rhmimaidl - Mathbox for Thierry Arnoux

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Theorem rhmimaidl 33868

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 2762	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		rhmimaidl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
3	1, 2	rhmf 20632	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐵)
4		fimass 6727	. . . . 5 ⊢ (𝐹:(Base‘𝑅)⟶𝐵 → (𝐹 “ 𝐼) ⊆ 𝐵)
5	3, 4	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ 𝐼) ⊆ 𝐵)
6	5	ad2antrr 739	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ⊆ 𝐵)
7	3	ffnd 6707	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
8	7	ad2antrr 739	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐹 Fn (Base‘𝑅))
9		rhmrcl1 20624	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	ad2antrr 739	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝑅 ∈ Ring)
11		eqid 2762	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
12	1, 11	ring0cl 20414	. . . . . 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 490	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐼 ∈ 𝑇)
15		rhmimaidl.t	. . . . . . 7 ⊢ 𝑇 = (LIdeal‘𝑅)
16	15, 11	lidl0cl 21414	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ 𝐼)
17	10, 14, 16	syl2anc 596	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ 𝐼)
18	8, 13, 17	fnfvimad 7237	. . . 4 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹‘(0_g‘𝑅)) ∈ (𝐹 “ 𝐼))
19	18	ne0d 4291	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ≠ ∅)
20		rhmghm 20631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
21	20	ad4antr 745	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
22	9	ad4antr 745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
23		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑧 ∈ (Base‘𝑅))
24	1, 15	lidlss 21405	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐼 ∈ 𝑇 → 𝐼 ⊆ (Base‘𝑅))
25	24	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐼 ⊆ (Base‘𝑅))
26		simplr 781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ 𝐼)
27	25, 26	sseldd 3935	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ (Base‘𝑅))
28		eqid 2762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._r‘𝑅) = (._r‘𝑅)
29	1, 28	ringcl 20395	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅))
30	22, 23, 27, 29	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅))
31		simpllr 788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ 𝐼)
32	25, 31	sseldd 3935	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ (Base‘𝑅))
33		eqid 2762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
34		eqid 2762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
35	1, 33, 34	ghmlin 19354	. . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
38		eqid 2762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._r‘𝑆) = (._r‘𝑆)
39	1, 28, 38	rhmmul 20637	. . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . 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 763	. . . . . . . . . . . . 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 732	. . . . . . . . . . . 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 764	. . . . . . . . . . 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 490	. . . . . . . . . . . . 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 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) → (𝐹‘𝑖) = 𝑎)
51	49, 50	oveq12d 7435	. . . . . . . . . . . 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 798	. . . . . . . . . . . 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 7435	. . . . . . . . . . 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 2797	. . . . . . . . . 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 755	. . . . . . . . . . 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 755	. . . . . . . . . . . 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 755	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . 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 796	. . . . . . . . . . . . 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 800	. . . . . . . . . . . . 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 21409	. . . . . . . . . . . . . . . . 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 1165	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ 𝑇 → ∀𝑧 ∈ (Base‘𝑅)∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
64	63	r19.21bi 3256	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
65	64	r19.21bi 3256	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
66	65	r19.21bi 3256	. . . . . . . . . . . . 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 854	. . . . . . . . . . . 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 3935	. . . . . . . . . . 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 7237	. . . . . . . . . 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 2863	. . . . . . . . 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 739	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐹:(Base‘𝑅)⟶𝐵)
72	71	ffund 6711	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → Fun 𝐹)
73	72	ad7antr 751	. . . . . . . . . 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 6717	. . . . . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ (Base‘𝑅)) = ran 𝐹)
78	77	eqeq1d 2764	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 “ (Base‘𝑅)) = 𝐵 ↔ ran 𝐹 = 𝐵))
79	78	biimpar 483	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝐹 “ (Base‘𝑅)) = 𝐵)
80	79	eleq2d 2848	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝑥 ∈ (𝐹 “ (Base‘𝑅)) ↔ 𝑥 ∈ 𝐵))
81	80	biimpar 483	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
82	81	adantlr 728	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
83	82	ad6antr 749	. . . . . . . . . 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 6947	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ (Base‘𝑅))) → ∃𝑧 ∈ (Base‘𝑅)(𝐹‘𝑧) = 𝑥)
85	73, 83, 84	syl2anc 596	. . . . . . . . 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 3172	. . . . . . . 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 747	. . . . . . . . 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 796	. . . . . . . . 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 6947	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑎 ∈ (𝐹 “ 𝐼)) → ∃𝑖 ∈ 𝐼 (𝐹‘𝑖) = 𝑎)
90	87, 88, 89	syl2anc 596	. . . . . . . 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 3172	. . . . . . 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 743	. . . . . . . 8 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → Fun 𝐹)
93		simpr 490	. . . . . . . 8 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → 𝑏 ∈ (𝐹 “ 𝐼))
94		fvelima 6947	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → ∃𝑗 ∈ 𝐼 (𝐹‘𝑗) = 𝑏)
95	92, 93, 94	syl2anc 596	. . . . . . 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 3172	. . . . . 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 472	. . . . 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 3207	. . . 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 3156	. . 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 21409	. . 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 1362	. 2 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)
103	102	3impa 1127	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 ⊆ wss 3902 ∅c0 4282 dom cdm 5659 ran crn 5660 “ cima 5662 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 Basecbs 17307 +_gcplusg 17348 ._rcmulr 17349 0_gc0g 17530 GrpHom cghm 19346 Ringcrg 20378 RingHom crh 20616 LIdealclidl 21399

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 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 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-ip 17366 df-0g 17532 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-mhm 18897 df-grp 19066 df-minusg 19067 df-sbg 19068 df-subg 19252 df-ghm 19347 df-mgp 20280 df-ur 20327 df-ring 20380 df-rhm 20619 df-subrg 20738 df-lmod 21052 df-lss 21122 df-sra 21363 df-rgmod 21364 df-lidl 21401

This theorem is used by: rhmpreimacnlem 34402 rhmpreimacn 34403

W3C validator