Theorem rhmpreimaprmidl 31627

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 19963	. . . 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 19964	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4		prmidlidl 31619	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
5	3, 4	sylan 580	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
6		eqid 2738	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	6	rhmpreimaidl 31603	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅))
8	5, 7	syldan 591	. . . 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 481	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑆 ∈ Ring)
11		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
12		eqid 2738	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
13	11, 12	prmidlnr 31614	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
14	3, 13	sylan 580	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
15		eqid 2738	. . . . . . . 8 ⊢ (1r‘𝑆) = (1r‘𝑆)
16	11, 15	pridln1 31618	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽)
17	10, 5, 14, 16	syl3anc 1370	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽)
18		eqid 2738	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
19	18, 15	rhm1 19974	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
20	19	ad2antrr 723	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
21		eqid 2738	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
22	21, 11	rhmf 19970	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
23	22	ffnd 6601	. . . . . . . . . 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 19807	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r‘𝑅) ∈ (Base‘𝑅))
27	26	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑅) ∈ (Base‘𝑅))
28		simpr 485	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (◡𝐹 “ 𝐽) = (Base‘𝑅))
29	27, 28	eleqtrrd 2842	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑅) ∈ (◡𝐹 “ 𝐽))
30		elpreima 6935	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → ((1r‘𝑅) ∈ (◡𝐹 “ 𝐽) ↔ ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽)))
31	30	biimpa 477	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ (1r‘𝑅) ∈ (◡𝐹 “ 𝐽)) → ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽))
32	24, 29, 31	syl2anc 584	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽))
33	32	simprd 496	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1r‘𝑅)) ∈ 𝐽)
34	20, 33	eqeltrrd 2840	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑆) ∈ 𝐽)
35	17, 34	mtand 813	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (◡𝐹 “ 𝐽) = (Base‘𝑅))
36	35	neqned 2950	. . . 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	ffvelrnd 6962	. . . . . . . 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	ffvelrnd 6962	. . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
47	21, 46, 12	rhmmul 19971	. . . . . . . . . 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 485	. . . . . . . . . 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 6935	. . . . . . . . . . 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 500	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) ∈ 𝐽)
53	49, 50, 52	syl2anc 584	. . . . . . . . 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 2840	. . . . . . . 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 31624	. . . . . . . 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 481	. . . . . . . . . 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 481	. . . . . . . . . 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 485	. . . . . . . . . 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 6936	. . . . . . . . 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 413	. . . . . . . 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 481	. . . . . . . . . 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 485	. . . . . . . . . 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 6936	. . . . . . . . 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 413	. . . . . . . 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 413	. . . . 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 467	. . . 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 3123	. . 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 31616	. . 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 2850	1 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  ._rcmulr 16963  1_rcur 19737  Ringcrg 19783  CRingccrg 19784   RingHom crh 19956  LIdealclidl 20432  PrmIdealcprmidl 31610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-ip 16980  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-ghm 18832  df-cmn 19388  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-rnghom 19959  df-subrg 20022  df-lmod 20125  df-lss 20194  df-lsp 20234  df-sra 20434  df-rgmod 20435  df-lidl 20436  df-rsp 20437  df-prmidl 31611

This theorem is referenced by:  rhmpreimacnlem  31834  rhmpreimacn  31835