| Step | Hyp | Ref
| Expression |
| 1 | | rhmpreimacn.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ CRing) |
| 2 | | rhmpreimacn.u |
. . . 4
⊢ 𝑈 = (Spec‘𝑆) |
| 3 | | rhmpreimacn.k |
. . . 4
⊢ 𝐾 = (TopOpen‘𝑈) |
| 4 | | rhmpreimacn.b |
. . . 4
⊢ 𝐵 = (PrmIdeal‘𝑆) |
| 5 | 2, 3, 4 | zartopon 33876 |
. . 3
⊢ (𝑆 ∈ CRing → 𝐾 ∈ (TopOn‘𝐵)) |
| 6 | 1, 5 | syl 17 |
. 2
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵)) |
| 7 | | rhmpreimacn.r |
. . 3
⊢ (𝜑 → 𝑅 ∈ CRing) |
| 8 | | rhmpreimacn.t |
. . . 4
⊢ 𝑇 = (Spec‘𝑅) |
| 9 | | rhmpreimacn.j |
. . . 4
⊢ 𝐽 = (TopOpen‘𝑇) |
| 10 | | rhmpreimacn.a |
. . . 4
⊢ 𝐴 = (PrmIdeal‘𝑅) |
| 11 | 8, 9, 10 | zartopon 33876 |
. . 3
⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝐴)) |
| 12 | 7, 11 | syl 17 |
. 2
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐴)) |
| 13 | 1 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑆 ∈ CRing) |
| 14 | | rhmpreimacn.f |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
| 15 | 14 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
| 16 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑖 ∈ 𝐵) |
| 17 | 16, 4 | eleqtrdi 2851 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑖 ∈ (PrmIdeal‘𝑆)) |
| 18 | 10 | rhmpreimaprmidl 33479 |
. . . 4
⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝑖 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝑖) ∈ 𝐴) |
| 19 | 13, 15, 17, 18 | syl21anc 838 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → (◡𝐹 “ 𝑖) ∈ 𝐴) |
| 20 | | rhmpreimacn.g |
. . 3
⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖)) |
| 21 | 19, 20 | fmptd 7134 |
. 2
⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
| 22 | 4 | fvexi 6920 |
. . . . . . 7
⊢ 𝐵 ∈ V |
| 23 | 22 | rabex 5339 |
. . . . . 6
⊢ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘} ∈ V |
| 24 | | sseq1 4009 |
. . . . . . . 8
⊢ (𝑙 = 𝑗 → (𝑙 ⊆ 𝑘 ↔ 𝑗 ⊆ 𝑘)) |
| 25 | 24 | rabbidv 3444 |
. . . . . . 7
⊢ (𝑙 = 𝑗 → {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘} = {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘}) |
| 26 | 25 | cbvmptv 5255 |
. . . . . 6
⊢ (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (𝑗 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘}) |
| 27 | 23, 26 | fnmpti 6711 |
. . . . 5
⊢ (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑆) |
| 28 | 14 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
| 29 | | rhmpreimacn.1 |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 = (Base‘𝑆)) |
| 30 | 29 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ran 𝐹 = (Base‘𝑆)) |
| 31 | | simplr 769 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑎 ∈ (LIdeal‘𝑅)) |
| 32 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝑆) =
(Base‘𝑆) |
| 33 | | eqid 2737 |
. . . . . . . . 9
⊢
(LIdeal‘𝑅) =
(LIdeal‘𝑅) |
| 34 | | eqid 2737 |
. . . . . . . . 9
⊢
(LIdeal‘𝑆) =
(LIdeal‘𝑆) |
| 35 | 32, 33, 34 | rhmimaidl 33460 |
. . . . . . . 8
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = (Base‘𝑆) ∧ 𝑎 ∈ (LIdeal‘𝑅)) → (𝐹 “ 𝑎) ∈ (LIdeal‘𝑆)) |
| 36 | 28, 30, 31, 35 | syl3anc 1373 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → (𝐹 “ 𝑎) ∈ (LIdeal‘𝑆)) |
| 37 | | fveqeq2 6915 |
. . . . . . . 8
⊢ (𝑏 = (𝐹 “ 𝑎) → (((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥) ↔ ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘(𝐹 “ 𝑎)) = (◡𝐺 “ 𝑥))) |
| 38 | 37 | adantl 481 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) ∧ 𝑏 = (𝐹 “ 𝑎)) → (((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥) ↔ ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘(𝐹 “ 𝑎)) = (◡𝐺 “ 𝑥))) |
| 39 | 7 | ad3antrrr 730 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑅 ∈ CRing) |
| 40 | 1 | ad3antrrr 730 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑆 ∈ CRing) |
| 41 | 24 | rabbidv 3444 |
. . . . . . . . . 10
⊢ (𝑙 = 𝑗 → {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘} = {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘}) |
| 42 | 41 | cbvmptv 5255 |
. . . . . . . . 9
⊢ (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (𝑗 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘}) |
| 43 | 8, 2, 10, 4, 9, 3,
20, 39, 40, 28, 30, 31, 42, 26 | rhmpreimacnlem 33883 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘(𝐹 “ 𝑎)) = (◡𝐺 “ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎))) |
| 44 | | simpr 484 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) |
| 45 | 44 | imaeq2d 6078 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → (◡𝐺 “ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎)) = (◡𝐺 “ 𝑥)) |
| 46 | 43, 45 | eqtrd 2777 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘(𝐹 “ 𝑎)) = (◡𝐺 “ 𝑥)) |
| 47 | 36, 38, 46 | rspcedvd 3624 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥)) |
| 48 | 10 | fvexi 6920 |
. . . . . . . . 9
⊢ 𝐴 ∈ V |
| 49 | 48 | rabex 5339 |
. . . . . . . 8
⊢ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘} ∈ V |
| 50 | 49, 42 | fnmpti 6711 |
. . . . . . 7
⊢ (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑅) |
| 51 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (Clsd‘𝐽)) |
| 52 | 7 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑅 ∈ CRing) |
| 53 | 8, 9, 10, 42 | zartopn 33874 |
. . . . . . . . . 10
⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝐴) ∧ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽))) |
| 54 | 53 | simprd 495 |
. . . . . . . . 9
⊢ (𝑅 ∈ CRing → ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽)) |
| 55 | 52, 54 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽)) |
| 56 | 51, 55 | eleqtrrd 2844 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})) |
| 57 | | fvelrnb 6969 |
. . . . . . . 8
⊢ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑅) → (𝑥 ∈ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) ↔ ∃𝑎 ∈ (LIdeal‘𝑅)((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥)) |
| 58 | 57 | biimpa 476 |
. . . . . . 7
⊢ (((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑅) ∧ 𝑥 ∈ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})) → ∃𝑎 ∈ (LIdeal‘𝑅)((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) |
| 59 | 50, 56, 58 | sylancr 587 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ∃𝑎 ∈ (LIdeal‘𝑅)((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) |
| 60 | 47, 59 | r19.29a 3162 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥)) |
| 61 | | fvelrnb 6969 |
. . . . . 6
⊢ ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑆) → ((◡𝐺 “ 𝑥) ∈ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) ↔ ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥))) |
| 62 | 61 | biimpar 477 |
. . . . 5
⊢ (((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑆) ∧ ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥)) → (◡𝐺 “ 𝑥) ∈ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})) |
| 63 | 27, 60, 62 | sylancr 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (◡𝐺 “ 𝑥) ∈ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})) |
| 64 | 2, 3, 4, 26 | zartopn 33874 |
. . . . . . 7
⊢ (𝑆 ∈ CRing → (𝐾 ∈ (TopOn‘𝐵) ∧ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾))) |
| 65 | 64 | simprd 495 |
. . . . . 6
⊢ (𝑆 ∈ CRing → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾)) |
| 66 | 1, 65 | syl 17 |
. . . . 5
⊢ (𝜑 → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾)) |
| 67 | 66 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾)) |
| 68 | 63, 67 | eleqtrd 2843 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾)) |
| 69 | 68 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾)) |
| 70 | | iscncl 23277 |
. . 3
⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐴)) → (𝐺 ∈ (𝐾 Cn 𝐽) ↔ (𝐺:𝐵⟶𝐴 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾)))) |
| 71 | 70 | biimpar 477 |
. 2
⊢ (((𝐾 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐴)) ∧ (𝐺:𝐵⟶𝐴 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾))) → 𝐺 ∈ (𝐾 Cn 𝐽)) |
| 72 | 6, 12, 21, 69, 71 | syl22anc 839 |
1
⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽)) |