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Theorem rhmpreimaprmidl 33479
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 20476 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
21ad2antlr 727 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑅 ∈ Ring)
3 rhmrcl2 20477 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4 prmidlidl 33472 . . . . . 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 2737 . . . . . 6 (LIdeal‘𝑅) = (LIdeal‘𝑅)
76rhmpreimaidl 21287 . . . . 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 714 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ (LIdeal‘𝑅))
103adantr 480 . . . . . . 7 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑆 ∈ Ring)
11 eqid 2737 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
12 eqid 2737 . . . . . . . . 9 (.r𝑆) = (.r𝑆)
1311, 12prmidlnr 33467 . . . . . . . 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 2737 . . . . . . . 8 (1r𝑆) = (1r𝑆)
1611, 15pridln1 33471 . . . . . . 7 ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1r𝑆) ∈ 𝐽)
1710, 5, 14, 16syl3anc 1373 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1r𝑆) ∈ 𝐽)
18 eqid 2737 . . . . . . . . 9 (1r𝑅) = (1r𝑅)
1918, 15rhm1 20489 . . . . . . . 8 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r𝑅)) = (1r𝑆))
2019ad2antrr 726 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (𝐹‘(1r𝑅)) = (1r𝑆))
21 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
2221, 11rhmf 20485 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
2322ffnd 6737 . . . . . . . . . 10 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
2423ad2antrr 726 . . . . . . . . 9 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → 𝐹 Fn (Base‘𝑅))
2521, 18ringidcl 20262 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
261, 25syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r𝑅) ∈ (Base‘𝑅))
2726ad2antrr 726 . . . . . . . . . 10 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝐹𝐽) = (Base‘𝑅)) → (1r𝑅) ∈ (Base‘𝑅))
28 simpr 484 . . . . . . . . . 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 7078 . . . . . . . . . 10 (𝐹 Fn (Base‘𝑅) → ((1r𝑅) ∈ (𝐹𝐽) ↔ ((1r𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r𝑅)) ∈ 𝐽)))
3130biimpa 476 . . . . . . . . 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 495 . . . . . . 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 816 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (𝐹𝐽) = (Base‘𝑅))
3635neqned 2947 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ≠ (Base‘𝑅))
3736adantll 714 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ≠ (Base‘𝑅))
38 simp-5l 785 . . . . . . . 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 784 . . . . . . . 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 786 . . . . . . . . . 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 776 . . . . . . . . 9 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 𝑎 ∈ (Base‘𝑅))
4341, 42ffvelcdmd 7105 . . . . . . . 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 769 . . . . . . . . 9 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → 𝑏 ∈ (Base‘𝑅))
4541, 44ffvelcdmd 7105 . . . . . . . 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 2737 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
4721, 46, 12rhmmul 20486 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(.r𝑅)𝑏)) = ((𝐹𝑎)(.r𝑆)(𝐹𝑏)))
4840, 42, 44, 47syl3anc 1373 . . . . . . . . 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 735 . . . . . . . . . 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 484 . . . . . . . . . 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 7078 . . . . . . . . . . 11 (𝐹 Fn (Base‘𝑅) → ((𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽) ↔ ((𝑎(.r𝑅)𝑏) ∈ (Base‘𝑅) ∧ (𝐹‘(𝑎(.r𝑅)𝑏)) ∈ 𝐽)))
5251simplbda 499 . . . . . . . . . 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 33476 . . . . . . . 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 1379 . . . . . . 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 480 . . . . . . . . . 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 480 . . . . . . . . . 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 484 . . . . . . . . . 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 7079 . . . . . . . . 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 412 . . . . . . . 8 ((((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r𝑅)𝑏) ∈ (𝐹𝐽)) → ((𝐹𝑎) ∈ 𝐽𝑎 ∈ (𝐹𝐽)))
6249adantr 480 . . . . . . . . . 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 776 . . . . . . . . . 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 484 . . . . . . . . . 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 7079 . . . . . . . . 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 412 . . . . . . . 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 967 . . . . . . 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 412 . . . . 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 466 . . . 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 3202 . . 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 33469 . . 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 839 . 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 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wral 3061  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  1rcur 20178  Ringcrg 20230  CRingccrg 20231   RingHom crh 20469  LIdealclidl 21216  PrmIdealcprmidl 33463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-rhm 20472  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-prmidl 33464
This theorem is referenced by:  ply1annprmidl  33750  rhmpreimacnlem  33883  rhmpreimacn  33884
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