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Theorem txsconn 30984
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 30970 . . 3 (𝑅 ∈ SConn → 𝑅 ∈ PConn)
2 sconnpconn 30970 . . 3 (𝑆 ∈ SConn → 𝑆 ∈ PConn)
3 txpconn 30975 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
41, 2, 3syl2an 494 . 2 ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ PConn)
5 simpll 789 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ SConn)
6 simprl 793 . . . . . . . . . 10 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
7 sconntop 30971 . . . . . . . . . . . . 13 (𝑅 ∈ SConn → 𝑅 ∈ Top)
87ad2antrr 761 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ Top)
9 eqid 2621 . . . . . . . . . . . . 13 𝑅 = 𝑅
109toptopon 20662 . . . . . . . . . . . 12 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
118, 10sylib 208 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ (TopOn‘ 𝑅))
12 sconntop 30971 . . . . . . . . . . . . 13 (𝑆 ∈ SConn → 𝑆 ∈ Top)
1312ad2antlr 762 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ Top)
14 eqid 2621 . . . . . . . . . . . . 13 𝑆 = 𝑆
1514toptopon 20662 . . . . . . . . . . . 12 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
1613, 15sylib 208 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ (TopOn‘ 𝑆))
17 tx1cn 21352 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
1811, 16, 17syl2anc 692 . . . . . . . . . 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 21010 . . . . . . . . . 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 692 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅))
21 simprr 795 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) = (𝑓‘1))
2221fveq2d 6162 . . . . . . . . . 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 22622 . . . . . . . . . . . . 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 21334 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
2611, 16, 25syl2anc 692 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
27 cnf2 20993 . . . . . . . . . . . 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 1323 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶( 𝑅 × 𝑆))
29 0elunit 12248 . . . . . . . . . . 11 0 ∈ (0[,]1)
30 fvco3 6242 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
3128, 29, 30sylancl 693 . . . . . . . . . 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 12249 . . . . . . . . . . 11 1 ∈ (0[,]1)
33 fvco3 6242 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
3428, 32, 33sylancl 693 . . . . . . . . . 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 2665 . . . . . . . . 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 30972 . . . . . . . . 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 1323 . . . . . . . 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 22733 . . . . . . . 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 208 . . . . . . 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 1073 . . . . . 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 3913 . . . . . 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 208 . . . . 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 791 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ SConn)
44 tx2cn 21353 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
4511, 16, 44syl2anc 692 . . . . . . . . . 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 21010 . . . . . . . . . 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 692 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
4821fveq2d 6162 . . . . . . . . . 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 6242 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
5028, 29, 49sylancl 693 . . . . . . . . . 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 6242 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
5228, 32, 51sylancl 693 . . . . . . . . . 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 2665 . . . . . . . . 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 30972 . . . . . . . . 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 1323 . . . . . . . 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 22733 . . . . . . . 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 208 . . . . . . 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 1073 . . . . . 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 3913 . . . . . 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 208 . . . . 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 eeanv 2181 . . . . . 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 2621 . . . . . . . . 9 ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)
66 eqid 2621 . . . . . . . . 9 ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) = ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)
67 simprl 793 . . . . . . . . 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 795 . . . . . . . . 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 30983 . . . . . . . 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 450 . . . . . . 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 1859 . . . . . 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 233 . . . . 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 714 . . . 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 642 . . 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 2962 . 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 30969 . 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 697 1 ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2908  c0 3897  {csn 4155   cuni 4409   class class class wbr 4623   × cxp 5082  cres 5086  ccom 5088  wf 5853  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  0cc0 9896  1c1 9897  [,]cicc 12136  Topctop 20638  TopOnctopon 20655   Cn ccn 20968   ×t ctx 21303  IIcii 22618  PHtpycphtpy 22707  phcphtpc 22708  PConncpconn 30962  SConncsconn 30963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-icc 12140  df-seq 12758  df-exp 12817  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690  df-cn 20971  df-cnp 20972  df-tx 21305  df-ii 22620  df-htpy 22709  df-phtpy 22710  df-phtpc 22731  df-pconn 30964  df-sconn 30965
This theorem is referenced by: (None)
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