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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmpreimaprmidl Structured version   Visualization version   GIF version

Theorem rhmpreimaprmidl 31621
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 19959 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
21ad2antlr 724 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑅 ∈ Ring)
3 rhmrcl2 19960 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4 prmidlidl 31613 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
53, 4sylan 580 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆))
6 eqid 2740 . . . . . 6 (LIdeal‘𝑅) = (LIdeal‘𝑅)
76rhmpreimaidl 31597 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹𝐽) ∈ (LIdeal‘𝑅))
85, 7syldan 591 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ (LIdeal‘𝑅))
98adantll 711 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ (LIdeal‘𝑅))
103adantr 481 . . . . . . 7 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑆 ∈ Ring)
11 eqid 2740 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
12 eqid 2740 . . . . . . . . 9 (.r𝑆) = (.r𝑆)
1311, 12prmidlnr 31608 . . . . . . . 8 ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
143, 13sylan 580 . . . . . . 7 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆))
15 eqid 2740 . . . . . . . 8 (1r𝑆) = (1r𝑆)
1611, 15pridln1 31612 . . . . . . 7 ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1r𝑆) ∈ 𝐽)
1710, 5, 14, 16syl3anc 1370 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1r𝑆) ∈ 𝐽)
18 eqid 2740 . . . . . . . . 9 (1r𝑅) = (1r𝑅)
1918, 15rhm1 19970 . . . . . . . 8 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r𝑅)) = (1r𝑆))
2019ad2antrr 723 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (𝐹‘(1r𝑅)) = (1r𝑆))
21 eqid 2740 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
2221, 11rhmf 19966 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
2322ffnd 6598 . . . . . . . . . 10 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
2423ad2antrr 723 . . . . . . . . 9 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → 𝐹 Fn (Base‘𝑅))
2521, 18ringidcl 19803 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
261, 25syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r𝑅) ∈ (Base‘𝑅))
2726ad2antrr 723 . . . . . . . . . 10 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (1r𝑅) ∈ (Base‘𝑅))
28 simpr 485 . . . . . . . . . 10 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (𝐹𝐽) = (Base‘𝑅))
2927, 28eleqtrrd 2844 . . . . . . . . 9 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (1r𝑅) ∈ (𝐹𝐽))
30 elpreima 6930 . . . . . . . . . 10 (𝐹 Fn (Base‘𝑅) → ((1r𝑅) ∈ (𝐹𝐽) ↔ ((1r𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r𝑅)) ∈ 𝐽)))
3130biimpa 477 . . . . . . . . 9 ((𝐹 Fn (Base‘𝑅) ∧ (1r𝑅) ∈ (𝐹𝐽)) → ((1r𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r𝑅)) ∈ 𝐽))
3224, 29, 31syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → ((1r𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r𝑅)) ∈ 𝐽))
3332simprd 496 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (𝐹‘(1r𝑅)) ∈ 𝐽)
3420, 33eqeltrrd 2842 . . . . . 6 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (1r𝑆) ∈ 𝐽)
3517, 34mtand 813 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (𝐹𝐽) = (Base‘𝑅))
3635neqned 2952 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ≠ (Base‘𝑅))
3736adantll 711 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ≠ (Base‘𝑅))
38 simp-5l 782 . . . . . . . 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 781 . . . . . . . 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 783 . . . . . . . . . 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 773 . . . . . . . . 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 6957 . . . . . . . 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 766 . . . . . . . . 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 6957 . . . . . . . 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 2740 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
4721, 46, 12rhmmul 19967 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(.r𝑅)𝑏)) = ((𝐹𝑎)(.r𝑆)(𝐹𝑏)))
4840, 42, 44, 47syl3anc 1370 . . . . . . . . 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 732 . . . . . . . . . 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 485 . . . . . . . . . 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 6930 . . . . . . . . . . 11 (𝐹 Fn (Base‘𝑅) → ((𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽) ↔ ((𝑎(.r𝑅)𝑏) ∈ (Base‘𝑅) ∧ (𝐹‘(𝑎(.r𝑅)𝑏)) ∈ 𝐽)))
5251simplbda 500 . . . . . . . . . 10 ((𝐹 Fn (Base‘𝑅) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → (𝐹‘(𝑎(.r𝑅)𝑏)) ∈ 𝐽)
5349, 50, 52syl2anc 584 . . . . . . . . 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 2842 . . . . . . . 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 31618 . . . . . . . 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 1376 . . . . . . 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 481 . . . . . . . . . 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 481 . . . . . . . . . 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 485 . . . . . . . . . 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 6931 . . . . . . . . 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 413 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → ((𝐹𝑎) ∈ 𝐽𝑎 ∈ (𝐹𝐽)))
6249adantr 481 . . . . . . . . . 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 773 . . . . . . . . . 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 485 . . . . . . . . . 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 6931 . . . . . . . . 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 413 . . . . . . . 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 413 . . . . 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 467 . . . 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 3117 . . 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 31610 . . 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 836 . 2 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ (PrmIdeal‘𝑅))
74 rhmpreimaprmidl.p . 2 𝑃 = (PrmIdeal‘𝑅)
7573, 74eleqtrrdi 2852 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 396  wo 844   = wceq 1542  wcel 2110  wne 2945  wral 3066  ccnv 5588  cima 5592   Fn wfn 6426  wf 6427  cfv 6431  (class class class)co 7269  Basecbs 16908  .rcmulr 16959  1rcur 19733  Ringcrg 19779  CRingccrg 19780   RingHom crh 19952  LIdealclidl 20428  PrmIdealcprmidl 31604
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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-cnex 10926  ax-resscn 10927  ax-1cn 10928  ax-icn 10929  ax-addcl 10930  ax-addrcl 10931  ax-mulcl 10932  ax-mulrcl 10933  ax-mulcom 10934  ax-addass 10935  ax-mulass 10936  ax-distr 10937  ax-i2m1 10938  ax-1ne0 10939  ax-1rid 10940  ax-rnegex 10941  ax-rrecex 10942  ax-cnre 10943  ax-pre-lttri 10944  ax-pre-lttrn 10945  ax-pre-ltadd 10946  ax-pre-mulgt0 10947
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-1st 7822  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-er 8479  df-map 8598  df-en 8715  df-dom 8716  df-sdom 8717  df-pnf 11010  df-mnf 11011  df-xr 11012  df-ltxr 11013  df-le 11014  df-sub 11205  df-neg 11206  df-nn 11972  df-2 12034  df-3 12035  df-4 12036  df-5 12037  df-6 12038  df-7 12039  df-8 12040  df-sets 16861  df-slot 16879  df-ndx 16891  df-base 16909  df-ress 16938  df-plusg 16971  df-mulr 16972  df-sca 16974  df-vsca 16975  df-ip 16976  df-0g 17148  df-mgm 18322  df-sgrp 18371  df-mnd 18382  df-mhm 18426  df-grp 18576  df-minusg 18577  df-sbg 18578  df-subg 18748  df-ghm 18828  df-cmn 19384  df-mgp 19717  df-ur 19734  df-ring 19781  df-cring 19782  df-rnghom 19955  df-subrg 20018  df-lmod 20121  df-lss 20190  df-lsp 20230  df-sra 20430  df-rgmod 20431  df-lidl 20432  df-rsp 20433  df-prmidl 31605
This theorem is referenced by:  rhmpreimacnlem  31828  rhmpreimacn  31829
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