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Theorem txsconnlem 31560
Description: Lemma for txsconn 31561. (Contributed by Mario Carneiro, 9-Mar-2015.)
Hypotheses
Ref Expression
txsconn.1 (𝜑𝑅 ∈ Top)
txsconn.2 (𝜑𝑆 ∈ Top)
txsconn.3 (𝜑𝐹 ∈ (II Cn (𝑅 ×t 𝑆)))
txsconn.5 𝐴 = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
txsconn.6 𝐵 = ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
txsconn.7 (𝜑𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})))
txsconn.8 (𝜑𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})))
Assertion
Ref Expression
txsconnlem (𝜑𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))

Proof of Theorem txsconnlem
Dummy variables 𝑥 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 txsconn.3 . 2 (𝜑𝐹 ∈ (II Cn (𝑅 ×t 𝑆)))
2 fconstmpt 5302 . . 3 ((0[,]1) × {(𝐹‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐹‘0))
3 iitopon 22902 . . . . 5 II ∈ (TopOn‘(0[,]1))
43a1i 11 . . . 4 (𝜑 → II ∈ (TopOn‘(0[,]1)))
5 txsconn.1 . . . . . 6 (𝜑𝑅 ∈ Top)
6 eqid 2771 . . . . . . 7 𝑅 = 𝑅
76toptopon 20942 . . . . . 6 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
85, 7sylib 208 . . . . 5 (𝜑𝑅 ∈ (TopOn‘ 𝑅))
9 txsconn.2 . . . . . 6 (𝜑𝑆 ∈ Top)
10 eqid 2771 . . . . . . 7 𝑆 = 𝑆
1110toptopon 20942 . . . . . 6 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
129, 11sylib 208 . . . . 5 (𝜑𝑆 ∈ (TopOn‘ 𝑆))
13 txtopon 21615 . . . . 5 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
148, 12, 13syl2anc 573 . . . 4 (𝜑 → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
15 cnf2 21274 . . . . . 6 ((II ∈ (TopOn‘(0[,]1)) ∧ (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)) ∧ 𝐹 ∈ (II Cn (𝑅 ×t 𝑆))) → 𝐹:(0[,]1)⟶( 𝑅 × 𝑆))
164, 14, 1, 15syl3anc 1476 . . . . 5 (𝜑𝐹:(0[,]1)⟶( 𝑅 × 𝑆))
17 0elunit 12497 . . . . 5 0 ∈ (0[,]1)
18 ffvelrn 6502 . . . . 5 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
1916, 17, 18sylancl 574 . . . 4 (𝜑 → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
204, 14, 19cnmptc 21686 . . 3 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐹‘0)) ∈ (II Cn (𝑅 ×t 𝑆)))
212, 20syl5eqel 2854 . 2 (𝜑 → ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)))
22 txsconn.5 . . . . . . . . . . 11 𝐴 = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
23 tx1cn 21633 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
248, 12, 23syl2anc 573 . . . . . . . . . . . 12 (𝜑 → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
25 cnco 21291 . . . . . . . . . . . 12 ((𝐹 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
261, 24, 25syl2anc 573 . . . . . . . . . . 11 (𝜑 → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
2722, 26syl5eqel 2854 . . . . . . . . . 10 (𝜑𝐴 ∈ (II Cn 𝑅))
28 fconstmpt 5302 . . . . . . . . . . 11 ((0[,]1) × {(𝐴‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐴‘0))
29 iiuni 22904 . . . . . . . . . . . . . . 15 (0[,]1) = II
3029, 6cnf 21271 . . . . . . . . . . . . . 14 (𝐴 ∈ (II Cn 𝑅) → 𝐴:(0[,]1)⟶ 𝑅)
3127, 30syl 17 . . . . . . . . . . . . 13 (𝜑𝐴:(0[,]1)⟶ 𝑅)
32 ffvelrn 6502 . . . . . . . . . . . . 13 ((𝐴:(0[,]1)⟶ 𝑅 ∧ 0 ∈ (0[,]1)) → (𝐴‘0) ∈ 𝑅)
3331, 17, 32sylancl 574 . . . . . . . . . . . 12 (𝜑 → (𝐴‘0) ∈ 𝑅)
344, 8, 33cnmptc 21686 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐴‘0)) ∈ (II Cn 𝑅))
3528, 34syl5eqel 2854 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐴‘0)}) ∈ (II Cn 𝑅))
3627, 35phtpycn 23002 . . . . . . . . 9 (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ ((II ×t II) Cn 𝑅))
37 txsconn.7 . . . . . . . . 9 (𝜑𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})))
3836, 37sseldd 3753 . . . . . . . 8 (𝜑𝐺 ∈ ((II ×t II) Cn 𝑅))
39 iitop 22903 . . . . . . . . . 10 II ∈ Top
4039, 39, 29, 29txunii 21617 . . . . . . . . 9 ((0[,]1) × (0[,]1)) = (II ×t II)
4140, 6cnf 21271 . . . . . . . 8 (𝐺 ∈ ((II ×t II) Cn 𝑅) → 𝐺:((0[,]1) × (0[,]1))⟶ 𝑅)
42 ffn 6184 . . . . . . . 8 (𝐺:((0[,]1) × (0[,]1))⟶ 𝑅𝐺 Fn ((0[,]1) × (0[,]1)))
4338, 41, 423syl 18 . . . . . . 7 (𝜑𝐺 Fn ((0[,]1) × (0[,]1)))
44 fnov 6919 . . . . . . 7 (𝐺 Fn ((0[,]1) × (0[,]1)) ↔ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
4543, 44sylib 208 . . . . . 6 (𝜑𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
4645, 38eqeltrrd 2851 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)) ∈ ((II ×t II) Cn 𝑅))
47 txsconn.6 . . . . . . . . . . 11 𝐵 = ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
48 tx2cn 21634 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
498, 12, 48syl2anc 573 . . . . . . . . . . . 12 (𝜑 → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
50 cnco 21291 . . . . . . . . . . . 12 ((𝐹 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
511, 49, 50syl2anc 573 . . . . . . . . . . 11 (𝜑 → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
5247, 51syl5eqel 2854 . . . . . . . . . 10 (𝜑𝐵 ∈ (II Cn 𝑆))
53 fconstmpt 5302 . . . . . . . . . . 11 ((0[,]1) × {(𝐵‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐵‘0))
5429, 10cnf 21271 . . . . . . . . . . . . . 14 (𝐵 ∈ (II Cn 𝑆) → 𝐵:(0[,]1)⟶ 𝑆)
5552, 54syl 17 . . . . . . . . . . . . 13 (𝜑𝐵:(0[,]1)⟶ 𝑆)
56 ffvelrn 6502 . . . . . . . . . . . . 13 ((𝐵:(0[,]1)⟶ 𝑆 ∧ 0 ∈ (0[,]1)) → (𝐵‘0) ∈ 𝑆)
5755, 17, 56sylancl 574 . . . . . . . . . . . 12 (𝜑 → (𝐵‘0) ∈ 𝑆)
584, 12, 57cnmptc 21686 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐵‘0)) ∈ (II Cn 𝑆))
5953, 58syl5eqel 2854 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐵‘0)}) ∈ (II Cn 𝑆))
6052, 59phtpycn 23002 . . . . . . . . 9 (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ ((II ×t II) Cn 𝑆))
61 txsconn.8 . . . . . . . . 9 (𝜑𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})))
6260, 61sseldd 3753 . . . . . . . 8 (𝜑𝐻 ∈ ((II ×t II) Cn 𝑆))
6340, 10cnf 21271 . . . . . . . 8 (𝐻 ∈ ((II ×t II) Cn 𝑆) → 𝐻:((0[,]1) × (0[,]1))⟶ 𝑆)
64 ffn 6184 . . . . . . . 8 (𝐻:((0[,]1) × (0[,]1))⟶ 𝑆𝐻 Fn ((0[,]1) × (0[,]1)))
6562, 63, 643syl 18 . . . . . . 7 (𝜑𝐻 Fn ((0[,]1) × (0[,]1)))
66 fnov 6919 . . . . . . 7 (𝐻 Fn ((0[,]1) × (0[,]1)) ↔ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
6765, 66sylib 208 . . . . . 6 (𝜑𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
6867, 62eqeltrrd 2851 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)) ∈ ((II ×t II) Cn 𝑆))
694, 4, 46, 68cnmpt2t 21697 . . . 4 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ ((II ×t II) Cn (𝑅 ×t 𝑆)))
7027, 35phtpyhtpy 23001 . . . . . . . . . 10 (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
7170, 37sseldd 3753 . . . . . . . . 9 (𝜑𝐺 ∈ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
724, 27, 35, 71htpyi 22993 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐺0) = (𝐴𝑠) ∧ (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠)))
7372simpld 482 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐴𝑠))
7422fveq1i 6334 . . . . . . . 8 (𝐴𝑠) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠)
75 fvco3 6419 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
7616, 75sylan 569 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
7774, 76syl5eq 2817 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
78 ffvelrn 6502 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (𝐹𝑠) ∈ ( 𝑅 × 𝑆))
7916, 78sylan 569 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑠) ∈ ( 𝑅 × 𝑆))
80 fvres 6350 . . . . . . . 8 ((𝐹𝑠) ∈ ( 𝑅 × 𝑆) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)) = (1st ‘(𝐹𝑠)))
8179, 80syl 17 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)) = (1st ‘(𝐹𝑠)))
8273, 77, 813eqtrd 2809 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (1st ‘(𝐹𝑠)))
8352, 59phtpyhtpy 23001 . . . . . . . . . 10 (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
8483, 61sseldd 3753 . . . . . . . . 9 (𝜑𝐻 ∈ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
854, 52, 59, 84htpyi 22993 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐻0) = (𝐵𝑠) ∧ (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠)))
8685simpld 482 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐵𝑠))
8747fveq1i 6334 . . . . . . . 8 (𝐵𝑠) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠)
88 fvco3 6419 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
8916, 88sylan 569 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
9087, 89syl5eq 2817 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
91 fvres 6350 . . . . . . . 8 ((𝐹𝑠) ∈ ( 𝑅 × 𝑆) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)) = (2nd ‘(𝐹𝑠)))
9279, 91syl 17 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)) = (2nd ‘(𝐹𝑠)))
9386, 90, 923eqtrd 2809 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (2nd ‘(𝐹𝑠)))
9482, 93opeq12d 4548 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
95 simpr 471 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
96 oveq12 6805 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 0) → (𝑥𝐺𝑦) = (𝑠𝐺0))
97 oveq12 6805 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 0) → (𝑥𝐻𝑦) = (𝑠𝐻0))
9896, 97opeq12d 4548 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 0) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
99 eqid 2771 . . . . . . 7 (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)
100 opex 5061 . . . . . . 7 ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ ∈ V
10198, 99, 100ovmpt2a 6942 . . . . . 6 ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
10295, 17, 101sylancl 574 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
103 1st2nd2 7358 . . . . . 6 ((𝐹𝑠) ∈ ( 𝑅 × 𝑆) → (𝐹𝑠) = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
10479, 103syl 17 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑠) = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
10594, 102, 1043eqtr4d 2815 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = (𝐹𝑠))
10672simprd 483 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠))
107 fvex 6344 . . . . . . . . 9 (𝐴‘0) ∈ V
108107fvconst2 6616 . . . . . . . 8 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
109108adantl 467 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
11022fveq1i 6334 . . . . . . . . 9 (𝐴‘0) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0)
111 fvco3 6419 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
11216, 17, 111sylancl 574 . . . . . . . . . 10 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
113 fvres 6350 . . . . . . . . . . 11 ((𝐹‘0) ∈ ( 𝑅 × 𝑆) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)) = (1st ‘(𝐹‘0)))
11419, 113syl 17 . . . . . . . . . 10 (𝜑 → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)) = (1st ‘(𝐹‘0)))
115112, 114eqtrd 2805 . . . . . . . . 9 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = (1st ‘(𝐹‘0)))
116110, 115syl5eq 2817 . . . . . . . 8 (𝜑 → (𝐴‘0) = (1st ‘(𝐹‘0)))
117116adantr 466 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴‘0) = (1st ‘(𝐹‘0)))
118106, 109, 1173eqtrd 2809 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (1st ‘(𝐹‘0)))
11985simprd 483 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠))
120 fvex 6344 . . . . . . . . 9 (𝐵‘0) ∈ V
121120fvconst2 6616 . . . . . . . 8 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
122121adantl 467 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
12347fveq1i 6334 . . . . . . . . 9 (𝐵‘0) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0)
124 fvco3 6419 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
12516, 17, 124sylancl 574 . . . . . . . . . 10 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
126 fvres 6350 . . . . . . . . . . 11 ((𝐹‘0) ∈ ( 𝑅 × 𝑆) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)) = (2nd ‘(𝐹‘0)))
12719, 126syl 17 . . . . . . . . . 10 (𝜑 → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)) = (2nd ‘(𝐹‘0)))
128125, 127eqtrd 2805 . . . . . . . . 9 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = (2nd ‘(𝐹‘0)))
129123, 128syl5eq 2817 . . . . . . . 8 (𝜑 → (𝐵‘0) = (2nd ‘(𝐹‘0)))
130129adantr 466 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵‘0) = (2nd ‘(𝐹‘0)))
131119, 122, 1303eqtrd 2809 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (2nd ‘(𝐹‘0)))
132118, 131opeq12d 4548 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
133 1elunit 12498 . . . . . 6 1 ∈ (0[,]1)
134 oveq12 6805 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 1) → (𝑥𝐺𝑦) = (𝑠𝐺1))
135 oveq12 6805 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 1) → (𝑥𝐻𝑦) = (𝑠𝐻1))
136134, 135opeq12d 4548 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 1) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
137 opex 5061 . . . . . . 7 ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ ∈ V
138136, 99, 137ovmpt2a 6942 . . . . . 6 ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
13995, 133, 138sylancl 574 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
140 fvex 6344 . . . . . . . 8 (𝐹‘0) ∈ V
141140fvconst2 6616 . . . . . . 7 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
142141adantl 467 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
14319adantr 466 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
144 1st2nd2 7358 . . . . . . 7 ((𝐹‘0) ∈ ( 𝑅 × 𝑆) → (𝐹‘0) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
145143, 144syl 17 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘0) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
146142, 145eqtrd 2805 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
147132, 139, 1463eqtr4d 2815 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = (((0[,]1) × {(𝐹‘0)})‘𝑠))
14827, 35, 37phtpyi 23003 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐺𝑠) = (𝐴‘0) ∧ (1𝐺𝑠) = (𝐴‘1)))
149148simpld 482 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐴‘0))
150149, 117eqtrd 2805 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (1st ‘(𝐹‘0)))
15152, 59, 61phtpyi 23003 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐵‘0) ∧ (1𝐻𝑠) = (𝐵‘1)))
152151simpld 482 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐵‘0))
153152, 130eqtrd 2805 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (2nd ‘(𝐹‘0)))
154150, 153opeq12d 4548 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
155 oveq12 6805 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (0𝐺𝑠))
156 oveq12 6805 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (0𝐻𝑠))
157155, 156opeq12d 4548 . . . . . . 7 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
158 opex 5061 . . . . . . 7 ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ ∈ V
159157, 99, 158ovmpt2a 6942 . . . . . 6 ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
16017, 95, 159sylancr 575 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
161154, 160, 1453eqtr4d 2815 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘0))
162148simprd 483 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐴‘1))
16322fveq1i 6334 . . . . . . . . . 10 (𝐴‘1) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1)
164 fvco3 6419 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
16516, 133, 164sylancl 574 . . . . . . . . . 10 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
166163, 165syl5eq 2817 . . . . . . . . 9 (𝜑 → (𝐴‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
167 ffvelrn 6502 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
16816, 133, 167sylancl 574 . . . . . . . . . 10 (𝜑 → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
169 fvres 6350 . . . . . . . . . 10 ((𝐹‘1) ∈ ( 𝑅 × 𝑆) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)) = (1st ‘(𝐹‘1)))
170168, 169syl 17 . . . . . . . . 9 (𝜑 → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)) = (1st ‘(𝐹‘1)))
171166, 170eqtrd 2805 . . . . . . . 8 (𝜑 → (𝐴‘1) = (1st ‘(𝐹‘1)))
172171adantr 466 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴‘1) = (1st ‘(𝐹‘1)))
173162, 172eqtrd 2805 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (1st ‘(𝐹‘1)))
174151simprd 483 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐵‘1))
17547fveq1i 6334 . . . . . . . . . 10 (𝐵‘1) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1)
176 fvco3 6419 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
17716, 133, 176sylancl 574 . . . . . . . . . 10 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
178175, 177syl5eq 2817 . . . . . . . . 9 (𝜑 → (𝐵‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
179 fvres 6350 . . . . . . . . . 10 ((𝐹‘1) ∈ ( 𝑅 × 𝑆) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)) = (2nd ‘(𝐹‘1)))
180168, 179syl 17 . . . . . . . . 9 (𝜑 → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)) = (2nd ‘(𝐹‘1)))
181178, 180eqtrd 2805 . . . . . . . 8 (𝜑 → (𝐵‘1) = (2nd ‘(𝐹‘1)))
182181adantr 466 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵‘1) = (2nd ‘(𝐹‘1)))
183174, 182eqtrd 2805 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (2nd ‘(𝐹‘1)))
184173, 183opeq12d 4548 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
185 oveq12 6805 . . . . . . . 8 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (1𝐺𝑠))
186 oveq12 6805 . . . . . . . 8 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (1𝐻𝑠))
187185, 186opeq12d 4548 . . . . . . 7 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
188 opex 5061 . . . . . . 7 ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ ∈ V
189187, 99, 188ovmpt2a 6942 . . . . . 6 ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
190133, 95, 189sylancr 575 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
191168adantr 466 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
192 1st2nd2 7358 . . . . . 6 ((𝐹‘1) ∈ ( 𝑅 × 𝑆) → (𝐹‘1) = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
193191, 192syl 17 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘1) = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
194184, 190, 1933eqtr4d 2815 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘1))
1951, 21, 69, 105, 147, 161, 194isphtpy2d 23006 . . 3 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})))
196195ne0d 4070 . 2 (𝜑 → (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅)
197 isphtpc 23013 . 2 (𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}) ↔ (𝐹 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅))
1981, 21, 196, 197syl3anbrc 1428 1 (𝜑𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  c0 4063  {csn 4317  cop 4323   cuni 4575   class class class wbr 4787  cmpt 4864   × cxp 5248  cres 5252  ccom 5254   Fn wfn 6025  wf 6026  cfv 6030  (class class class)co 6796  cmpt2 6798  1st c1st 7317  2nd c2nd 7318  0cc0 10142  1c1 10143  [,]cicc 12383  Topctop 20918  TopOnctopon 20935   Cn ccn 21249   ×t ctx 21584  IIcii 22898   Htpy chtpy 22986  PHtpycphtpy 22987  phcphtpc 22988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-sup 8508  df-inf 8509  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-q 11997  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-icc 12387  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-topgen 16312  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-top 20919  df-topon 20936  df-bases 20971  df-cn 21252  df-cnp 21253  df-tx 21586  df-ii 22900  df-htpy 22989  df-phtpy 22990  df-phtpc 23011
This theorem is referenced by:  txsconn  31561
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