Proof of Theorem cnmpopc
Step | Hyp | Ref
| Expression |
1 | | eqid 2740 |
. 2
⊢ ∪ (𝑂
×t 𝐽) =
∪ (𝑂 ×t 𝐽) |
2 | | eqid 2740 |
. 2
⊢ ∪ 𝐾 =
∪ 𝐾 |
3 | | cnmpopc.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
4 | | cnmpopc.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ ℝ) |
5 | | iccssre 13160 |
. . . . . 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 3927 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
9 | | icccld 23928 |
. . . . . . 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 6774 |
. . . . . 6
⊢
(Clsd‘𝑅) =
(Clsd‘(topGen‘ran (,))) |
13 | 10, 12 | eleqtrrdi 2852 |
. . . . 5
⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑅)) |
14 | | ssun1 4111 |
. . . . . 6
⊢ (𝐴[,]𝐵) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) |
15 | | iccsplit 13216 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ (𝐴[,]𝐶)) → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))) |
16 | 3, 4, 7, 15 | syl3anc 1370 |
. . . . . 6
⊢ (𝜑 → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))) |
17 | 14, 16 | sseqtrrid 3979 |
. . . . 5
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶)) |
18 | | uniretop 23924 |
. . . . . . 7
⊢ ℝ =
∪ (topGen‘ran (,)) |
19 | 11 | unieqi 4858 |
. . . . . . 7
⊢ ∪ 𝑅 =
∪ (topGen‘ran (,)) |
20 | 18, 19 | eqtr4i 2771 |
. . . . . 6
⊢ ℝ =
∪ 𝑅 |
21 | 20 | restcldi 22322 |
. . . . 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 1370 |
. . . 4
⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))) |
23 | | cnmpopc.o |
. . . . 5
⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶)) |
24 | 23 | fveq2i 6774 |
. . . 4
⊢
(Clsd‘𝑂) =
(Clsd‘(𝑅
↾t (𝐴[,]𝐶))) |
25 | 22, 24 | eleqtrrdi 2852 |
. . 3
⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑂)) |
26 | | cnmpopc.j |
. . . . 5
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
27 | | toponuni 22061 |
. . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
28 | 26, 27 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
29 | | topontop 22060 |
. . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
30 | | eqid 2740 |
. . . . . 6
⊢ ∪ 𝐽 =
∪ 𝐽 |
31 | 30 | topcld 22184 |
. . . . 5
⊢ (𝐽 ∈ Top → ∪ 𝐽
∈ (Clsd‘𝐽)) |
32 | 26, 29, 31 | 3syl 18 |
. . . 4
⊢ (𝜑 → ∪ 𝐽
∈ (Clsd‘𝐽)) |
33 | 28, 32 | eqeltrd 2841 |
. . 3
⊢ (𝜑 → 𝑋 ∈ (Clsd‘𝐽)) |
34 | | txcld 22752 |
. . 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 23928 |
. . . . . . 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 2852 |
. . . . 5
⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑅)) |
39 | | ssun2 4112 |
. . . . . 6
⊢ (𝐵[,]𝐶) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) |
40 | 39, 16 | sseqtrrid 3979 |
. . . . 5
⊢ (𝜑 → (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶)) |
41 | 20 | restcldi 22322 |
. . . . 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 1370 |
. . . 4
⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))) |
43 | 42, 24 | eleqtrrdi 2852 |
. . 3
⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑂)) |
44 | | txcld 22752 |
. . 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 5619 |
. . . 4
⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋)) |
47 | | xpundir 5657 |
. . . 4
⊢ (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) |
48 | 46, 47 | eqtrdi 2796 |
. . 3
⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋))) |
49 | | retopon 23925 |
. . . . . . . 8
⊢
(topGen‘ran (,)) ∈ (TopOn‘ℝ) |
50 | 11, 49 | eqeltri 2837 |
. . . . . . 7
⊢ 𝑅 ∈
(TopOn‘ℝ) |
51 | | resttopon 22310 |
. . . . . . 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 2845 |
. . . . 5
⊢ (𝜑 → 𝑂 ∈ (TopOn‘(𝐴[,]𝐶))) |
54 | | txtopon 22740 |
. . . . 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 22061 |
. . . 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 2782 |
. 2
⊢ (𝜑 → (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) = ∪ (𝑂 ×t 𝐽)) |
59 | | cnmpopc.m |
. . . . . . . . . 10
⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵)) |
60 | 17, 6 | sstrd 3936 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
61 | | resttopon 22310 |
. . . . . . . . . . 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 2845 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (TopOn‘(𝐴[,]𝐵))) |
64 | | txtopon 22740 |
. . . . . . . . 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 22390 |
. . . . . . . . . 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 22065 |
. . . . . . . . 9
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
70 | 68, 69 | sylib 217 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
71 | | elicc2 13143 |
. . . . . . . . . . . . . . 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 477 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)) |
74 | 73 | simp3d 1143 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) |
75 | 74 | 3adant3 1131 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → 𝑥 ≤ 𝐵) |
76 | 75 | iftrued 4473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐷) |
77 | 76 | mpoeq3dva 7346 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷)) |
78 | 77, 66 | eqeltrd 2841 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾)) |
79 | | cnf2 22398 |
. . . . . . . 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 1370 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶∪ 𝐾) |
81 | | eqid 2740 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) |
82 | 81 | fmpo 7901 |
. . . . . . 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 233 |
. . . . . 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 3936 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ) |
86 | | resttopon 22310 |
. . . . . . . . . . 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 2845 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ (TopOn‘(𝐵[,]𝐶))) |
89 | | txtopon 22740 |
. . . . . . . . 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 13143 |
. . . . . . . . . . . . . . . . . 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 477 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶)) |
94 | 93 | simp2d 1142 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ≤ 𝑥) |
95 | 94 | biantrud 532 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥))) |
96 | 93 | simp1d 1141 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 ∈ ℝ) |
97 | 8 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ∈ ℝ) |
98 | 96, 97 | letri3d 11117 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 = 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥))) |
99 | 95, 98 | bitr4d 281 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ 𝑥 = 𝐵)) |
100 | 99 | 3adant3 1131 |
. . . . . . . . . . . 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 647 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → 𝐷 = 𝐸) |
103 | 102 | ifeq1d 4484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = if(𝑥 ≤ 𝐵, 𝐸, 𝐸)) |
104 | | ifid 4505 |
. . . . . . . . . . . . . . 15
⊢ if(𝑥 ≤ 𝐵, 𝐸, 𝐸) = 𝐸 |
105 | 103, 104 | eqtrdi 2796 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸) |
106 | 105 | expr 457 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)) |
107 | 106 | 3adant2 1130 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)) |
108 | 100, 107 | sylbid 239 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)) |
109 | | iffalse 4474 |
. . . . . . . . . . 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 7346 |
. . . . . . . . 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 2841 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾)) |
114 | | cnf2 22398 |
. . . . . . . 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 1370 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶∪ 𝐾) |
116 | | eqid 2740 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) |
117 | 116 | fmpo 7901 |
. . . . . . 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 233 |
. . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) |
119 | | ralun 4131 |
. . . . . 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 | 16 | raleqdv 3347 |
. . . . 5
⊢ (𝜑 → (∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)) |
122 | 120, 121 | mpbird 256 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) |
123 | | eqid 2740 |
. . . . 5
⊢ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) |
124 | 123 | fmpo 7901 |
. . . 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) |
125 | 122, 124 | sylib 217 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾) |
126 | 57 | feq2d 6584 |
. . 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)) |
127 | 125, 126 | mpbid 231 |
. 2
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):∪ (𝑂 ×t 𝐽)⟶∪ 𝐾) |
128 | | ssid 3948 |
. . . 4
⊢ 𝑋 ⊆ 𝑋 |
129 | | resmpo 7388 |
. . . 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))) |
130 | 17, 128, 129 | 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))) |
131 | | retop 23923 |
. . . . . . . . . 10
⊢
(topGen‘ran (,)) ∈ Top |
132 | 11, 131 | eqeltri 2837 |
. . . . . . . . 9
⊢ 𝑅 ∈ Top |
133 | | ovex 7304 |
. . . . . . . . 9
⊢ (𝐴[,]𝐶) ∈ V |
134 | | resttop 22309 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐶) ∈ V) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top) |
135 | 132, 133,
134 | mp2an 689 |
. . . . . . . 8
⊢ (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top |
136 | 23, 135 | eqeltri 2837 |
. . . . . . 7
⊢ 𝑂 ∈ Top |
137 | 136 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 𝑂 ∈ Top) |
138 | | ovexd 7306 |
. . . . . 6
⊢ (𝜑 → (𝐴[,]𝐵) ∈ V) |
139 | | txrest 22780 |
. . . . . 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))) |
140 | 137, 26, 138, 33, 139 | syl22anc 836 |
. . . . 5
⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋))) |
141 | 132 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ Top) |
142 | | ovexd 7306 |
. . . . . . . 8
⊢ (𝜑 → (𝐴[,]𝐶) ∈ V) |
143 | | restabs 22314 |
. . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵))) |
144 | 141, 17, 142, 143 | syl3anc 1370 |
. . . . . . 7
⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵))) |
145 | 23 | oveq1i 7281 |
. . . . . . 7
⊢ (𝑂 ↾t (𝐴[,]𝐵)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) |
146 | 144, 145,
59 | 3eqtr4g 2805 |
. . . . . 6
⊢ (𝜑 → (𝑂 ↾t (𝐴[,]𝐵)) = 𝑀) |
147 | 28 | oveq2d 7287 |
. . . . . . 7
⊢ (𝜑 → (𝐽 ↾t 𝑋) = (𝐽 ↾t ∪ 𝐽)) |
148 | 30 | restid 17142 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ↾t ∪ 𝐽) =
𝐽) |
149 | 26, 148 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐽 ↾t ∪ 𝐽) =
𝐽) |
150 | 147, 149 | eqtrd 2780 |
. . . . . 6
⊢ (𝜑 → (𝐽 ↾t 𝑋) = 𝐽) |
151 | 146, 150 | oveq12d 7289 |
. . . . 5
⊢ (𝜑 → ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋)) = (𝑀 ×t 𝐽)) |
152 | 140, 151 | eqtrd 2780 |
. . . 4
⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = (𝑀 ×t 𝐽)) |
153 | 152 | oveq1d 7286 |
. . 3
⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾) = ((𝑀 ×t 𝐽) Cn 𝐾)) |
154 | 78, 130, 153 | 3eltr4d 2856 |
. 2
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾)) |
155 | | resmpo 7388 |
. . . 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))) |
156 | 40, 128, 155 | 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))) |
157 | | ovexd 7306 |
. . . . . 6
⊢ (𝜑 → (𝐵[,]𝐶) ∈ V) |
158 | | txrest 22780 |
. . . . . 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))) |
159 | 137, 26, 157, 33, 158 | syl22anc 836 |
. . . . 5
⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋))) |
160 | | restabs 22314 |
. . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶))) |
161 | 141, 40, 142, 160 | syl3anc 1370 |
. . . . . . 7
⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶))) |
162 | 23 | oveq1i 7281 |
. . . . . . 7
⊢ (𝑂 ↾t (𝐵[,]𝐶)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) |
163 | 161, 162,
84 | 3eqtr4g 2805 |
. . . . . 6
⊢ (𝜑 → (𝑂 ↾t (𝐵[,]𝐶)) = 𝑁) |
164 | 163, 150 | oveq12d 7289 |
. . . . 5
⊢ (𝜑 → ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)) = (𝑁 ×t 𝐽)) |
165 | 159, 164 | eqtrd 2780 |
. . . 4
⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = (𝑁 ×t 𝐽)) |
166 | 165 | oveq1d 7286 |
. . 3
⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾) = ((𝑁 ×t 𝐽) Cn 𝐾)) |
167 | 113, 156,
166 | 3eltr4d 2856 |
. 2
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾)) |
168 | 1, 2, 35, 45, 58, 127, 154, 167 | paste 22443 |
1
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾)) |