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Theorem txsconn 33203
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 33189 . . 3 (𝑅 ∈ SConn → 𝑅 ∈ PConn)
2 sconnpconn 33189 . . 3 (𝑆 ∈ SConn → 𝑆 ∈ PConn)
3 txpconn 33194 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
41, 2, 3syl2an 596 . 2 ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ PConn)
5 simpll 764 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ SConn)
6 simprl 768 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
7 sconntop 33190 . . . . . . . . . . . . 13 (𝑅 ∈ SConn → 𝑅 ∈ Top)
87ad2antrr 723 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ Top)
9 eqid 2738 . . . . . . . . . . . . 13 𝑅 = 𝑅
109toptopon 22066 . . . . . . . . . . . 12 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
118, 10sylib 217 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ (TopOn‘ 𝑅))
12 sconntop 33190 . . . . . . . . . . . . 13 (𝑆 ∈ SConn → 𝑆 ∈ Top)
1312ad2antlr 724 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ Top)
14 eqid 2738 . . . . . . . . . . . . 13 𝑆 = 𝑆
1514toptopon 22066 . . . . . . . . . . . 12 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
1613, 15sylib 217 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ (TopOn‘ 𝑆))
17 tx1cn 22760 . . . . . . . . . . 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 22417 . . . . . . . . . 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 770 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) = (𝑓‘1))
2221fveq2d 6778 . . . . . . . . . 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 24042 . . . . . . . . . . . . 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 22742 . . . . . . . . . . . . 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 22400 . . . . . . . . . . . 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 1370 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶( 𝑅 × 𝑆))
29 0elunit 13201 . . . . . . . . . . 11 0 ∈ (0[,]1)
30 fvco3 6867 . . . . . . . . . . 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 13202 . . . . . . . . . . 11 1 ∈ (0[,]1)
33 fvco3 6867 . . . . . . . . . . 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 2788 . . . . . . . . 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 33191 . . . . . . . . 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 1370 . . . . . . . 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 24157 . . . . . . . 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 217 . . . . . . 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 1143 . . . . . 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 4280 . . . . . 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 217 . . . . 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 766 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ SConn)
44 tx2cn 22761 . . . . . . . . . . 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 22417 . . . . . . . . . 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 6778 . . . . . . . . . 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 6867 . . . . . . . . . . 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 6867 . . . . . . . . . . 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 2788 . . . . . . . . 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 33191 . . . . . . . . 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 1370 . . . . . . . 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 24157 . . . . . . . 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 217 . . . . . . 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 1143 . . . . . 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 4280 . . . . . 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 217 . . . . 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 1959 . . . . . 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 481 . . . . . . . . 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 481 . . . . . . . . 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 481 . . . . . . . . 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 2738 . . . . . . . . 9 ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)
66 eqid 2738 . . . . . . . . 9 ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) = ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)
67 simprl 768 . . . . . . . . 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 770 . . . . . . . . 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 33202 . . . . . . . 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 413 . . . . . . 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 1937 . . . . . 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, 71syl5bir 242 . . . . 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 696 . . . 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 457 . . 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 3103 . 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 33188 . 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 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  wne 2943  wral 3064  c0 4256  {csn 4561   cuni 4839   class class class wbr 5074   × cxp 5587  cres 5591  ccom 5593  wf 6429  cfv 6433  (class class class)co 7275  1st c1st 7829  2nd c2nd 7830  0cc0 10871  1c1 10872  [,]cicc 13082  Topctop 22042  TopOnctopon 22059   Cn ccn 22375   ×t ctx 22711  IIcii 24038  PHtpycphtpy 24131  phcphtpc 24132  PConncpconn 33181  SConncsconn 33182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-icc 13086  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-cnp 22379  df-tx 22713  df-ii 24040  df-htpy 24133  df-phtpy 24134  df-phtpc 24155  df-pconn 33183  df-sconn 33184
This theorem is referenced by: (None)
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