| Step | Hyp | Ref
| Expression |
| 1 | | rhmrcl1 20476 |
. . . 4
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring) |
| 2 | 1 | ad2antlr 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)) |
| 5 | 3, 4 | sylan 580 |
. . . . 5
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ∈ (LIdeal‘𝑆)) |
| 6 | | eqid 2737 |
. . . . . 6
⊢
(LIdeal‘𝑅) =
(LIdeal‘𝑅) |
| 7 | 6 | rhmpreimaidl 21287 |
. . . . 5
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅)) |
| 8 | 5, 7 | syldan 591 |
. . . 4
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅)) |
| 9 | 8 | adantll 714 |
. . 3
⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅)) |
| 10 | 3 | adantr 480 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝑆 ∈ Ring) |
| 11 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝑆) =
(Base‘𝑆) |
| 12 | | eqid 2737 |
. . . . . . . . 9
⊢
(.r‘𝑆) = (.r‘𝑆) |
| 13 | 11, 12 | prmidlnr 33467 |
. . . . . . . 8
⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆)) |
| 14 | 3, 13 | sylan 580 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → 𝐽 ≠ (Base‘𝑆)) |
| 15 | | eqid 2737 |
. . . . . . . 8
⊢
(1r‘𝑆) = (1r‘𝑆) |
| 16 | 11, 15 | pridln1 33471 |
. . . . . . 7
⊢ ((𝑆 ∈ Ring ∧ 𝐽 ∈ (LIdeal‘𝑆) ∧ 𝐽 ≠ (Base‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽) |
| 17 | 10, 5, 14, 16 | syl3anc 1373 |
. . . . . 6
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (1r‘𝑆) ∈ 𝐽) |
| 18 | | eqid 2737 |
. . . . . . . . 9
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 19 | 18, 15 | rhm1 20489 |
. . . . . . . 8
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆)) |
| 20 | 19 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆)) |
| 21 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(Base‘𝑅) =
(Base‘𝑅) |
| 22 | 21, 11 | rhmf 20485 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) |
| 23 | 22 | ffnd 6737 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅)) |
| 24 | 23 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → 𝐹 Fn (Base‘𝑅)) |
| 25 | 21, 18 | ringidcl 20262 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ (Base‘𝑅)) |
| 26 | 1, 25 | syl 17 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r‘𝑅) ∈ (Base‘𝑅)) |
| 27 | 26 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑅) ∈ (Base‘𝑅)) |
| 28 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (◡𝐹 “ 𝐽) = (Base‘𝑅)) |
| 29 | 27, 28 | eleqtrrd 2844 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑅) ∈ (◡𝐹 “ 𝐽)) |
| 30 | | elpreima 7078 |
. . . . . . . . . 10
⊢ (𝐹 Fn (Base‘𝑅) →
((1r‘𝑅)
∈ (◡𝐹 “ 𝐽) ↔ ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽))) |
| 31 | 30 | biimpa 476 |
. . . . . . . . 9
⊢ ((𝐹 Fn (Base‘𝑅) ∧
(1r‘𝑅)
∈ (◡𝐹 “ 𝐽)) → ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽)) |
| 32 | 24, 29, 31 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (𝐹‘(1r‘𝑅)) ∈ 𝐽)) |
| 33 | 32 | simprd 495 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (𝐹‘(1r‘𝑅)) ∈ 𝐽) |
| 34 | 20, 33 | eqeltrrd 2842 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (◡𝐹 “ 𝐽) = (Base‘𝑅)) → (1r‘𝑆) ∈ 𝐽) |
| 35 | 17, 34 | mtand 816 |
. . . . 5
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ¬ (◡𝐹 “ 𝐽) = (Base‘𝑅)) |
| 36 | 35 | neqned 2947 |
. . . 4
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ (Base‘𝑅)) |
| 37 | 36 | adantll 714 |
. . 3
⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ≠ (Base‘𝑅)) |
| 38 | | simp-5l 785 |
. . . . . . . 8
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → 𝑆 ∈ CRing) |
| 39 | | simp-4r 784 |
. . . . . . . 8
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → 𝐽 ∈ (PrmIdeal‘𝑆)) |
| 40 | | simp-5r 786 |
. . . . . . . . . 10
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
| 41 | 40, 22 | syl 17 |
. . . . . . . . 9
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → 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(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → 𝑎 ∈ (Base‘𝑅)) |
| 43 | 41, 42 | ffvelcdmd 7105 |
. . . . . . . 8
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑎) ∈ (Base‘𝑆)) |
| 44 | | simplr 769 |
. . . . . . . . 9
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → 𝑏 ∈ (Base‘𝑅)) |
| 45 | 41, 44 | ffvelcdmd 7105 |
. . . . . . . 8
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝐹‘𝑏) ∈ (Base‘𝑆)) |
| 46 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 47 | 21, 46, 12 | rhmmul 20486 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏))) |
| 48 | 40, 42, 44, 47 | syl3anc 1373 |
. . . . . . . . 9
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) = ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏))) |
| 49 | 23 | ad5antlr 735 |
. . . . . . . . . 10
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → 𝐹 Fn (Base‘𝑅)) |
| 50 | | simpr 484 |
. . . . . . . . . 10
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) |
| 51 | | elpreima 7078 |
. . . . . . . . . . 11
⊢ (𝐹 Fn (Base‘𝑅) → ((𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽) ↔ ((𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝑅) ∧ (𝐹‘(𝑎(.r‘𝑅)𝑏)) ∈ 𝐽))) |
| 52 | 51 | simplbda 499 |
. . . . . . . . . 10
⊢ ((𝐹 Fn (Base‘𝑅) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) ∈ 𝐽) |
| 53 | 49, 50, 52 | syl2anc 584 |
. . . . . . . . 9
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝐹‘(𝑎(.r‘𝑅)𝑏)) ∈ 𝐽) |
| 54 | 48, 53 | eqeltrrd 2842 |
. . . . . . . 8
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)) ∈ 𝐽) |
| 55 | 11, 12 | prmidlc 33476 |
. . . . . . . 8
⊢ (((𝑆 ∈ CRing ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ ((𝐹‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑎)(.r‘𝑆)(𝐹‘𝑏)) ∈ 𝐽)) → ((𝐹‘𝑎) ∈ 𝐽 ∨ (𝐹‘𝑏) ∈ 𝐽)) |
| 56 | 38, 39, 43, 45, 54, 55 | syl23anc 1379 |
. . . . . . 7
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → ((𝐹‘𝑎) ∈ 𝐽 ∨ (𝐹‘𝑏) ∈ 𝐽)) |
| 57 | 49 | adantr 480 |
. . . . . . . . . 10
⊢
(((((((𝑆 ∈
CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑎) ∈ 𝐽) → 𝐹 Fn (Base‘𝑅)) |
| 58 | 42 | adantr 480 |
. . . . . . . . . 10
⊢
(((((((𝑆 ∈
CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑎) ∈ 𝐽) → 𝑎 ∈ (Base‘𝑅)) |
| 59 | | simpr 484 |
. . . . . . . . . 10
⊢
(((((((𝑆 ∈
CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑎) ∈ 𝐽) → (𝐹‘𝑎) ∈ 𝐽) |
| 60 | 57, 58, 59 | elpreimad 7079 |
. . . . . . . . 9
⊢
(((((((𝑆 ∈
CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑎) ∈ 𝐽) → 𝑎 ∈ (◡𝐹 “ 𝐽)) |
| 61 | 60 | ex 412 |
. . . . . . . 8
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → ((𝐹‘𝑎) ∈ 𝐽 → 𝑎 ∈ (◡𝐹 “ 𝐽))) |
| 62 | 49 | adantr 480 |
. . . . . . . . . 10
⊢
(((((((𝑆 ∈
CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑏) ∈ 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(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑏) ∈ 𝐽) → 𝑏 ∈ (Base‘𝑅)) |
| 64 | | simpr 484 |
. . . . . . . . . 10
⊢
(((((((𝑆 ∈
CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑏) ∈ 𝐽) → (𝐹‘𝑏) ∈ 𝐽) |
| 65 | 62, 63, 64 | elpreimad 7079 |
. . . . . . . . 9
⊢
(((((((𝑆 ∈
CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) ∧ (𝐹‘𝑏) ∈ 𝐽) → 𝑏 ∈ (◡𝐹 “ 𝐽)) |
| 66 | 65 | ex 412 |
. . . . . . . 8
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → ((𝐹‘𝑏) ∈ 𝐽 → 𝑏 ∈ (◡𝐹 “ 𝐽))) |
| 67 | 61, 66 | orim12d 967 |
. . . . . . 7
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (((𝐹‘𝑎) ∈ 𝐽 ∨ (𝐹‘𝑏) ∈ 𝐽) → (𝑎 ∈ (◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (◡𝐹 “ 𝐽)))) |
| 68 | 56, 67 | mpd 15 |
. . . . . 6
⊢
((((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽)) → (𝑎 ∈ (◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (◡𝐹 “ 𝐽))) |
| 69 | 68 | ex 412 |
. . . . 5
⊢
(((((𝑆 ∈ CRing
∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) → ((𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽) → (𝑎 ∈ (◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (◡𝐹 “ 𝐽)))) |
| 70 | 69 | anasss 466 |
. . . 4
⊢ ((((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽) → (𝑎 ∈ (◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (◡𝐹 “ 𝐽)))) |
| 71 | 70 | ralrimivva 3202 |
. . 3
⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)((𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽) → (𝑎 ∈ (◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (◡𝐹 “ 𝐽)))) |
| 72 | 21, 46 | prmidl2 33469 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ (◡𝐹 “ 𝐽) ∈ (LIdeal‘𝑅)) ∧ ((◡𝐹 “ 𝐽) ≠ (Base‘𝑅) ∧ ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)((𝑎(.r‘𝑅)𝑏) ∈ (◡𝐹 “ 𝐽) → (𝑎 ∈ (◡𝐹 “ 𝐽) ∨ 𝑏 ∈ (◡𝐹 “ 𝐽))))) → (◡𝐹 “ 𝐽) ∈ (PrmIdeal‘𝑅)) |
| 73 | 2, 9, 37, 71, 72 | syl22anc 839 |
. 2
⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ (PrmIdeal‘𝑅)) |
| 74 | | rhmpreimaprmidl.p |
. 2
⊢ 𝑃 = (PrmIdeal‘𝑅) |
| 75 | 73, 74 | eleqtrrdi 2852 |
1
⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃) |