rhmimaidl - Mathbox for Thierry Arnoux

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Theorem rhmimaidl 33197

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 2725	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		rhmimaidl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
3	1, 2	rhmf 20428	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐵)
4		fimass 6738	. . . . 5 ⊢ (𝐹:(Base‘𝑅)⟶𝐵 → (𝐹 “ 𝐼) ⊆ 𝐵)
5	3, 4	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ 𝐼) ⊆ 𝐵)
6	5	ad2antrr 724	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ⊆ 𝐵)
7	3	ffnd 6718	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
8	7	ad2antrr 724	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐹 Fn (Base‘𝑅))
9		rhmrcl1 20419	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	9	ad2antrr 724	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝑅 ∈ Ring)
11		eqid 2725	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
12	1, 11	ring0cl 20207	. . . . . 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 483	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐼 ∈ 𝑇)
15		rhmimaidl.t	. . . . . . 7 ⊢ 𝑇 = (LIdeal‘𝑅)
16	15, 11	lidl0cl 21120	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ 𝐼)
17	10, 14, 16	syl2anc 582	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (0_g‘𝑅) ∈ 𝐼)
18	8, 13, 17	fnfvimad 7242	. . . 4 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹‘(0_g‘𝑅)) ∈ (𝐹 “ 𝐼))
19	18	ne0d 4331	. . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ≠ ∅)
20		rhmghm 20427	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
21	20	ad4antr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
22	9	ad4antr 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
23		simpr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑧 ∈ (Base‘𝑅))
24	1, 15	lidlss 21112	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐼 ∈ 𝑇 → 𝐼 ⊆ (Base‘𝑅))
25	24	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐼 ⊆ (Base‘𝑅))
26		simplr 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ 𝐼)
27	25, 26	sseldd 3973	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ (Base‘𝑅))
28		eqid 2725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._r‘𝑅) = (._r‘𝑅)
29	1, 28	ringcl 20194	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅))
30	22, 23, 27, 29	syl3anc 1368	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑧(._r‘𝑅)𝑖) ∈ (Base‘𝑅))
31		simpllr 774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ 𝐼)
32	25, 31	sseldd 3973	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ (Base‘𝑅))
33		eqid 2725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
34		eqid 2725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
35	1, 33, 34	ghmlin 19179	. . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼 ∈ 𝑇) ∧ 𝑗 ∈ 𝐼) ∧ 𝑖 ∈ 𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
38		eqid 2725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._r‘𝑆) = (._r‘𝑆)
39	1, 28, 38	rhmmul 20429	. . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . 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 753	. . . . . . . . . . . . . . . 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 748	. . . . . . . . . . . . . . 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 748	. . . . . . . . . . . . . 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 748	. . . . . . . . . . . . 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 717	. . . . . . . . . . . 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 749	. . . . . . . . . . 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 483	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 7434	. . . . . . . . . . . 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 784	. . . . . . . . . . . 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 7434	. . . . . . . . . . 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 2765	. . . . . . . . . 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 740	. . . . . . . . . . 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 740	. . . . . . . . . . . 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 740	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . 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 786	. . . . . . . . . . . . 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 21115	. . . . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ 𝑇 → ∀𝑧 ∈ (Base‘𝑅)∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
64	63	r19.21bi 3239	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
65	64	r19.21bi 3239	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑇 ∧ 𝑧 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ 𝐼) → ∀𝑗 ∈ 𝐼 ((𝑧(._r‘𝑅)𝑖)(+_g‘𝑅)𝑗) ∈ 𝐼)
66	65	r19.21bi 3239	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 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 3973	. . . . . . . . . . 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 7242	. . . . . . . . . 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 2826	. . . . . . . . 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 724	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → 𝐹:(Base‘𝑅)⟶𝐵)
72	71	ffund 6721	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → Fun 𝐹)
73	72	ad7antr 736	. . . . . . . . . 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 6728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
75	74	imaeq2d 6058	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ dom 𝐹) = (𝐹 “ (Base‘𝑅)))
76		imadmrn 6068	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
77	75, 76	eqtr3di 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ (Base‘𝑅)) = ran 𝐹)
78	77	eqeq1d 2727	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 “ (Base‘𝑅)) = 𝐵 ↔ ran 𝐹 = 𝐵))
79	78	biimpar 476	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝐹 “ (Base‘𝑅)) = 𝐵)
80	79	eleq2d 2811	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝑥 ∈ (𝐹 “ (Base‘𝑅)) ↔ 𝑥 ∈ 𝐵))
81	80	biimpar 476	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
82	81	adantlr 713	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
83	82	ad6antr 734	. . . . . . . . . 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 6959	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ (𝐹 “ (Base‘𝑅))) → ∃𝑧 ∈ (Base‘𝑅)(𝐹‘𝑧) = 𝑥)
85	73, 83, 84	syl2anc 582	. . . . . . . . 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 3152	. . . . . . . 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 732	. . . . . . . . 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 782	. . . . . . . . 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 6959	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑎 ∈ (𝐹 “ 𝐼)) → ∃𝑖 ∈ 𝐼 (𝐹‘𝑖) = 𝑎)
90	87, 88, 89	syl2anc 582	. . . . . . . 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 3152	. . . . . . 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 728	. . . . . . . 8 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → Fun 𝐹)
93		simpr 483	. . . . . . . 8 ⊢ ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) ∧ 𝑥 ∈ 𝐵) ∧ 𝑎 ∈ (𝐹 “ 𝐼)) ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → 𝑏 ∈ (𝐹 “ 𝐼))
94		fvelima 6959	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (𝐹 “ 𝐼)) → ∃𝑗 ∈ 𝐼 (𝐹‘𝑗) = 𝑏)
95	92, 93, 94	syl2anc 582	. . . . . . 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 3152	. . . . . 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 465	. . . . 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 3191	. . . 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 3136	. . 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 21115	. . 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 1340	. 2 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)
103	102	3impa 1107	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ⊆ wss 3939 ∅c0 4318 dom cdm 5672 ran crn 5673 “ cima 5675 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +_gcplusg 17232 ._rcmulr 17233 0_gc0g 17420 GrpHom cghm 19171 Ringcrg 20177 RingHom crh 20412 LIdealclidl 21106

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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-ghm 19172 df-mgp 20079 df-ur 20126 df-ring 20179 df-rhm 20415 df-subrg 20512 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-lidl 21108

This theorem is referenced by: rhmpreimacnlem 33542 rhmpreimacn 33543

W3C validator