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Theorem txsconnlem 32561
Description: Lemma for txsconn 32562. (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 5591 . . 3 ((0[,]1) × {(𝐹‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐹‘0))
3 iitopon 23482 . . . . 5 II ∈ (TopOn‘(0[,]1))
43a1i 11 . . . 4 (𝜑 → II ∈ (TopOn‘(0[,]1)))
5 txsconn.1 . . . . . 6 (𝜑𝑅 ∈ Top)
6 eqid 2822 . . . . . . 7 𝑅 = 𝑅
76toptopon 21520 . . . . . 6 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
85, 7sylib 221 . . . . 5 (𝜑𝑅 ∈ (TopOn‘ 𝑅))
9 txsconn.2 . . . . . 6 (𝜑𝑆 ∈ Top)
10 eqid 2822 . . . . . . 7 𝑆 = 𝑆
1110toptopon 21520 . . . . . 6 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
129, 11sylib 221 . . . . 5 (𝜑𝑆 ∈ (TopOn‘ 𝑆))
13 txtopon 22194 . . . . 5 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
148, 12, 13syl2anc 587 . . . 4 (𝜑 → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
15 cnf2 21852 . . . . . 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 1368 . . . . 5 (𝜑𝐹:(0[,]1)⟶( 𝑅 × 𝑆))
17 0elunit 12847 . . . . 5 0 ∈ (0[,]1)
18 ffvelrn 6831 . . . . 5 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
1916, 17, 18sylancl 589 . . . 4 (𝜑 → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
204, 14, 19cnmptc 22265 . . 3 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐹‘0)) ∈ (II Cn (𝑅 ×t 𝑆)))
212, 20eqeltrid 2918 . 2 (𝜑 → ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)))
22 txsconn.5 . . . . . . . . . . 11 𝐴 = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
23 tx1cn 22212 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
248, 12, 23syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
25 cnco 21869 . . . . . . . . . . . 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 587 . . . . . . . . . . 11 (𝜑 → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
2722, 26eqeltrid 2918 . . . . . . . . . 10 (𝜑𝐴 ∈ (II Cn 𝑅))
28 fconstmpt 5591 . . . . . . . . . . 11 ((0[,]1) × {(𝐴‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐴‘0))
29 iiuni 23484 . . . . . . . . . . . . . . 15 (0[,]1) = II
3029, 6cnf 21849 . . . . . . . . . . . . . 14 (𝐴 ∈ (II Cn 𝑅) → 𝐴:(0[,]1)⟶ 𝑅)
3127, 30syl 17 . . . . . . . . . . . . 13 (𝜑𝐴:(0[,]1)⟶ 𝑅)
32 ffvelrn 6831 . . . . . . . . . . . . 13 ((𝐴:(0[,]1)⟶ 𝑅 ∧ 0 ∈ (0[,]1)) → (𝐴‘0) ∈ 𝑅)
3331, 17, 32sylancl 589 . . . . . . . . . . . 12 (𝜑 → (𝐴‘0) ∈ 𝑅)
344, 8, 33cnmptc 22265 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐴‘0)) ∈ (II Cn 𝑅))
3528, 34eqeltrid 2918 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐴‘0)}) ∈ (II Cn 𝑅))
3627, 35phtpycn 23586 . . . . . . . . 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 3943 . . . . . . . 8 (𝜑𝐺 ∈ ((II ×t II) Cn 𝑅))
39 iitop 23483 . . . . . . . . . 10 II ∈ Top
4039, 39, 29, 29txunii 22196 . . . . . . . . 9 ((0[,]1) × (0[,]1)) = (II ×t II)
4140, 6cnf 21849 . . . . . . . 8 (𝐺 ∈ ((II ×t II) Cn 𝑅) → 𝐺:((0[,]1) × (0[,]1))⟶ 𝑅)
42 ffn 6494 . . . . . . . 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 7266 . . . . . . 7 (𝐺 Fn ((0[,]1) × (0[,]1)) ↔ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
4543, 44sylib 221 . . . . . 6 (𝜑𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
4645, 38eqeltrrd 2915 . . . . 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 22213 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
498, 12, 48syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
50 cnco 21869 . . . . . . . . . . . 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 587 . . . . . . . . . . 11 (𝜑 → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
5247, 51eqeltrid 2918 . . . . . . . . . 10 (𝜑𝐵 ∈ (II Cn 𝑆))
53 fconstmpt 5591 . . . . . . . . . . 11 ((0[,]1) × {(𝐵‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐵‘0))
5429, 10cnf 21849 . . . . . . . . . . . . . 14 (𝐵 ∈ (II Cn 𝑆) → 𝐵:(0[,]1)⟶ 𝑆)
5552, 54syl 17 . . . . . . . . . . . . 13 (𝜑𝐵:(0[,]1)⟶ 𝑆)
56 ffvelrn 6831 . . . . . . . . . . . . 13 ((𝐵:(0[,]1)⟶ 𝑆 ∧ 0 ∈ (0[,]1)) → (𝐵‘0) ∈ 𝑆)
5755, 17, 56sylancl 589 . . . . . . . . . . . 12 (𝜑 → (𝐵‘0) ∈ 𝑆)
584, 12, 57cnmptc 22265 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐵‘0)) ∈ (II Cn 𝑆))
5953, 58eqeltrid 2918 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐵‘0)}) ∈ (II Cn 𝑆))
6052, 59phtpycn 23586 . . . . . . . . 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 3943 . . . . . . . 8 (𝜑𝐻 ∈ ((II ×t II) Cn 𝑆))
6340, 10cnf 21849 . . . . . . . 8 (𝐻 ∈ ((II ×t II) Cn 𝑆) → 𝐻:((0[,]1) × (0[,]1))⟶ 𝑆)
64 ffn 6494 . . . . . . . 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 7266 . . . . . . 7 (𝐻 Fn ((0[,]1) × (0[,]1)) ↔ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
6765, 66sylib 221 . . . . . 6 (𝜑𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
6867, 62eqeltrrd 2915 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)) ∈ ((II ×t II) Cn 𝑆))
694, 4, 46, 68cnmpt2t 22276 . . . 4 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ ((II ×t II) Cn (𝑅 ×t 𝑆)))
7027, 35phtpyhtpy 23585 . . . . . . . . . 10 (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
7170, 37sseldd 3943 . . . . . . . . 9 (𝜑𝐺 ∈ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
724, 27, 35, 71htpyi 23577 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐺0) = (𝐴𝑠) ∧ (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠)))
7372simpld 498 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐴𝑠))
7422fveq1i 6653 . . . . . . . 8 (𝐴𝑠) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠)
75 fvco3 6742 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
7616, 75sylan 583 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
7774, 76syl5eq 2869 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
78 ffvelrn 6831 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (𝐹𝑠) ∈ ( 𝑅 × 𝑆))
7916, 78sylan 583 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑠) ∈ ( 𝑅 × 𝑆))
80 fvres 6671 . . . . . . . 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 2861 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (1st ‘(𝐹𝑠)))
8352, 59phtpyhtpy 23585 . . . . . . . . . 10 (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
8483, 61sseldd 3943 . . . . . . . . 9 (𝜑𝐻 ∈ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
854, 52, 59, 84htpyi 23577 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐻0) = (𝐵𝑠) ∧ (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠)))
8685simpld 498 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐵𝑠))
8747fveq1i 6653 . . . . . . . 8 (𝐵𝑠) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠)
88 fvco3 6742 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
8916, 88sylan 583 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
9087, 89syl5eq 2869 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
91 fvres 6671 . . . . . . . 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 2861 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (2nd ‘(𝐹𝑠)))
9482, 93opeq12d 4786 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
95 simpr 488 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
96 oveq12 7149 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 0) → (𝑥𝐺𝑦) = (𝑠𝐺0))
97 oveq12 7149 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 0) → (𝑥𝐻𝑦) = (𝑠𝐻0))
9896, 97opeq12d 4786 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 0) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
99 eqid 2822 . . . . . . 7 (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)
100 opex 5333 . . . . . . 7 ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ ∈ V
10198, 99, 100ovmpoa 7289 . . . . . 6 ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
10295, 17, 101sylancl 589 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
103 1st2nd2 7714 . . . . . 6 ((𝐹𝑠) ∈ ( 𝑅 × 𝑆) → (𝐹𝑠) = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
10479, 103syl 17 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑠) = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
10594, 102, 1043eqtr4d 2867 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = (𝐹𝑠))
10672simprd 499 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠))
107 fvex 6665 . . . . . . . . 9 (𝐴‘0) ∈ V
108107fvconst2 6948 . . . . . . . 8 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
109108adantl 485 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
11022fveq1i 6653 . . . . . . . . 9 (𝐴‘0) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0)
111 fvco3 6742 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
11216, 17, 111sylancl 589 . . . . . . . . . 10 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
113 fvres 6671 . . . . . . . . . . 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 2857 . . . . . . . . 9 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = (1st ‘(𝐹‘0)))
116110, 115syl5eq 2869 . . . . . . . 8 (𝜑 → (𝐴‘0) = (1st ‘(𝐹‘0)))
117116adantr 484 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴‘0) = (1st ‘(𝐹‘0)))
118106, 109, 1173eqtrd 2861 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (1st ‘(𝐹‘0)))
11985simprd 499 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠))
120 fvex 6665 . . . . . . . . 9 (𝐵‘0) ∈ V
121120fvconst2 6948 . . . . . . . 8 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
122121adantl 485 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
12347fveq1i 6653 . . . . . . . . 9 (𝐵‘0) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0)
124 fvco3 6742 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
12516, 17, 124sylancl 589 . . . . . . . . . 10 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
126 fvres 6671 . . . . . . . . . . 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 2857 . . . . . . . . 9 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = (2nd ‘(𝐹‘0)))
129123, 128syl5eq 2869 . . . . . . . 8 (𝜑 → (𝐵‘0) = (2nd ‘(𝐹‘0)))
130129adantr 484 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵‘0) = (2nd ‘(𝐹‘0)))
131119, 122, 1303eqtrd 2861 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (2nd ‘(𝐹‘0)))
132118, 131opeq12d 4786 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
133 1elunit 12848 . . . . . 6 1 ∈ (0[,]1)
134 oveq12 7149 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 1) → (𝑥𝐺𝑦) = (𝑠𝐺1))
135 oveq12 7149 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 1) → (𝑥𝐻𝑦) = (𝑠𝐻1))
136134, 135opeq12d 4786 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 1) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
137 opex 5333 . . . . . . 7 ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ ∈ V
138136, 99, 137ovmpoa 7289 . . . . . 6 ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
13995, 133, 138sylancl 589 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
140 fvex 6665 . . . . . . . 8 (𝐹‘0) ∈ V
141140fvconst2 6948 . . . . . . 7 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
142141adantl 485 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
14319adantr 484 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
144 1st2nd2 7714 . . . . . . 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 2857 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
147132, 139, 1463eqtr4d 2867 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = (((0[,]1) × {(𝐹‘0)})‘𝑠))
14827, 35, 37phtpyi 23587 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐺𝑠) = (𝐴‘0) ∧ (1𝐺𝑠) = (𝐴‘1)))
149148simpld 498 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐴‘0))
150149, 117eqtrd 2857 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (1st ‘(𝐹‘0)))
15152, 59, 61phtpyi 23587 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐵‘0) ∧ (1𝐻𝑠) = (𝐵‘1)))
152151simpld 498 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐵‘0))
153152, 130eqtrd 2857 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (2nd ‘(𝐹‘0)))
154150, 153opeq12d 4786 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
155 oveq12 7149 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (0𝐺𝑠))
156 oveq12 7149 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (0𝐻𝑠))
157155, 156opeq12d 4786 . . . . . . 7 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
158 opex 5333 . . . . . . 7 ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ ∈ V
159157, 99, 158ovmpoa 7289 . . . . . 6 ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
16017, 95, 159sylancr 590 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
161154, 160, 1453eqtr4d 2867 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘0))
162148simprd 499 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐴‘1))
16322fveq1i 6653 . . . . . . . . . 10 (𝐴‘1) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1)
164 fvco3 6742 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
16516, 133, 164sylancl 589 . . . . . . . . . 10 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
166163, 165syl5eq 2869 . . . . . . . . 9 (𝜑 → (𝐴‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
167 ffvelrn 6831 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
16816, 133, 167sylancl 589 . . . . . . . . . 10 (𝜑 → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
169 fvres 6671 . . . . . . . . . 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 2857 . . . . . . . 8 (𝜑 → (𝐴‘1) = (1st ‘(𝐹‘1)))
172171adantr 484 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴‘1) = (1st ‘(𝐹‘1)))
173162, 172eqtrd 2857 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (1st ‘(𝐹‘1)))
174151simprd 499 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐵‘1))
17547fveq1i 6653 . . . . . . . . . 10 (𝐵‘1) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1)
176 fvco3 6742 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
17716, 133, 176sylancl 589 . . . . . . . . . 10 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
178175, 177syl5eq 2869 . . . . . . . . 9 (𝜑 → (𝐵‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
179 fvres 6671 . . . . . . . . . 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 2857 . . . . . . . 8 (𝜑 → (𝐵‘1) = (2nd ‘(𝐹‘1)))
182181adantr 484 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵‘1) = (2nd ‘(𝐹‘1)))
183174, 182eqtrd 2857 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (2nd ‘(𝐹‘1)))
184173, 183opeq12d 4786 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
185 oveq12 7149 . . . . . . . 8 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (1𝐺𝑠))
186 oveq12 7149 . . . . . . . 8 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (1𝐻𝑠))
187185, 186opeq12d 4786 . . . . . . 7 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
188 opex 5333 . . . . . . 7 ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ ∈ V
189187, 99, 188ovmpoa 7289 . . . . . 6 ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
190133, 95, 189sylancr 590 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
191168adantr 484 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
192 1st2nd2 7714 . . . . . 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 2867 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘1))
1951, 21, 69, 105, 147, 161, 194isphtpy2d 23590 . . 3 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})))
196195ne0d 4273 . 2 (𝜑 → (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅)
197 isphtpc 23597 . 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 1340 1 (𝜑𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wne 3011  c0 4265  {csn 4539  cop 4545   cuni 4813   class class class wbr 5042  cmpt 5122   × cxp 5530  cres 5534  ccom 5536   Fn wfn 6329  wf 6330  cfv 6334  (class class class)co 7140  cmpo 7142  1st c1st 7673  2nd c2nd 7674  0cc0 10526  1c1 10527  [,]cicc 12729  Topctop 21496  TopOnctopon 21513   Cn ccn 21827   ×t ctx 22163  IIcii 23478   Htpy chtpy 23570  PHtpycphtpy 23571  phcphtpc 23572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-inf 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-icc 12733  df-seq 13365  df-exp 13426  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-topgen 16708  df-psmet 20081  df-xmet 20082  df-met 20083  df-bl 20084  df-mopn 20085  df-top 21497  df-topon 21514  df-bases 21549  df-cn 21830  df-cnp 21831  df-tx 22165  df-ii 23480  df-htpy 23573  df-phtpy 23574  df-phtpc 23595
This theorem is referenced by:  txsconn  32562
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