MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ptunhmeo Structured version   Visualization version   GIF version

Theorem ptunhmeo 23031
Description: Define a homeomorphism from a binary product of indexed product topologies to an indexed product topology on the union of the index sets. This is the topological analogue of (𝐴𝐵) · (𝐴𝐶) = 𝐴↑(𝐵 + 𝐶). (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
ptunhmeo.x 𝑋 = 𝐾
ptunhmeo.y 𝑌 = 𝐿
ptunhmeo.j 𝐽 = (∏t𝐹)
ptunhmeo.k 𝐾 = (∏t‘(𝐹𝐴))
ptunhmeo.l 𝐿 = (∏t‘(𝐹𝐵))
ptunhmeo.g 𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
ptunhmeo.c (𝜑𝐶𝑉)
ptunhmeo.f (𝜑𝐹:𝐶⟶Top)
ptunhmeo.u (𝜑𝐶 = (𝐴𝐵))
ptunhmeo.i (𝜑 → (𝐴𝐵) = ∅)
Assertion
Ref Expression
ptunhmeo (𝜑𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ptunhmeo
Dummy variables 𝑓 𝑘 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptunhmeo.g . . . . 5 𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
2 vex 3445 . . . . . . . 8 𝑥 ∈ V
3 vex 3445 . . . . . . . 8 𝑦 ∈ V
42, 3op1std 7886 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
52, 3op2ndd 7887 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
64, 5uneq12d 4109 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st𝑧) ∪ (2nd𝑧)) = (𝑥𝑦))
76mpompt 7428 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
81, 7eqtr4i 2768 . . . 4 𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧)))
9 xp1st 7908 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
109adantl 482 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) ∈ 𝑋)
11 ixpeq2 8747 . . . . . . . . . . . . 13 (∀𝑛𝐴 ((𝐹𝐴)‘𝑛) = (𝐹𝑛) → X𝑛𝐴 ((𝐹𝐴)‘𝑛) = X𝑛𝐴 (𝐹𝑛))
12 fvres 6830 . . . . . . . . . . . . . 14 (𝑛𝐴 → ((𝐹𝐴)‘𝑛) = (𝐹𝑛))
1312unieqd 4864 . . . . . . . . . . . . 13 (𝑛𝐴 ((𝐹𝐴)‘𝑛) = (𝐹𝑛))
1411, 13mprg 3068 . . . . . . . . . . . 12 X𝑛𝐴 ((𝐹𝐴)‘𝑛) = X𝑛𝐴 (𝐹𝑛)
15 ptunhmeo.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑉)
16 ssun1 4117 . . . . . . . . . . . . . . 15 𝐴 ⊆ (𝐴𝐵)
17 ptunhmeo.u . . . . . . . . . . . . . . 15 (𝜑𝐶 = (𝐴𝐵))
1816, 17sseqtrrid 3984 . . . . . . . . . . . . . 14 (𝜑𝐴𝐶)
1915, 18ssexd 5263 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ V)
20 ptunhmeo.f . . . . . . . . . . . . . 14 (𝜑𝐹:𝐶⟶Top)
2120, 18fssresd 6678 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝐴):𝐴⟶Top)
22 ptunhmeo.k . . . . . . . . . . . . . 14 𝐾 = (∏t‘(𝐹𝐴))
2322ptuni 22817 . . . . . . . . . . . . 13 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → X𝑛𝐴 ((𝐹𝐴)‘𝑛) = 𝐾)
2419, 21, 23syl2anc 584 . . . . . . . . . . . 12 (𝜑X𝑛𝐴 ((𝐹𝐴)‘𝑛) = 𝐾)
2514, 24eqtr3id 2791 . . . . . . . . . . 11 (𝜑X𝑛𝐴 (𝐹𝑛) = 𝐾)
26 ptunhmeo.x . . . . . . . . . . 11 𝑋 = 𝐾
2725, 26eqtr4di 2795 . . . . . . . . . 10 (𝜑X𝑛𝐴 (𝐹𝑛) = 𝑋)
2827adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
2910, 28eleqtrrd 2841 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛))
30 xp2nd 7909 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
3130adantl 482 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ 𝑌)
3217eqcomd 2743 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐵) = 𝐶)
33 ptunhmeo.i . . . . . . . . . . . . . 14 (𝜑 → (𝐴𝐵) = ∅)
34 uneqdifeq 4435 . . . . . . . . . . . . . 14 ((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3518, 33, 34syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3632, 35mpbid 231 . . . . . . . . . . . 12 (𝜑 → (𝐶𝐴) = 𝐵)
3736ixpeq1d 8745 . . . . . . . . . . 11 (𝜑X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = X𝑛𝐵 (𝐹𝑛))
38 ixpeq2 8747 . . . . . . . . . . . . . 14 (∀𝑛𝐵 ((𝐹𝐵)‘𝑛) = (𝐹𝑛) → X𝑛𝐵 ((𝐹𝐵)‘𝑛) = X𝑛𝐵 (𝐹𝑛))
39 fvres 6830 . . . . . . . . . . . . . . 15 (𝑛𝐵 → ((𝐹𝐵)‘𝑛) = (𝐹𝑛))
4039unieqd 4864 . . . . . . . . . . . . . 14 (𝑛𝐵 ((𝐹𝐵)‘𝑛) = (𝐹𝑛))
4138, 40mprg 3068 . . . . . . . . . . . . 13 X𝑛𝐵 ((𝐹𝐵)‘𝑛) = X𝑛𝐵 (𝐹𝑛)
42 ssun2 4118 . . . . . . . . . . . . . . . 16 𝐵 ⊆ (𝐴𝐵)
4342, 17sseqtrrid 3984 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
4415, 43ssexd 5263 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ V)
4520, 43fssresd 6678 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐵):𝐵⟶Top)
46 ptunhmeo.l . . . . . . . . . . . . . . 15 𝐿 = (∏t‘(𝐹𝐵))
4746ptuni 22817 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → X𝑛𝐵 ((𝐹𝐵)‘𝑛) = 𝐿)
4844, 45, 47syl2anc 584 . . . . . . . . . . . . 13 (𝜑X𝑛𝐵 ((𝐹𝐵)‘𝑛) = 𝐿)
4941, 48eqtr3id 2791 . . . . . . . . . . . 12 (𝜑X𝑛𝐵 (𝐹𝑛) = 𝐿)
50 ptunhmeo.y . . . . . . . . . . . 12 𝑌 = 𝐿
5149, 50eqtr4di 2795 . . . . . . . . . . 11 (𝜑X𝑛𝐵 (𝐹𝑛) = 𝑌)
5237, 51eqtrd 2777 . . . . . . . . . 10 (𝜑X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = 𝑌)
5352adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = 𝑌)
5431, 53eleqtrrd 2841 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ X𝑛 ∈ (𝐶𝐴) (𝐹𝑛))
5518adantr 481 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → 𝐴𝐶)
56 undifixp 8770 . . . . . . . 8 (((1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛) ∧ (2nd𝑧) ∈ X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) ∧ 𝐴𝐶) → ((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛))
5729, 54, 55, 56syl3anc 1370 . . . . . . 7 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛))
58 ixpfn 8739 . . . . . . 7 (((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛) → ((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶)
5957, 58syl 17 . . . . . 6 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶)
60 dffn5 6867 . . . . . 6 (((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶 ↔ ((1st𝑧) ∪ (2nd𝑧)) = (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)))
6159, 60sylib 217 . . . . 5 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) = (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)))
6261mpteq2dva 5187 . . . 4 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))))
638, 62eqtrid 2789 . . 3 (𝜑𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))))
64 ptunhmeo.j . . . 4 𝐽 = (∏t𝐹)
65 pttop 22805 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → (∏t‘(𝐹𝐴)) ∈ Top)
6619, 21, 65syl2anc 584 . . . . . . 7 (𝜑 → (∏t‘(𝐹𝐴)) ∈ Top)
6722, 66eqeltrid 2842 . . . . . 6 (𝜑𝐾 ∈ Top)
6826toptopon 22138 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑋))
6967, 68sylib 217 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑋))
70 pttop 22805 . . . . . . . 8 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → (∏t‘(𝐹𝐵)) ∈ Top)
7144, 45, 70syl2anc 584 . . . . . . 7 (𝜑 → (∏t‘(𝐹𝐵)) ∈ Top)
7246, 71eqeltrid 2842 . . . . . 6 (𝜑𝐿 ∈ Top)
7350toptopon 22138 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘𝑌))
7472, 73sylib 217 . . . . 5 (𝜑𝐿 ∈ (TopOn‘𝑌))
75 txtopon 22814 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
7669, 74, 75syl2anc 584 . . . 4 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
7717eleq2d 2823 . . . . . . 7 (𝜑 → (𝑘𝐶𝑘 ∈ (𝐴𝐵)))
7877biimpa 477 . . . . . 6 ((𝜑𝑘𝐶) → 𝑘 ∈ (𝐴𝐵))
79 elun 4094 . . . . . 6 (𝑘 ∈ (𝐴𝐵) ↔ (𝑘𝐴𝑘𝐵))
8078, 79sylib 217 . . . . 5 ((𝜑𝑘𝐶) → (𝑘𝐴𝑘𝐵))
81 ixpfn 8739 . . . . . . . . . . 11 ((1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛) → (1st𝑧) Fn 𝐴)
8229, 81syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
8382adantlr 712 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
8451adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛𝐵 (𝐹𝑛) = 𝑌)
8531, 84eleqtrrd 2841 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ X𝑛𝐵 (𝐹𝑛))
86 ixpfn 8739 . . . . . . . . . . 11 ((2nd𝑧) ∈ X𝑛𝐵 (𝐹𝑛) → (2nd𝑧) Fn 𝐵)
8785, 86syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
8887adantlr 712 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
8933ad2antrr 723 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴𝐵) = ∅)
90 simplr 766 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘𝐴)
91 fvun1 6898 . . . . . . . . 9 (((1st𝑧) Fn 𝐴 ∧ (2nd𝑧) Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑘𝐴)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((1st𝑧)‘𝑘))
9283, 88, 89, 90, 91syl112anc 1373 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((1st𝑧)‘𝑘))
9392mpteq2dva 5187 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧)‘𝑘)))
9476adantr 481 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
954mpompt 7428 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
9669adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐾 ∈ (TopOn‘𝑋))
9774adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐿 ∈ (TopOn‘𝑌))
9896, 97cnmpt1st 22891 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
9995, 98eqeltrid 2842 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
10019adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐴 ∈ V)
10121adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → (𝐹𝐴):𝐴⟶Top)
102 simpr 485 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝑘𝐴)
10326, 22ptpjcn 22834 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top ∧ 𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn ((𝐹𝐴)‘𝑘)))
104100, 101, 102, 103syl3anc 1370 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn ((𝐹𝐴)‘𝑘)))
105 fvres 6830 . . . . . . . . . . 11 (𝑘𝐴 → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
106105adantl 482 . . . . . . . . . 10 ((𝜑𝑘𝐴) → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
107106oveq2d 7331 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝐾 Cn ((𝐹𝐴)‘𝑘)) = (𝐾 Cn (𝐹𝑘)))
108104, 107eleqtrd 2840 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn (𝐹𝑘)))
109 fveq1 6810 . . . . . . . 8 (𝑓 = (1st𝑧) → (𝑓𝑘) = ((1st𝑧)‘𝑘))
11094, 99, 96, 108, 109cnmpt11 22886 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
11193, 110eqeltrd 2838 . . . . . 6 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
11282adantlr 712 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
11387adantlr 712 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
11433ad2antrr 723 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴𝐵) = ∅)
115 simplr 766 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘𝐵)
116 fvun2 6899 . . . . . . . . 9 (((1st𝑧) Fn 𝐴 ∧ (2nd𝑧) Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑘𝐵)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((2nd𝑧)‘𝑘))
117112, 113, 114, 115, 116syl112anc 1373 . . . . . . . 8 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((2nd𝑧)‘𝑘))
118117mpteq2dva 5187 . . . . . . 7 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd𝑧)‘𝑘)))
11976adantr 481 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
1205mpompt 7428 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
12169adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐾 ∈ (TopOn‘𝑋))
12274adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐿 ∈ (TopOn‘𝑌))
123121, 122cnmpt2nd 22892 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
124120, 123eqeltrid 2842 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
12544adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐵 ∈ V)
12645adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → (𝐹𝐵):𝐵⟶Top)
127 simpr 485 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝑘𝐵)
12850, 46ptpjcn 22834 . . . . . . . . . 10 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top ∧ 𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn ((𝐹𝐵)‘𝑘)))
129125, 126, 127, 128syl3anc 1370 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn ((𝐹𝐵)‘𝑘)))
130 fvres 6830 . . . . . . . . . . 11 (𝑘𝐵 → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
131130adantl 482 . . . . . . . . . 10 ((𝜑𝑘𝐵) → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
132131oveq2d 7331 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝐿 Cn ((𝐹𝐵)‘𝑘)) = (𝐿 Cn (𝐹𝑘)))
133129, 132eleqtrd 2840 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn (𝐹𝑘)))
134 fveq1 6810 . . . . . . . 8 (𝑓 = (2nd𝑧) → (𝑓𝑘) = ((2nd𝑧)‘𝑘))
135119, 124, 122, 133, 134cnmpt11 22886 . . . . . . 7 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
136118, 135eqeltrd 2838 . . . . . 6 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
137111, 136jaodan 955 . . . . 5 ((𝜑 ∧ (𝑘𝐴𝑘𝐵)) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
13880, 137syldan 591 . . . 4 ((𝜑𝑘𝐶) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
13964, 76, 15, 20, 138ptcn 22850 . . 3 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
14063, 139eqeltrd 2838 . 2 (𝜑𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
14126, 50, 64, 22, 46, 1, 15, 20, 17, 33ptuncnv 23030 . . 3 (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))
142 pttop 22805 . . . . . . 7 ((𝐶𝑉𝐹:𝐶⟶Top) → (∏t𝐹) ∈ Top)
14315, 20, 142syl2anc 584 . . . . . 6 (𝜑 → (∏t𝐹) ∈ Top)
14464, 143eqeltrid 2842 . . . . 5 (𝜑𝐽 ∈ Top)
145 eqid 2737 . . . . . 6 𝐽 = 𝐽
146145toptopon 22138 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
147144, 146sylib 217 . . . 4 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
148145, 64, 22ptrescn 22862 . . . . 5 ((𝐶𝑉𝐹:𝐶⟶Top ∧ 𝐴𝐶) → (𝑧 𝐽 ↦ (𝑧𝐴)) ∈ (𝐽 Cn 𝐾))
14915, 20, 18, 148syl3anc 1370 . . . 4 (𝜑 → (𝑧 𝐽 ↦ (𝑧𝐴)) ∈ (𝐽 Cn 𝐾))
150145, 64, 46ptrescn 22862 . . . . 5 ((𝐶𝑉𝐹:𝐶⟶Top ∧ 𝐵𝐶) → (𝑧 𝐽 ↦ (𝑧𝐵)) ∈ (𝐽 Cn 𝐿))
15115, 20, 43, 150syl3anc 1370 . . . 4 (𝜑 → (𝑧 𝐽 ↦ (𝑧𝐵)) ∈ (𝐽 Cn 𝐿))
152147, 149, 151cnmpt1t 22888 . . 3 (𝜑 → (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
153141, 152eqeltrd 2838 . 2 (𝜑𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
154 ishmeo 22982 . 2 (𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽) ↔ (𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽) ∧ 𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿))))
155140, 153, 154sylanbrc 583 1 (𝜑𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1540  wcel 2105  Vcvv 3441  cdif 3894  cun 3895  cin 3896  wss 3897  c0 4267  cop 4577   cuni 4850  cmpt 5170   × cxp 5605  ccnv 5606  cres 5609   Fn wfn 6460  wf 6461  cfv 6465  (class class class)co 7315  cmpo 7317  1st c1st 7874  2nd c2nd 7875  Xcixp 8733  tcpt 17219  Topctop 22114  TopOnctopon 22131   Cn ccn 22447   ×t ctx 22783  Homeochmeo 22976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-ov 7318  df-oprab 7319  df-mpo 7320  df-om 7758  df-1st 7876  df-2nd 7877  df-1o 8344  df-er 8546  df-map 8665  df-ixp 8734  df-en 8782  df-dom 8783  df-fin 8785  df-fi 9240  df-topgen 17224  df-pt 17225  df-top 22115  df-topon 22132  df-bases 22168  df-cn 22450  df-cnp 22451  df-tx 22785  df-hmeo 22978
This theorem is referenced by:  xpstopnlem1  23032  ptcmpfi  23036
  Copyright terms: Public domain W3C validator