . . . . . . . . . . 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 799	. . . . . . . . . . . . 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 21314	. . . . . . . . . . . . . . . . 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 1163	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ 𝑇 → ∀𝑧 ∈ (Base‘𝑅)∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
64	63	r19.21bi 3263	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
65	64	r19.21bi 3263	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
66	65	r19.21bi 3263	. . . . . . . . . . . . 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 853	. . . . . . . . . . . 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 3946	. . . . . . . . . . 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 7230	. . . . . . . . . 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 2870	. . . . . . . . 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 738	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐹:(Base‘𝑅)⟶𝐵)
72	71	ffund 6708	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → Fun 𝐹)
73	72	ad7antr 750	. . . . . . . . . 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 6714	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
75	74	imaeq2d 6060	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ dom 𝐹) = (𝐹 “ (Base‘𝑅)))
76		imadmrn 6070	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
77	75, 76	eqtr3di 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ (Base‘𝑅)) = ran 𝐹)
78	77	eqeq1d 2771	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 “ (Base‘𝑅)) = 𝐵 ↔ ran 𝐹 = 𝐵))
79	78	biimpar 482	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝐹 “ (Base‘𝑅)) = 𝐵)
80	79	eleq2d 2855	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝑥 ∈ (𝐹 “ (Base‘𝑅)) ↔ 𝑥 ∈ 𝐵))
81	80	biimpar 482	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
82	81	adantlr 727	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
83	82	ad6antr 748	. . . . . . . . . 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 6944	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ (Base‘𝑅))) → ∃𝑧 ∈ (Base‘𝑅)(𝐹‘𝑧) = 𝑥)
85	73, 83, 84	syl2anc 595	. . . . . . . . 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 3179	. . . . . . . 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 746	. . . . . . . . 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 795	. . . . . . . . 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 6944	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑎 ∈ (𝐹 “ 𝐼)) → ∃𝑖 ∈ 𝐼 (𝐹‘𝑖) = 𝑎)
90	87, 88, 89	syl2anc 595	. . . . . . . 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 3179	. . . . . . 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 742	. . . . . . . 8 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → Fun 𝐹)
93		simpr 489	. . . . . . . 8 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → 𝑏 ∈ (𝐹 “ 𝐼))
94		fvelima 6944	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → ∃𝑗 ∈ 𝐼 (𝐹‘𝑗) = 𝑏)
95	92, 93, 94	syl2anc 595	. . . . . . 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 3179	. . . . . 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 471	. . . . 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 3214	. . . 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 3163	. . 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 21314	. . 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 1360	. 2 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)
103	102	3impa 1125	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ⊆ wss 3913 ∅c0 4294 dom cdm 5659 ran crn 5660 “ cima 5662 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7408 Basecbs 17265 +_gcplusg 17306 ._rcmulr 17307 0_gc0g 17488 GrpHom cghm 19279 Ringcrg 20311 RingHom crh 20547 LIdealclidl 21304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-ip 17324 df-0g 17490 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-mhm 18837 df-grp 18999 df-minusg 19000 df-sbg 19001 df-subg 19185 df-ghm 19280 df-mgp 20213 df-ur 20260 df-ring 20313 df-rhm 20550 df-subrg 20651 df-lmod 20957 df-lss 21027 df-sra 21268 df-rgmod 21269 df-lidl 21306

This theorem is referenced by: rhmpreimacnlem 34215 rhmpreimacn 34216

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