Proof of Theorem cnmpopc
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2736 | . 2
⊢ ∪ (𝑂
×t 𝐽) =
∪ (𝑂 ×t 𝐽) | 
| 2 |  | eqid 2736 | . 2
⊢ ∪ 𝐾 =
∪ 𝐾 | 
| 3 |  | cnmpopc.a | . . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| 4 |  | cnmpopc.c | . . . . . 6
⊢ (𝜑 → 𝐶 ∈ ℝ) | 
| 5 |  | iccssre 13470 | . . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴[,]𝐶) ⊆ ℝ) | 
| 6 | 3, 4, 5 | syl2anc 584 | . . . . 5
⊢ (𝜑 → (𝐴[,]𝐶) ⊆ ℝ) | 
| 7 |  | cnmpopc.b | . . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶)) | 
| 8 | 6, 7 | sseldd 3983 | . . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| 9 |  | icccld 24788 | . . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran
(,)))) | 
| 10 | 3, 8, 9 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran
(,)))) | 
| 11 |  | cnmpopc.r | . . . . . . 7
⊢ 𝑅 = (topGen‘ran
(,)) | 
| 12 | 11 | fveq2i 6908 | . . . . . 6
⊢
(Clsd‘𝑅) =
(Clsd‘(topGen‘ran (,))) | 
| 13 | 10, 12 | eleqtrrdi 2851 | . . . . 5
⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑅)) | 
| 14 |  | ssun1 4177 | . . . . . 6
⊢ (𝐴[,]𝐵) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) | 
| 15 |  | iccsplit 13526 | . . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ (𝐴[,]𝐶)) → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))) | 
| 16 | 3, 4, 7, 15 | syl3anc 1372 | . . . . . 6
⊢ (𝜑 → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))) | 
| 17 | 14, 16 | sseqtrrid 4026 | . . . . 5
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶)) | 
| 18 |  | uniretop 24784 | . . . . . . 7
⊢ ℝ =
∪ (topGen‘ran (,)) | 
| 19 | 11 | unieqi 4918 | . . . . . . 7
⊢ ∪ 𝑅 =
∪ (topGen‘ran (,)) | 
| 20 | 18, 19 | eqtr4i 2767 | . . . . . 6
⊢ ℝ =
∪ 𝑅 | 
| 21 | 20 | restcldi 23182 | . . . . 5
⊢ (((𝐴[,]𝐶) ⊆ ℝ ∧ (𝐴[,]𝐵) ∈ (Clsd‘𝑅) ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶)) → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))) | 
| 22 | 6, 13, 17, 21 | syl3anc 1372 | . . . 4
⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))) | 
| 23 |  | cnmpopc.o | . . . . 5
⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶)) | 
| 24 | 23 | fveq2i 6908 | . . . 4
⊢
(Clsd‘𝑂) =
(Clsd‘(𝑅
↾t (𝐴[,]𝐶))) | 
| 25 | 22, 24 | eleqtrrdi 2851 | . . 3
⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑂)) | 
| 26 |  | cnmpopc.j | . . . . 5
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | 
| 27 |  | toponuni 22921 | . . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 28 | 26, 27 | syl 17 | . . . 4
⊢ (𝜑 → 𝑋 = ∪ 𝐽) | 
| 29 |  | topontop 22920 | . . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | 
| 30 |  | eqid 2736 | . . . . . 6
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 31 | 30 | topcld 23044 | . . . . 5
⊢ (𝐽 ∈ Top → ∪ 𝐽
∈ (Clsd‘𝐽)) | 
| 32 | 26, 29, 31 | 3syl 18 | . . . 4
⊢ (𝜑 → ∪ 𝐽
∈ (Clsd‘𝐽)) | 
| 33 | 28, 32 | eqeltrd 2840 | . . 3
⊢ (𝜑 → 𝑋 ∈ (Clsd‘𝐽)) | 
| 34 |  | txcld 23612 | . . 3
⊢ (((𝐴[,]𝐵) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽))) | 
| 35 | 25, 33, 34 | syl2anc 584 | . 2
⊢ (𝜑 → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽))) | 
| 36 |  | icccld 24788 | . . . . . . 7
⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran
(,)))) | 
| 37 | 8, 4, 36 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran
(,)))) | 
| 38 | 37, 12 | eleqtrrdi 2851 | . . . . 5
⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑅)) | 
| 39 |  | ssun2 4178 | . . . . . 6
⊢ (𝐵[,]𝐶) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) | 
| 40 | 39, 16 | sseqtrrid 4026 | . . . . 5
⊢ (𝜑 → (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶)) | 
| 41 | 20 | restcldi 23182 | . . . . 5
⊢ (((𝐴[,]𝐶) ⊆ ℝ ∧ (𝐵[,]𝐶) ∈ (Clsd‘𝑅) ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶)) → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))) | 
| 42 | 6, 38, 40, 41 | syl3anc 1372 | . . . 4
⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))) | 
| 43 | 42, 24 | eleqtrrdi 2851 | . . 3
⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑂)) | 
| 44 |  | txcld 23612 | . . 3
⊢ (((𝐵[,]𝐶) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽))) | 
| 45 | 43, 33, 44 | syl2anc 584 | . 2
⊢ (𝜑 → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽))) | 
| 46 | 16 | xpeq1d 5713 | . . . 4
⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋)) | 
| 47 |  | xpundir 5754 | . . . 4
⊢ (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) | 
| 48 | 46, 47 | eqtrdi 2792 | . . 3
⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋))) | 
| 49 |  | retopon 24785 | . . . . . . . 8
⊢
(topGen‘ran (,)) ∈ (TopOn‘ℝ) | 
| 50 | 11, 49 | eqeltri 2836 | . . . . . . 7
⊢ 𝑅 ∈
(TopOn‘ℝ) | 
| 51 |  | resttopon 23170 | . . . . . . 7
⊢ ((𝑅 ∈ (TopOn‘ℝ)
∧ (𝐴[,]𝐶) ⊆ ℝ) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶))) | 
| 52 | 50, 6, 51 | sylancr 587 | . . . . . 6
⊢ (𝜑 → (𝑅 ↾t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶))) | 
| 53 | 23, 52 | eqeltrid 2844 | . . . . 5
⊢ (𝜑 → 𝑂 ∈ (TopOn‘(𝐴[,]𝐶))) | 
| 54 |  | txtopon 23600 | . . . . 5
⊢ ((𝑂 ∈ (TopOn‘(𝐴[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋))) | 
| 55 | 53, 26, 54 | syl2anc 584 | . . . 4
⊢ (𝜑 → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋))) | 
| 56 |  | toponuni 22921 | . . . 4
⊢ ((𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)) → ((𝐴[,]𝐶) × 𝑋) = ∪ (𝑂 ×t 𝐽)) | 
| 57 | 55, 56 | syl 17 | . . 3
⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = ∪ (𝑂 ×t 𝐽)) | 
| 58 | 48, 57 | eqtr3d 2778 | . 2
⊢ (𝜑 → (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) = ∪ (𝑂 ×t 𝐽)) | 
| 59 |  | cnmpopc.m | . . . . . . . . . 10
⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵)) | 
| 60 | 17, 6 | sstrd 3993 | . . . . . . . . . . 11
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) | 
| 61 |  | resttopon 23170 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ (TopOn‘ℝ)
∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝑅 ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵))) | 
| 62 | 50, 60, 61 | sylancr 587 | . . . . . . . . . 10
⊢ (𝜑 → (𝑅 ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵))) | 
| 63 | 59, 62 | eqeltrid 2844 | . . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (TopOn‘(𝐴[,]𝐵))) | 
| 64 |  | txtopon 23600 | . . . . . . . . 9
⊢ ((𝑀 ∈ (TopOn‘(𝐴[,]𝐵)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋))) | 
| 65 | 63, 26, 64 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → (𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋))) | 
| 66 |  | cnmpopc.d | . . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾)) | 
| 67 |  | cntop2 23250 | . . . . . . . . . 10
⊢ ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾) → 𝐾 ∈ Top) | 
| 68 | 66, 67 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Top) | 
| 69 |  | toptopon2 22925 | . . . . . . . . 9
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | 
| 70 | 68, 69 | sylib 218 | . . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) | 
| 71 |  | elicc2 13453 | . . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) | 
| 72 | 3, 8, 71 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) | 
| 73 | 72 | biimpa 476 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)) | 
| 74 | 73 | simp3d 1144 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) | 
| 75 | 74 | 3adant3 1132 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → 𝑥 ≤ 𝐵) | 
| 76 | 75 | iftrued 4532 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐷) | 
| 77 | 76 | mpoeq3dva 7511 | . . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷)) | 
| 78 | 77, 66 | eqeltrd 2840 | . . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾)) | 
| 79 |  | cnf2 23258 | . . . . . . . 8
⊢ (((𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)
∧ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾)) → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶∪ 𝐾) | 
| 80 | 65, 70, 78, 79 | syl3anc 1372 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶∪ 𝐾) | 
| 81 |  | eqid 2736 | . . . . . . . 8
⊢ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) | 
| 82 | 81 | fmpo 8094 | . . . . . . 7
⊢
(∀𝑥 ∈
(𝐴[,]𝐵)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶∪ 𝐾) | 
| 83 | 80, 82 | sylibr 234 | . . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) | 
| 84 |  | cnmpopc.n | . . . . . . . . . 10
⊢ 𝑁 = (𝑅 ↾t (𝐵[,]𝐶)) | 
| 85 | 40, 6 | sstrd 3993 | . . . . . . . . . . 11
⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ) | 
| 86 |  | resttopon 23170 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ (TopOn‘ℝ)
∧ (𝐵[,]𝐶) ⊆ ℝ) → (𝑅 ↾t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶))) | 
| 87 | 50, 85, 86 | sylancr 587 | . . . . . . . . . 10
⊢ (𝜑 → (𝑅 ↾t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶))) | 
| 88 | 84, 87 | eqeltrid 2844 | . . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ (TopOn‘(𝐵[,]𝐶))) | 
| 89 |  | txtopon 23600 | . . . . . . . . 9
⊢ ((𝑁 ∈ (TopOn‘(𝐵[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋))) | 
| 90 | 88, 26, 89 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋))) | 
| 91 |  | elicc2 13453 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶))) | 
| 92 | 8, 4, 91 | syl2anc 584 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶))) | 
| 93 | 92 | biimpa 476 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶)) | 
| 94 | 93 | simp2d 1143 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ≤ 𝑥) | 
| 95 | 94 | biantrud 531 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥))) | 
| 96 | 93 | simp1d 1142 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 ∈ ℝ) | 
| 97 | 8 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ∈ ℝ) | 
| 98 | 96, 97 | letri3d 11404 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 = 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥))) | 
| 99 | 95, 98 | bitr4d 282 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ 𝑥 = 𝐵)) | 
| 100 | 99 | 3adant3 1132 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝐵 ↔ 𝑥 = 𝐵)) | 
| 101 |  | cnmpopc.q | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑥 = 𝐵 ∧ 𝑦 ∈ 𝑋)) → 𝐷 = 𝐸) | 
| 102 | 101 | ancom2s 650 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → 𝐷 = 𝐸) | 
| 103 | 102 | ifeq1d 4544 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = if(𝑥 ≤ 𝐵, 𝐸, 𝐸)) | 
| 104 |  | ifid 4565 | . . . . . . . . . . . . . . 15
⊢ if(𝑥 ≤ 𝐵, 𝐸, 𝐸) = 𝐸 | 
| 105 | 103, 104 | eqtrdi 2792 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸) | 
| 106 | 105 | expr 456 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)) | 
| 107 | 106 | 3adant2 1131 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)) | 
| 108 | 100, 107 | sylbid 240 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)) | 
| 109 |  | iffalse 4533 | . . . . . . . . . . 11
⊢ (¬
𝑥 ≤ 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸) | 
| 110 | 108, 109 | pm2.61d1 180 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸) | 
| 111 | 110 | mpoeq3dva 7511 | . . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸)) | 
| 112 |  | cnmpopc.e | . . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾)) | 
| 113 | 111, 112 | eqeltrd 2840 | . . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾)) | 
| 114 |  | cnf2 23258 | . . . . . . . 8
⊢ (((𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)
∧ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾)) → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶∪ 𝐾) | 
| 115 | 90, 70, 113, 114 | syl3anc 1372 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶∪ 𝐾) | 
| 116 |  | eqid 2736 | . . . . . . . 8
⊢ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) | 
| 117 | 116 | fmpo 8094 | . . . . . . 7
⊢
(∀𝑥 ∈
(𝐵[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ↔ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶∪ 𝐾) | 
| 118 | 115, 117 | sylibr 234 | . . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) | 
| 119 |  | ralun 4197 | . . . . . 6
⊢
((∀𝑥 ∈
(𝐴[,]𝐵)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ∧ ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) → ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) | 
| 120 | 83, 118, 119 | syl2anc 584 | . . . . 5
⊢ (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) | 
| 121 | 120, 16 | raleqtrrdv 3329 | . . . 4
⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) | 
| 122 |  | eqid 2736 | . . . . 5
⊢ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) | 
| 123 | 122 | fmpo 8094 | . . . 4
⊢
(∀𝑥 ∈
(𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾) | 
| 124 | 121, 123 | sylib 218 | . . 3
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾) | 
| 125 | 57 | feq2d 6721 | . . 3
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):∪ (𝑂 ×t 𝐽)⟶∪ 𝐾)) | 
| 126 | 124, 125 | mpbid 232 | . 2
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):∪ (𝑂 ×t 𝐽)⟶∪ 𝐾) | 
| 127 |  | ssid 4005 | . . . 4
⊢ 𝑋 ⊆ 𝑋 | 
| 128 |  | resmpo 7554 | . . . 4
⊢ (((𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ 𝑋 ⊆ 𝑋) → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))) | 
| 129 | 17, 127, 128 | sylancl 586 | . . 3
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))) | 
| 130 |  | retop 24783 | . . . . . . . . . 10
⊢
(topGen‘ran (,)) ∈ Top | 
| 131 | 11, 130 | eqeltri 2836 | . . . . . . . . 9
⊢ 𝑅 ∈ Top | 
| 132 |  | ovex 7465 | . . . . . . . . 9
⊢ (𝐴[,]𝐶) ∈ V | 
| 133 |  | resttop 23169 | . . . . . . . . 9
⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐶) ∈ V) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top) | 
| 134 | 131, 132,
133 | mp2an 692 | . . . . . . . 8
⊢ (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top | 
| 135 | 23, 134 | eqeltri 2836 | . . . . . . 7
⊢ 𝑂 ∈ Top | 
| 136 | 135 | a1i 11 | . . . . . 6
⊢ (𝜑 → 𝑂 ∈ Top) | 
| 137 |  | ovexd 7467 | . . . . . 6
⊢ (𝜑 → (𝐴[,]𝐵) ∈ V) | 
| 138 |  | txrest 23640 | . . . . . 6
⊢ (((𝑂 ∈ Top ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ ((𝐴[,]𝐵) ∈ V ∧ 𝑋 ∈ (Clsd‘𝐽))) → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋))) | 
| 139 | 136, 26, 137, 33, 138 | syl22anc 838 | . . . . 5
⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋))) | 
| 140 | 131 | a1i 11 | . . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ Top) | 
| 141 |  | ovexd 7467 | . . . . . . . 8
⊢ (𝜑 → (𝐴[,]𝐶) ∈ V) | 
| 142 |  | restabs 23174 | . . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵))) | 
| 143 | 140, 17, 141, 142 | syl3anc 1372 | . . . . . . 7
⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵))) | 
| 144 | 23 | oveq1i 7442 | . . . . . . 7
⊢ (𝑂 ↾t (𝐴[,]𝐵)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) | 
| 145 | 143, 144,
59 | 3eqtr4g 2801 | . . . . . 6
⊢ (𝜑 → (𝑂 ↾t (𝐴[,]𝐵)) = 𝑀) | 
| 146 | 28 | oveq2d 7448 | . . . . . . 7
⊢ (𝜑 → (𝐽 ↾t 𝑋) = (𝐽 ↾t ∪ 𝐽)) | 
| 147 | 30 | restid 17479 | . . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ↾t ∪ 𝐽) =
𝐽) | 
| 148 | 26, 147 | syl 17 | . . . . . . 7
⊢ (𝜑 → (𝐽 ↾t ∪ 𝐽) =
𝐽) | 
| 149 | 146, 148 | eqtrd 2776 | . . . . . 6
⊢ (𝜑 → (𝐽 ↾t 𝑋) = 𝐽) | 
| 150 | 145, 149 | oveq12d 7450 | . . . . 5
⊢ (𝜑 → ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋)) = (𝑀 ×t 𝐽)) | 
| 151 | 139, 150 | eqtrd 2776 | . . . 4
⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = (𝑀 ×t 𝐽)) | 
| 152 | 151 | oveq1d 7447 | . . 3
⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾) = ((𝑀 ×t 𝐽) Cn 𝐾)) | 
| 153 | 78, 129, 152 | 3eltr4d 2855 | . 2
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾)) | 
| 154 |  | resmpo 7554 | . . . 4
⊢ (((𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ 𝑋 ⊆ 𝑋) → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))) | 
| 155 | 40, 127, 154 | sylancl 586 | . . 3
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))) | 
| 156 |  | ovexd 7467 | . . . . . 6
⊢ (𝜑 → (𝐵[,]𝐶) ∈ V) | 
| 157 |  | txrest 23640 | . . . . . 6
⊢ (((𝑂 ∈ Top ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ ((𝐵[,]𝐶) ∈ V ∧ 𝑋 ∈ (Clsd‘𝐽))) → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋))) | 
| 158 | 136, 26, 156, 33, 157 | syl22anc 838 | . . . . 5
⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋))) | 
| 159 |  | restabs 23174 | . . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶))) | 
| 160 | 140, 40, 141, 159 | syl3anc 1372 | . . . . . . 7
⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶))) | 
| 161 | 23 | oveq1i 7442 | . . . . . . 7
⊢ (𝑂 ↾t (𝐵[,]𝐶)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) | 
| 162 | 160, 161,
84 | 3eqtr4g 2801 | . . . . . 6
⊢ (𝜑 → (𝑂 ↾t (𝐵[,]𝐶)) = 𝑁) | 
| 163 | 162, 149 | oveq12d 7450 | . . . . 5
⊢ (𝜑 → ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)) = (𝑁 ×t 𝐽)) | 
| 164 | 158, 163 | eqtrd 2776 | . . . 4
⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = (𝑁 ×t 𝐽)) | 
| 165 | 164 | oveq1d 7447 | . . 3
⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾) = ((𝑁 ×t 𝐽) Cn 𝐾)) | 
| 166 | 113, 155,
165 | 3eltr4d 2855 | . 2
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾)) | 
| 167 | 1, 2, 35, 45, 58, 126, 153, 166 | paste 23303 | 1
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾)) |