Theorem rhmpreimaprmidl 33534

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 20447	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
2	1	ad2antlr 733	. . 3 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑅 ∈ Ring)
3		rhmrcl2 20448	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4		prmidlidl 33527	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
5	3, 4	sylan 586	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
6		eqid 2739	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	6	rhmpreimaidl 21270	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅))
8	5, 7	syldan 597	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅))
9	8	adantll 720	. . 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 2739	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
12		eqid 2739	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
13	11, 12	prmidlnr 33522	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
14	3, 13	sylan 586	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
15		eqid 2739	. . . . . . . 8 ⊢ (1r‘𝑆) = (1r‘𝑆)
16	11, 15	pridln1 33526	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽)
17	10, 5, 14, 16	syl3anc 1379	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽)
18		eqid 2739	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
19	18, 15	rhm1 20460	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
20	19	ad2antrr 732	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
21		eqid 2739	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
22	21, 11	rhmf 20455	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
23	22	ffnd 6656	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
24	23	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → 𝐹 Fn (Base‘𝑅))
25	21, 18	ringidcl 20237	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r‘𝑅) ∈ (Base‘𝑅))
27	26	ad2antrr 732	. . . . . . . . . 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 6999	. . . . . . . . . 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 590	. . . . . . . 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 821	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (◡𝐹 “ 𝐽) = (Base‘𝑅))
36	35	neqned 2941	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ (Base‘𝑅))
37	36	adantll 720	. . 3 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ (Base‘𝑅))
38		simp-5l 790	. . . . . . . 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 789	. . . . . . . 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 791	. . . . . . . . . 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 781	. . . . . . . . 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 7026	. . . . . . . 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 774	. . . . . . . . 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 7026	. . . . . . . 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 2739	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
47	21, 46, 12	rhmmul 20457	. . . . . . . . . 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 1379	. . . . . . . . 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 741	. . . . . . . . . 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 6999	. . . . . . . . . . 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 590	. . . . . . . . 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 33531	. . . . . . . 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 1385	. . . . . . 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 7000	. . . . . . . . 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 781	. . . . . . . . . 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 7000	. . . . . . . . 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 972	. . . . . . 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 3182	. . 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 33524	. . 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 844	. 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 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ^◡ccnv 5617   “ cima 5621   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  ._rcmulr 17212  1_rcur 20153  Ringcrg 20205  CRingccrg 20206   RingHom crh 20440  LIdealclidl 21199  PrmIdealcprmidl 33518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  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 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-grp 18903  df-minusg 18904  df-sbg 18905  df-subg 19090  df-ghm 19179  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-rhm 20443  df-subrg 20542  df-lmod 20852  df-lss 20922  df-lsp 20962  df-sra 21163  df-rgmod 21164  df-lidl 21201  df-rsp 21202  df-prmidl 33519

This theorem is referenced by:  ply1annprmidl  33891  rhmpreimacnlem  34068  rhmpreimacn  34069