Theorem rhmpreimaprmidl 33481

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 20410	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
2	1	ad2antlr 727	. . 3 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑅 ∈ Ring)
3		rhmrcl2 20411	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4		prmidlidl 33474	. . . . . 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 2734	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	6	rhmpreimaidl 21230	. . . . 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 714	. . 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 2734	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
12		eqid 2734	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
13	11, 12	prmidlnr 33469	. . . . . . . 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 2734	. . . . . . . 8 ⊢ (1r‘𝑆) = (1r‘𝑆)
16	11, 15	pridln1 33473	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽)
17	10, 5, 14, 16	syl3anc 1373	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽)
18		eqid 2734	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
19	18, 15	rhm1 20422	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
20	19	ad2antrr 726	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
21		eqid 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
22	21, 11	rhmf 20418	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
23	22	ffnd 6661	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
24	23	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → 𝐹 Fn (Base‘𝑅))
25	21, 18	ringidcl 20198	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r‘𝑅) ∈ (Base‘𝑅))
27	26	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘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 2837	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑅) ∈ (◡𝐹 “ 𝐽))
30		elpreima 7001	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → ((1r‘𝑅) ∈ (◡𝐹 “ 𝐽) ↔ ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽)))
31	30	biimpa 476	. . . . . . . . 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 495	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1r‘𝑅)) ∈ 𝐽)
34	20, 33	eqeltrrd 2835	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑆) ∈ 𝐽)
35	17, 34	mtand 815	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (◡𝐹 “ 𝐽) = (Base‘𝑅))
36	35	neqned 2937	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ (Base‘𝑅))
37	36	adantll 714	. . 3 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ (Base‘𝑅))
38		simp-5l 784	. . . . . . . 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 783	. . . . . . . 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 785	. . . . . . . . . 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 775	. . . . . . . . 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 7028	. . . . . . . 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 768	. . . . . . . . 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 7028	. . . . . . . 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 2734	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
47	21, 46, 12	rhmmul 20419	. . . . . . . . . 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 1373	. . . . . . . . 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 735	. . . . . . . . . 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 7001	. . . . . . . . . . 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 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 2835	. . . . . . . 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 33478	. . . . . . . 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 1379	. . . . . . 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 7002	. . . . . . . . 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 775	. . . . . . . . . 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 7002	. . . . . . . . 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 966	. . . . . . 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 3177	. . 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 33471	. . 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 838	. 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 2845	1 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ^◡ccnv 5621   “ cima 5625   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  ._rcmulr 17176  1_rcur 20114  Ringcrg 20166  CRingccrg 20167   RingHom crh 20403  LIdealclidl 21159  PrmIdealcprmidl 33465

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-sca 17191  df-vsca 17192  df-ip 17193  df-0g 17359  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19051  df-ghm 19140  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-cring 20169  df-rhm 20406  df-subrg 20501  df-lmod 20811  df-lss 20881  df-lsp 20921  df-sra 21123  df-rgmod 21124  df-lidl 21161  df-rsp 21162  df-prmidl 33466

This theorem is referenced by:  ply1annprmidl  33813  rhmpreimacnlem  33990  rhmpreimacn  33991