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)
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