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Theorem rhmpreimaprmidl 33459
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 20493 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
21ad2antlr 727 . . 3 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑅 ∈ Ring)
3 rhmrcl2 20494 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
4 prmidlidl 33452 . . . . . 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 2735 . . . . . 6 (LIdeal‘𝑅) = (LIdeal‘𝑅)
76rhmpreimaidl 21305 . . . . 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 2735 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
12 eqid 2735 . . . . . . . . 9 (.r𝑆) = (.r𝑆)
1311, 12prmidlnr 33447 . . . . . . . 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 2735 . . . . . . . 8 (1r𝑆) = (1r𝑆)
1611, 15pridln1 33451 . . . . . . 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 2735 . . . . . . . . 9 (1r𝑅) = (1r𝑅)
1918, 15rhm1 20506 . . . . . . . 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 2735 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
2221, 11rhmf 20502 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
2322ffnd 6738 . . . . . . . . . 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 20280 . . . . . . . . . . . 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 2842 . . . . . . . . 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 2840 . . . . . 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 2945 . . . 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 2735 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
4721, 46, 12rhmmul 20503 . . . . . . . . . 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 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 2840 . . . . . . . 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 33456 . . . . . . . 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 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 966 . . . . . . 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 3200 . . 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 33449 . . 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 2850 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 847   = wceq 1537  wcel 2106  wne 2938  wral 3059  ccnv 5688  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  1rcur 20199  Ringcrg 20251  CRingccrg 20252   RingHom crh 20486  LIdealclidl 21234  PrmIdealcprmidl 33443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-rhm 20489  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-prmidl 33444
This theorem is referenced by:  ply1annprmidl  33715  rhmpreimacnlem  33845  rhmpreimacn  33846
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