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Theorem txsconn 35246
Description: The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
Assertion
Ref Expression
txsconn ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)

Proof of Theorem txsconn
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sconnpconn 35232 . . 3 (𝑅 ∈ SConn → 𝑅 ∈ PConn)
2 sconnpconn 35232 . . 3 (𝑆 ∈ SConn → 𝑆 ∈ PConn)
3 txpconn 35237 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
41, 2, 3syl2an 596 . 2 ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ PConn)
5 simpll 767 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ SConn)
6 simprl 771 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
7 sconntop 35233 . . . . . . . . . . . . 13 (𝑅 ∈ SConn → 𝑅 ∈ Top)
87ad2antrr 726 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ Top)
9 eqid 2737 . . . . . . . . . . . . 13 𝑅 = 𝑅
109toptopon 22923 . . . . . . . . . . . 12 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
118, 10sylib 218 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ (TopOn‘ 𝑅))
12 sconntop 35233 . . . . . . . . . . . . 13 (𝑆 ∈ SConn → 𝑆 ∈ Top)
1312ad2antlr 727 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ Top)
14 eqid 2737 . . . . . . . . . . . . 13 𝑆 = 𝑆
1514toptopon 22923 . . . . . . . . . . . 12 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
1613, 15sylib 218 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ (TopOn‘ 𝑆))
17 tx1cn 23617 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
1811, 16, 17syl2anc 584 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
19 cnco 23274 . . . . . . . . . 10 ((𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅))
206, 18, 19syl2anc 584 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅))
21 simprr 773 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) = (𝑓‘1))
2221fveq2d 6910 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
23 iitopon 24905 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
2423a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → II ∈ (TopOn‘(0[,]1)))
25 txtopon 23599 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
2611, 16, 25syl2anc 584 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
27 cnf2 23257 . . . . . . . . . . . 12 ((II ∈ (TopOn‘(0[,]1)) ∧ (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)) ∧ 𝑓 ∈ (II Cn (𝑅 ×t 𝑆))) → 𝑓:(0[,]1)⟶( 𝑅 × 𝑆))
2824, 26, 6, 27syl3anc 1373 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶( 𝑅 × 𝑆))
29 0elunit 13509 . . . . . . . . . . 11 0 ∈ (0[,]1)
30 fvco3 7008 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
3128, 29, 30sylancl 586 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
32 1elunit 13510 . . . . . . . . . . 11 1 ∈ (0[,]1)
33 fvco3 7008 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
3428, 32, 33sylancl 586 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
3522, 31, 343eqtr4d 2787 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1))
36 sconnpht 35234 . . . . . . . . 9 ((𝑅 ∈ SConn ∧ ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1)) → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)( ≃ph𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))
375, 20, 35, 36syl3anc 1373 . . . . . . . 8 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)( ≃ph𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))
38 isphtpc 25026 . . . . . . . 8 (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)( ≃ph𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}) ↔ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ ((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑅) ∧ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
3937, 38sylib 218 . . . . . . 7 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ ((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑅) ∧ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
4039simp3d 1145 . . . . . 6 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅)
41 n0 4353 . . . . . 6 ((((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))
4240, 41sylib 218 . . . . 5 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ∃𝑔 𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))
43 simplr 769 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ SConn)
44 tx2cn 23618 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
4511, 16, 44syl2anc 584 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
46 cnco 23274 . . . . . . . . . 10 ((𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
476, 45, 46syl2anc 584 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
4821fveq2d 6910 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
49 fvco3 7008 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
5028, 29, 49sylancl 586 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
51 fvco3 7008 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
5228, 32, 51sylancl 586 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
5348, 50, 523eqtr4d 2787 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1))
54 sconnpht 35234 . . . . . . . . 9 ((𝑆 ∈ SConn ∧ ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1)) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)( ≃ph𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))
5543, 47, 53, 54syl3anc 1373 . . . . . . . 8 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)( ≃ph𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))
56 isphtpc 25026 . . . . . . . 8 (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)( ≃ph𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}) ↔ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ ((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑆) ∧ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
5755, 56sylib 218 . . . . . . 7 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ ((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑆) ∧ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
5857simp3d 1145 . . . . . 6 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅)
59 n0 4353 . . . . . 6 ((((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅ ↔ ∃ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))
6058, 59sylib 218 . . . . 5 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ∃ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))
61 exdistrv 1955 . . . . . 6 (∃𝑔(𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))) ↔ (∃𝑔 𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∃ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))))
628adantr 480 . . . . . . . . 9 ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))) → 𝑅 ∈ Top)
6313adantr 480 . . . . . . . . 9 ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))) → 𝑆 ∈ Top)
646adantr 480 . . . . . . . . 9 ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
65 eqid 2737 . . . . . . . . 9 ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)
66 eqid 2737 . . . . . . . . 9 ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) = ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)
67 simprl 771 . . . . . . . . 9 ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))) → 𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))
68 simprr 773 . . . . . . . . 9 ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))) → ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))
6962, 63, 64, 65, 66, 67, 68txsconnlem 35245 . . . . . . . 8 ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))
7069ex 412 . . . . . . 7 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
7170exlimdvv 1934 . . . . . 6 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (∃𝑔(𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
7261, 71biimtrrid 243 . . . . 5 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((∃𝑔 𝑔 ∈ (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)})) ∧ ∃ ∈ (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
7342, 60, 72mp2and 699 . . . 4 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))
7473expr 456 . . 3 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ 𝑓 ∈ (II Cn (𝑅 ×t 𝑆))) → ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
7574ralrimiva 3146 . 2 ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → ∀𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
76 issconn 35231 . 2 ((𝑅 ×t 𝑆) ∈ SConn ↔ ((𝑅 ×t 𝑆) ∈ PConn ∧ ∀𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))))
774, 75, 76sylanbrc 583 1 ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  c0 4333  {csn 4626   cuni 4907   class class class wbr 5143   × cxp 5683  cres 5687  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  0cc0 11155  1c1 11156  [,]cicc 13390  Topctop 22899  TopOnctopon 22916   Cn ccn 23232   ×t ctx 23568  IIcii 24901  PHtpycphtpy 25000  phcphtpc 25001  PConncpconn 35224  SConncsconn 35225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-icc 13394  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-cnp 23236  df-tx 23570  df-ii 24903  df-htpy 25002  df-phtpy 25003  df-phtpc 25024  df-pconn 35226  df-sconn 35227
This theorem is referenced by: (None)
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