Theorem rhmpreimaprmidl 31133

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 19527	. . . 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 19528	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4		prmidlidl 31125	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
5	3, 4	sylan 584	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
6		eqid 2759	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	6	rhmpreimaidl 31109	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅))
8	5, 7	syldan 595	. . . 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 485	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑆 ∈ Ring)
11		eqid 2759	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
12		eqid 2759	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
13	11, 12	prmidlnr 31120	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
14	3, 13	sylan 584	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
15		eqid 2759	. . . . . . . 8 ⊢ (1r‘𝑆) = (1r‘𝑆)
16	11, 15	pridln1 31124	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽)
17	10, 5, 14, 16	syl3anc 1369	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽)
18		eqid 2759	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
19	18, 15	rhm1 19538	. . . . . . . 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 2759	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
22	21, 11	rhmf 19534	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
23	22	ffnd 6492	. . . . . . . . . 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 19374	. . . . . . . . . . . 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 489	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (◡𝐹 “ 𝐽) = (Base‘𝑅))
29	27, 28	eleqtrrd 2854	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑅) ∈ (◡𝐹 “ 𝐽))
30		elpreima 6812	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑅) → ((1r‘𝑅) ∈ (◡𝐹 “ 𝐽) ↔ ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽)))
31	30	biimpa 481	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ (1r‘𝑅) ∈ (◡𝐹 “ 𝐽)) → ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽))
32	24, 29, 31	syl2anc 588	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽))
33	32	simprd 500	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1r‘𝑅)) ∈ 𝐽)
34	20, 33	eqeltrrd 2852	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑆) ∈ 𝐽)
35	17, 34	mtand 816	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (◡𝐹 “ 𝐽) = (Base‘𝑅))
36	35	neqned 2956	. . . 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 785	. . . . . . . 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 784	. . . . . . . 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 786	. . . . . . . . . 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 776	. . . . . . . . 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 6836	. . . . . . . 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 769	. . . . . . . . 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 6836	. . . . . . . 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 2759	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
47	21, 46, 12	rhmmul 19535	. . . . . . . . . 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 1369	. . . . . . . . 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 489	. . . . . . . . . 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 6812	. . . . . . . . . . 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 504	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑅) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) ∈ 𝐽)
53	49, 50, 52	syl2anc 588	. . . . . . . . 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 2852	. . . . . . . 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 31130	. . . . . . . 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 1375	. . . . . . 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 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 Fn (Base‘𝑅))
58	42	adantr 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) → 𝑎 ∈ (Base‘𝑅))
59		simpr 489	. . . . . . . . . 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 6813	. . . . . . . . 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 417	. . . . . . . 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 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 Fn (Base‘𝑅))
63		simpllr 776	. . . . . . . . . 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 489	. . . . . . . . . 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 6813	. . . . . . . . 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 417	. . . . . . . 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 963	. . . . . . 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 417	. . . . 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 471	. . . 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 3118	. . 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 31122	. . 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 2862	1 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112   ≠ wne 2949  ∀wral 3068  ^◡ccnv 5516   “ cima 5520   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7143  Basecbs 16526  ._rcmulr 16609  1_rcur 19304  Ringcrg 19350  CRingccrg 19351   RingHom crh 19520  LIdealclidl 19995  PrmIdealcprmidl 31116

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-map 8411  df-en 8521  df-dom 8522  df-sdom 8523  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-nn 11660  df-2 11722  df-3 11723  df-4 11724  df-5 11725  df-6 11726  df-7 11727  df-8 11728  df-ndx 16529  df-slot 16530  df-base 16532  df-sets 16533  df-ress 16534  df-plusg 16621  df-mulr 16622  df-sca 16624  df-vsca 16625  df-ip 16626  df-0g 16758  df-mgm 17903  df-sgrp 17952  df-mnd 17963  df-mhm 18007  df-grp 18157  df-minusg 18158  df-sbg 18159  df-subg 18328  df-ghm 18408  df-cmn 18960  df-mgp 19293  df-ur 19305  df-ring 19352  df-cring 19353  df-rnghom 19523  df-subrg 19586  df-lmod 19689  df-lss 19757  df-lsp 19797  df-sra 19997  df-rgmod 19998  df-lidl 19999  df-rsp 20000  df-prmidl 31117

This theorem is referenced by:  rhmpreimacnlem  31340  rhmpreimacn  31341