Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmpreimaprmidl Structured version   Visualization version   GIF version

Theorem rhmpreimaprmidl 31035
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)) → (𝐹𝐽) ∈ 𝑃)

Proof of Theorem rhmpreimaprmidl
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rhmrcl1 19470 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
21ad2antlr 726 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑅 ∈ Ring)
3 rhmrcl2 19471 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4 prmidlidl 31027 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
53, 4sylan 583 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
6 eqid 2801 . . . . . 6 (LIdeal‘𝑅) = (LIdeal‘𝑅)
76rhmpreimaidl 31014 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹𝐽) ∈ (LIdeal‘𝑅))
85, 7syldan 594 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ (LIdeal‘𝑅))
98adantll 713 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ (LIdeal‘𝑅))
103adantr 484 . . . . . . 7 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑆 ∈ Ring)
11 eqid 2801 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
12 eqid 2801 . . . . . . . . 9 (.r𝑆) = (.r𝑆)
1311, 12prmidlnr 31022 . . . . . . . 8 ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
143, 13sylan 583 . . . . . . 7 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
15 eqid 2801 . . . . . . . 8 (1r𝑆) = (1r𝑆)
1611, 15pridln1 31026 . . . . . . 7 ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1r𝑆) ∈ 𝐽)
1710, 5, 14, 16syl3anc 1368 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1r𝑆) ∈ 𝐽)
18 eqid 2801 . . . . . . . . 9 (1r𝑅) = (1r𝑅)
1918, 15rhm1 19481 . . . . . . . 8 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r𝑅)) = (1r𝑆))
2019ad2antrr 725 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (𝐹‘(1r𝑅)) = (1r𝑆))
21 eqid 2801 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
2221, 11rhmf 19477 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
2322ffnd 6492 . . . . . . . . . 10 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
2423ad2antrr 725 . . . . . . . . 9 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → 𝐹 Fn (Base‘𝑅))
2521, 18ringidcl 19317 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
261, 25syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r𝑅) ∈ (Base‘𝑅))
2726ad2antrr 725 . . . . . . . . . 10 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (1r𝑅) ∈ (Base‘𝑅))
28 simpr 488 . . . . . . . . . 10 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (𝐹𝐽) = (Base‘𝑅))
2927, 28eleqtrrd 2896 . . . . . . . . 9 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (1r𝑅) ∈ (𝐹𝐽))
30 elpreima 6809 . . . . . . . . . 10 (𝐹 Fn (Base‘𝑅) → ((1r𝑅) ∈ (𝐹𝐽) ↔ ((1r𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r𝑅)) ∈ 𝐽)))
3130biimpa 480 . . . . . . . . 9 ((𝐹 Fn (Base‘𝑅) ∧ (1r𝑅) ∈ (𝐹𝐽)) → ((1r𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r𝑅)) ∈ 𝐽))
3224, 29, 31syl2anc 587 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → ((1r𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r𝑅)) ∈ 𝐽))
3332simprd 499 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (𝐹‘(1r𝑅)) ∈ 𝐽)
3420, 33eqeltrrd 2894 . . . . . 6 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (1r𝑆) ∈ 𝐽)
3517, 34mtand 815 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (𝐹𝐽) = (Base‘𝑅))
3635neqned 2997 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ≠ (Base‘𝑅))
3736adantll 713 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ≠ (Base‘𝑅))
38 simp-5l 784 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 𝑆 ∈ CRing)
39 simp-4r 783 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 𝐽 ∈ (PrmIdeal‘𝑆))
40 simp-5r 785 . . . . . . . . . 10 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
4140, 22syl 17 . . . . . . . . 9 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 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(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 𝑎 ∈ (Base‘𝑅))
4341, 42ffvelrnd 6833 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (𝐹𝑎) ∈ (Base‘𝑆))
44 simplr 768 . . . . . . . . 9 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 𝑏 ∈ (Base‘𝑅))
4541, 44ffvelrnd 6833 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (𝐹𝑏) ∈ (Base‘𝑆))
46 eqid 2801 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
4721, 46, 12rhmmul 19478 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(.r𝑅)𝑏)) = ((𝐹𝑎)(.r𝑆)(𝐹𝑏)))
4840, 42, 44, 47syl3anc 1368 . . . . . . . . 9 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (𝐹‘(𝑎(.r𝑅)𝑏)) = ((𝐹𝑎)(.r𝑆)(𝐹𝑏)))
4923ad5antlr 734 . . . . . . . . . 10 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 𝐹 Fn (Base‘𝑅))
50 simpr 488 . . . . . . . . . 10 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽))
51 elpreima 6809 . . . . . . . . . . 11 (𝐹 Fn (Base‘𝑅) → ((𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽) ↔ ((𝑎(.r𝑅)𝑏) ∈ (Base‘𝑅) ∧ (𝐹‘(𝑎(.r𝑅)𝑏)) ∈ 𝐽)))
5251simplbda 503 . . . . . . . . . 10 ((𝐹 Fn (Base‘𝑅) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (𝐹‘(𝑎(.r𝑅)𝑏)) ∈ 𝐽)
5349, 50, 52syl2anc 587 . . . . . . . . 9 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (𝐹‘(𝑎(.r𝑅)𝑏)) ∈ 𝐽)
5448, 53eqeltrrd 2894 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → ((𝐹𝑎)(.r𝑆)(𝐹𝑏)) ∈ 𝐽)
5511, 12prmidlc 31032 . . . . . . . 8 (((𝑆 ∈ CRing ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ ((𝐹𝑎) ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ (Base‘𝑆) ∧ ((𝐹𝑎)(.r𝑆)(𝐹𝑏)) ∈ 𝐽)) → ((𝐹𝑎) ∈ 𝐽 ∨ (𝐹𝑏) ∈ 𝐽))
5638, 39, 43, 45, 54, 55syl23anc 1374 . . . . . . 7 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → ((𝐹𝑎) ∈ 𝐽 ∨ (𝐹𝑏) ∈ 𝐽))
5749adantr 484 . . . . . . . . . 10 (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) ∧ (𝐹𝑎) ∈ 𝐽) → 𝐹 Fn (Base‘𝑅))
5842adantr 484 . . . . . . . . . 10 (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) ∧ (𝐹𝑎) ∈ 𝐽) → 𝑎 ∈ (Base‘𝑅))
59 simpr 488 . . . . . . . . . 10 (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) ∧ (𝐹𝑎) ∈ 𝐽) → (𝐹𝑎) ∈ 𝐽)
6057, 58, 59elpreimad 6810 . . . . . . . . 9 (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) ∧ (𝐹𝑎) ∈ 𝐽) → 𝑎 ∈ (𝐹𝐽))
6160ex 416 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → ((𝐹𝑎) ∈ 𝐽𝑎 ∈ (𝐹𝐽)))
6249adantr 484 . . . . . . . . . 10 (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) ∧ (𝐹𝑏) ∈ 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(.r𝑅)𝑏) ∈ (𝐹𝐽)) ∧ (𝐹𝑏) ∈ 𝐽) → 𝑏 ∈ (Base‘𝑅))
64 simpr 488 . . . . . . . . . 10 (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) ∧ (𝐹𝑏) ∈ 𝐽) → (𝐹𝑏) ∈ 𝐽)
6562, 63, 64elpreimad 6810 . . . . . . . . 9 (((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) ∧ (𝐹𝑏) ∈ 𝐽) → 𝑏 ∈ (𝐹𝐽))
6665ex 416 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → ((𝐹𝑏) ∈ 𝐽𝑏 ∈ (𝐹𝐽)))
6761, 66orim12d 962 . . . . . . 7 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (((𝐹𝑎) ∈ 𝐽 ∨ (𝐹𝑏) ∈ 𝐽) → (𝑎 ∈ (𝐹𝐽) ∨ 𝑏 ∈ (𝐹𝐽))))
6856, 67mpd 15 . . . . . 6 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (𝑎 ∈ (𝐹𝐽) ∨ 𝑏 ∈ (𝐹𝐽)))
6968ex 416 . . . . 5 (((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) → ((𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽) → (𝑎 ∈ (𝐹𝐽) ∨ 𝑏 ∈ (𝐹𝐽))))
7069anasss 470 . . . 4 ((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽) → (𝑎 ∈ (𝐹𝐽) ∨ 𝑏 ∈ (𝐹𝐽))))
7170ralrimivva 3159 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)((𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽) → (𝑎 ∈ (𝐹𝐽) ∨ 𝑏 ∈ (𝐹𝐽))))
7221, 46prmidl2 31024 . . 3 (((𝑅 ∈ Ring ∧ (𝐹𝐽) ∈ (LIdeal‘𝑅)) ∧ ((𝐹𝐽) ≠ (Base‘𝑅) ∧ ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)((𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽) → (𝑎 ∈ (𝐹𝐽) ∨ 𝑏 ∈ (𝐹𝐽))))) → (𝐹𝐽) ∈ (PrmIdeal‘𝑅))
732, 9, 37, 71, 72syl22anc 837 . 2 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ (PrmIdeal‘𝑅))
74 rhmpreimaprmidl.p . 2 𝑃 = (PrmIdeal‘𝑅)
7573, 74eleqtrrdi 2904 1 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  wral 3109  ccnv 5522  cima 5526   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  Basecbs 16478  .rcmulr 16561  1rcur 19247  Ringcrg 19293  CRingccrg 19294   RingHom crh 19463  LIdealclidl 19938  PrmIdealcprmidl 31018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  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 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-grp 18101  df-minusg 18102  df-sbg 18103  df-subg 18271  df-ghm 18351  df-cmn 18903  df-mgp 19236  df-ur 19248  df-ring 19295  df-cring 19296  df-rnghom 19466  df-subrg 19529  df-lmod 19632  df-lss 19700  df-lsp 19740  df-sra 19940  df-rgmod 19941  df-lidl 19942  df-rsp 19943  df-prmidl 31019
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator