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Theorem txsconn 32776
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 32762 . . 3 (𝑅 ∈ SConn → 𝑅 ∈ PConn)
2 sconnpconn 32762 . . 3 (𝑆 ∈ SConn → 𝑆 ∈ PConn)
3 txpconn 32767 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
41, 2, 3syl2an 599 . 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 32763 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . 13 𝑅 = 𝑅
109toptopon 21670 . . . . . . . . . . . 12 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
118, 10sylib 221 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ (TopOn‘ 𝑅))
12 sconntop 32763 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . 13 𝑆 = 𝑆
1514toptopon 21670 . . . . . . . . . . . 12 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
1613, 15sylib 221 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ (TopOn‘ 𝑆))
17 tx1cn 22362 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
1811, 16, 17syl2anc 587 . . . . . . . . . 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 22019 . . . . . . . . . 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 587 . . . . . . . . 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 6680 . . . . . . . . . 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 23633 . . . . . . . . . . . . 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 22344 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
2611, 16, 25syl2anc 587 . . . . . . . . . . . 12 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
27 cnf2 22002 . . . . . . . . . . . 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 1372 . . . . . . . . . . 11 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶( 𝑅 × 𝑆))
29 0elunit 12945 . . . . . . . . . . 11 0 ∈ (0[,]1)
30 fvco3 6769 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
3128, 29, 30sylancl 589 . . . . . . . . . 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 12946 . . . . . . . . . . 11 1 ∈ (0[,]1)
33 fvco3 6769 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
3428, 32, 33sylancl 589 . . . . . . . . . 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 2783 . . . . . . . . 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 32764 . . . . . . . . 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 1372 . . . . . . . 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 23748 . . . . . . . 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 221 . . . . . . 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 4235 . . . . . 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 221 . . . . 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 22363 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
4511, 16, 44syl2anc 587 . . . . . . . . . 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 22019 . . . . . . . . . 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 587 . . . . . . . . 9 (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
4821fveq2d 6680 . . . . . . . . . 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 6769 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘0)))
5028, 29, 49sylancl 589 . . . . . . . . . 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 6769 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝑓‘1)))
5228, 32, 51sylancl 589 . . . . . . . . . 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 2783 . . . . . . . . 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 32764 . . . . . . . . 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 1372 . . . . . . . 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 23748 . . . . . . . 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 221 . . . . . . 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 4235 . . . . . 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 221 . . . . 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 1963 . . . . . 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 484 . . . . . . . . 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 484 . . . . . . . . 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 484 . . . . . . . . 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 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 32775 . . . . . . . 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 416 . . . . . . 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 1941 . . . . . 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 246 . . . . 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 460 . . 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 3096 . 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 32761 . 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 586 1 ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  wne 2934  wral 3053  c0 4211  {csn 4516   cuni 4796   class class class wbr 5030   × cxp 5523  cres 5527  ccom 5529  wf 6335  cfv 6339  (class class class)co 7172  1st c1st 7714  2nd c2nd 7715  0cc0 10617  1c1 10618  [,]cicc 12826  Topctop 21646  TopOnctopon 21663   Cn ccn 21977   ×t ctx 22313  IIcii 23629  PHtpycphtpy 23722  phcphtpc 23723  PConncpconn 32754  SConncsconn 32755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-cnex 10673  ax-resscn 10674  ax-1cn 10675  ax-icn 10676  ax-addcl 10677  ax-addrcl 10678  ax-mulcl 10679  ax-mulrcl 10680  ax-mulcom 10681  ax-addass 10682  ax-mulass 10683  ax-distr 10684  ax-i2m1 10685  ax-1ne0 10686  ax-1rid 10687  ax-rnegex 10688  ax-rrecex 10689  ax-cnre 10690  ax-pre-lttri 10691  ax-pre-lttrn 10692  ax-pre-ltadd 10693  ax-pre-mulgt0 10694  ax-pre-sup 10695
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-om 7602  df-1st 7716  df-2nd 7717  df-wrecs 7978  df-recs 8039  df-rdg 8077  df-er 8322  df-map 8441  df-en 8558  df-dom 8559  df-sdom 8560  df-sup 8981  df-inf 8982  df-pnf 10757  df-mnf 10758  df-xr 10759  df-ltxr 10760  df-le 10761  df-sub 10952  df-neg 10953  df-div 11378  df-nn 11719  df-2 11781  df-3 11782  df-n0 11979  df-z 12065  df-uz 12327  df-q 12433  df-rp 12475  df-xneg 12592  df-xadd 12593  df-xmul 12594  df-icc 12830  df-seq 13463  df-exp 13524  df-cj 14550  df-re 14551  df-im 14552  df-sqrt 14686  df-abs 14687  df-topgen 16822  df-psmet 20211  df-xmet 20212  df-met 20213  df-bl 20214  df-mopn 20215  df-top 21647  df-topon 21664  df-bases 21699  df-cn 21980  df-cnp 21981  df-tx 22315  df-ii 23631  df-htpy 23724  df-phtpy 23725  df-phtpc 23746  df-pconn 32756  df-sconn 32757
This theorem is referenced by: (None)
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