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Theorem txsconn 34687
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 34673 . . 3 (𝑅 ∈ SConn → 𝑅 ∈ PConn)
2 sconnpconn 34673 . . 3 (𝑆 ∈ SConn → 𝑆 ∈ PConn)
3 txpconn 34678 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
41, 2, 3syl2an 595 . 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 34674 . . . . . . . . . . . . 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 2724 . . . . . . . . . . . . 13 𝑅 = 𝑅
109toptopon 22729 . . . . . . . . . . . 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 34674 . . . . . . . . . . . . 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 2724 . . . . . . . . . . . . 13 𝑆 = 𝑆
1514toptopon 22729 . . . . . . . . . . . 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 23423 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
1811, 16, 17syl2anc 583 . . . . . . . . . 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 23080 . . . . . . . . . 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 583 . . . . . . . . 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 6885 . . . . . . . . . 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 24709 . . . . . . . . . . . . 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 23405 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
2611, 16, 25syl2anc 583 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
27 cnf2 23063 . . . . . . . . . . . 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 1368 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶( 𝑅 × 𝑆))
29 0elunit 13442 . . . . . . . . . . 11 0 ∈ (0[,]1)
30 fvco3 6980 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
3128, 29, 30sylancl 585 . . . . . . . . . 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 13443 . . . . . . . . . . 11 1 ∈ (0[,]1)
33 fvco3 6980 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
3428, 32, 33sylancl 585 . . . . . . . . . 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 2774 . . . . . . . . 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 34675 . . . . . . . . 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 1368 . . . . . . . 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 24830 . . . . . . . 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 1141 . . . . . 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 4338 . . . . . 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 23424 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
4511, 16, 44syl2anc 583 . . . . . . . . . 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 23080 . . . . . . . . . 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 583 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
4821fveq2d 6885 . . . . . . . . . 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 6980 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
5028, 29, 49sylancl 585 . . . . . . . . . 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 6980 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
5228, 32, 51sylancl 585 . . . . . . . . . 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 2774 . . . . . . . . 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 34675 . . . . . . . . 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 1368 . . . . . . . 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 24830 . . . . . . . 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 1141 . . . . . 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 4338 . . . . . 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 1951 . . . . . 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 2724 . . . . . . . . 9 ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)
66 eqid 2724 . . . . . . . . 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 34686 . . . . . . . 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 1929 . . . . . 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 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 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 3138 . 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 34672 . 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 582 1 ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2932  wral 3053  c0 4314  {csn 4620   cuni 4899   class class class wbr 5138   × cxp 5664  cres 5668  ccom 5670  wf 6529  cfv 6533  (class class class)co 7401  1st c1st 7966  2nd c2nd 7967  0cc0 11105  1c1 11106  [,]cicc 13323  Topctop 22705  TopOnctopon 22722   Cn ccn 23038   ×t ctx 23374  IIcii 24705  PHtpycphtpy 24804  phcphtpc 24805  PConncpconn 34665  SConncsconn 34666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-sup 9432  df-inf 9433  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-icc 13327  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-topgen 17385  df-psmet 21215  df-xmet 21216  df-met 21217  df-bl 21218  df-mopn 21219  df-top 22706  df-topon 22723  df-bases 22759  df-cn 23041  df-cnp 23042  df-tx 23376  df-ii 24707  df-htpy 24806  df-phtpy 24807  df-phtpc 24828  df-pconn 34667  df-sconn 34668
This theorem is referenced by: (None)
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