rhmpreimaprmidl - Mathbox for Thierry Arnoux

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Theorem rhmpreimaprmidl 32845

Description: The preimage of a prime ideal by a ring homomorphism is a prime ideal. (Contributed by Thierry Arnoux, 29-Jun-2024.)

**Hypothesis**
Ref	Expression
rhmpreimaprmidl.p	⊢ 𝑃 = (PrmIdeal‘𝑅)

**Assertion**
Ref	Expression
rhmpreimaprmidl	⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ 𝑃)

Proof of Theorem rhmpreimaprmidl Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmrcl1 20368	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
2	1	ad2antlr 724	. . 3 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑅 ∈ Ring)
3		rhmrcl2 20369	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4		prmidlidl 32837	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
5	3, 4	sylan 579	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
6		eqid 2731	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	6	rhmpreimaidl 32812	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅))
8	5, 7	syldan 590	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅))
9	8	adantll 711	. . 3 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅))
10	3	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑆 ∈ Ring)
11		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
12		eqid 2731	. . . . . . . . 9 ⊢ (._r‘𝑆) = (._r‘𝑆)
13	11, 12	prmidlnr 32832	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
14	3, 13	sylan 579	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
15		eqid 2731	. . . . . . . 8 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
16	11, 15	pridln1 32836	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1_r‘𝑆) ∈ 𝐽)
17	10, 5, 14, 16	syl3anc 1370	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1_r‘𝑆) ∈ 𝐽)
18		eqid 2731	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
19	18, 15	rhm1 20381	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1_r‘𝑅)) = (1_r‘𝑆))
20	19	ad2antrr 723	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (^◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1_r‘𝑅)) = (1_r‘𝑆))
21		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
22	21, 11	rhmf 20377	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
23	22	ffnd 6718	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
24	23	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (^◡𝐹 “ 𝐽) = (Base‘𝑅)) → 𝐹 Fn (Base‘𝑅))
25	21, 18	ringidcl 20155	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1_r‘𝑅) ∈ (Base‘𝑅))
27	26	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (^◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1_r‘𝑅) ∈ (Base‘𝑅))
28		simpr 484	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (^◡𝐹 “ 𝐽) = (Base‘𝑅)) → (^◡𝐹 “ 𝐽) = (Base‘𝑅))
29	27, 28	eleqtrrd 2835	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (^◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1_r‘𝑅) ∈ (^◡𝐹 “ 𝐽))
30		elpreima 7059	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → ((1_r‘𝑅) ∈ (^◡𝐹 “ 𝐽) ↔ ((1_r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1_r‘𝑅)) ∈ 𝐽)))
31	30	biimpa 476	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ (1_r‘𝑅) ∈ (^◡𝐹 “ 𝐽)) → ((1_r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1_r‘𝑅)) ∈ 𝐽))
32	24, 29, 31	syl2anc 583	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (^◡𝐹 “ 𝐽) = (Base‘𝑅)) → ((1_r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1_r‘𝑅)) ∈ 𝐽))
33	32	simprd 495	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (^◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1_r‘𝑅)) ∈ 𝐽)
34	20, 33	eqeltrrd 2833	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (^◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1_r‘𝑆) ∈ 𝐽)
35	17, 34	mtand 813	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (^◡𝐹 “ 𝐽) = (Base‘𝑅))
36	35	neqned 2946	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ≠ (Base‘𝑅))
37	36	adantll 711	. . 3 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ≠ (Base‘𝑅))
38		simp-5l 782	. . . . . . . 8 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → 𝑆 ∈ CRing)
39		simp-4r 781	. . . . . . . 8 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → 𝐽 ∈ (PrmIdeal‘𝑆))
40		simp-5r 783	. . . . . . . . . 10 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
41	40, 22	syl 17	. . . . . . . . 9 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42		simpllr 773	. . . . . . . . 9 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅))
43	41, 42	ffvelcdmd 7087	. . . . . . . 8 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑎) ∈ (Base‘𝑆))
44		simplr 766	. . . . . . . . 9 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → 𝑏 ∈ (Base‘𝑅))
45	41, 44	ffvelcdmd 7087	. . . . . . . 8 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ (Base‘𝑆))
46		eqid 2731	. . . . . . . . . . 11 ⊢ (._r‘𝑅) = (._r‘𝑅)
47	21, 46, 12	rhmmul 20378	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(._r‘𝑅)𝑏)) = ((𝐹‘𝑎)(._r‘𝑆)(𝐹‘𝑏)))
48	40, 42, 44, 47	syl3anc 1370	. . . . . . . . 9 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘(𝑎(._r‘𝑅)𝑏)) = ((𝐹‘𝑎)(._r‘𝑆)(𝐹‘𝑏)))
49	23	ad5antlr 732	. . . . . . . . . 10 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → 𝐹 Fn (Base‘𝑅))
50		simpr 484	. . . . . . . . . 10 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽))
51		elpreima 7059	. . . . . . . . . . 11 ⊢ (𝐹 Fn (Base‘𝑅) → ((𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽) ↔ ((𝑎(._r‘𝑅)𝑏) ∈ (Base‘𝑅) ∧ (𝐹‘(𝑎(._r‘𝑅)𝑏)) ∈ 𝐽)))
52	51	simplbda 499	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘(𝑎(._r‘𝑅)𝑏)) ∈ 𝐽)
53	49, 50, 52	syl2anc 583	. . . . . . . . 9 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → (𝐹‘(𝑎(._r‘𝑅)𝑏)) ∈ 𝐽)
54	48, 53	eqeltrrd 2833	. . . . . . . 8 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → ((𝐹‘𝑎)(._r‘𝑆)(𝐹‘𝑏)) ∈ 𝐽)
55	11, 12	prmidlc 32842	. . . . . . . 8 ⊢ (((𝑆 ∈ CRing ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ ((𝐹‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑎)(._r‘𝑆)(𝐹‘𝑏)) ∈ 𝐽)) → ((𝐹‘𝑎) ∈ 𝐽 ∨ (𝐹‘𝑏) ∈ 𝐽))
56	38, 39, 43, 45, 54, 55	syl23anc 1376	. . . . . . 7 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → ((𝐹‘𝑎) ∈ 𝐽 ∨ (𝐹‘𝑏) ∈ 𝐽))
57	49	adantr 480	. . . . . . . . . 10 ⊢ (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑎) ∈ 𝐽) → 𝐹 Fn (Base‘𝑅))
58	42	adantr 480	. . . . . . . . . 10 ⊢ (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑎) ∈ 𝐽) → 𝑎 ∈ (Base‘𝑅))
59		simpr 484	. . . . . . . . . 10 ⊢ (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑎) ∈ 𝐽) → (𝐹‘𝑎) ∈ 𝐽)
60	57, 58, 59	elpreimad 7060	. . . . . . . . 9 ⊢ (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑎) ∈ 𝐽) → 𝑎 ∈ (^◡𝐹 “ 𝐽))
61	60	ex 412	. . . . . . . 8 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → ((𝐹‘𝑎) ∈ 𝐽 → 𝑎 ∈ (^◡𝐹 “ 𝐽)))
62	49	adantr 480	. . . . . . . . . 10 ⊢ (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑏) ∈ 𝐽) → 𝐹 Fn (Base‘𝑅))
63		simpllr 773	. . . . . . . . . 10 ⊢ (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑏) ∈ 𝐽) → 𝑏 ∈ (Base‘𝑅))
64		simpr 484	. . . . . . . . . 10 ⊢ (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑏) ∈ 𝐽) → (𝐹‘𝑏) ∈ 𝐽)
65	62, 63, 64	elpreimad 7060	. . . . . . . . 9 ⊢ (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑏) ∈ 𝐽) → 𝑏 ∈ (^◡𝐹 “ 𝐽))
66	65	ex 412	. . . . . . . 8 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → ((𝐹‘𝑏) ∈ 𝐽 → 𝑏 ∈ (^◡𝐹 “ 𝐽)))
67	61, 66	orim12d 962	. . . . . . 7 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → (((𝐹‘𝑎) ∈ 𝐽 ∨ (𝐹‘𝑏) ∈ 𝐽) → (𝑎 ∈ (^◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (^◡𝐹 “ 𝐽))))
68	56, 67	mpd 15	. . . . . 6 ⊢ ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽)) → (𝑎 ∈ (^◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (^◡𝐹 “ 𝐽)))
69	68	ex 412	. . . . 5 ⊢ (((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) → ((𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽) → (𝑎 ∈ (^◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (^◡𝐹 “ 𝐽))))
70	69	anasss 466	. . . 4 ⊢ ((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽) → (𝑎 ∈ (^◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (^◡𝐹 “ 𝐽))))
71	70	ralrimivva 3199	. . 3 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)((𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽) → (𝑎 ∈ (^◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (^◡𝐹 “ 𝐽))))
72	21, 46	prmidl2 32834	. . 3 ⊢ (((𝑅 ∈ Ring ∧ (^◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅)) ∧ ((^◡𝐹 “ 𝐽) ≠ (Base‘𝑅) ∧ ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)((𝑎(._r‘𝑅)𝑏) ∈ (^◡𝐹 “ 𝐽) → (𝑎 ∈ (^◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (^◡𝐹 “ 𝐽))))) → (^◡𝐹 “ 𝐽) ∈ (PrmIdeal‘𝑅))
73	2, 9, 37, 71, 72	syl22anc 836	. 2 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ (PrmIdeal‘𝑅))
74		rhmpreimaprmidl.p	. 2 ⊢ 𝑃 = (PrmIdeal‘𝑅)
75	73, 74	eleqtrrdi 2843	1 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (^◡𝐹 “ 𝐽) ∈ 𝑃)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ^◡ccnv 5675 “ cima 5679 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Basecbs 17149 ._rcmulr 17203 1_rcur 20076 Ringcrg 20128 CRingccrg 20129 RingHom crh 20361 LIdealclidl 20929 PrmIdealcprmidl 32828

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-ghm 19129 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrg 20460 df-lmod 20617 df-lss 20688 df-lsp 20728 df-sra 20931 df-rgmod 20932 df-lidl 20933 df-rsp 20934 df-prmidl 32829

This theorem is referenced by: ply1annprmidl 33058 rhmpreimacnlem 33163 rhmpreimacn 33164

W3C validator