| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sconnpconn |
⊢ ( 𝑅 ∈ SConn → 𝑅 ∈ PConn ) |
| 2 |
|
sconnpconn |
⊢ ( 𝑆 ∈ SConn → 𝑆 ∈ PConn ) |
| 3 |
|
txpconn |
⊢ ( ( 𝑅 ∈ PConn ∧ 𝑆 ∈ PConn ) → ( 𝑅 ×t 𝑆 ) ∈ PConn ) |
| 4 |
1 2 3
|
syl2an |
⊢ ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) → ( 𝑅 ×t 𝑆 ) ∈ PConn ) |
| 5 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑅 ∈ SConn ) |
| 6 |
|
simprl |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ) |
| 7 |
|
sconntop |
⊢ ( 𝑅 ∈ SConn → 𝑅 ∈ Top ) |
| 8 |
7
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑅 ∈ Top ) |
| 9 |
|
eqid |
⊢ ∪ 𝑅 = ∪ 𝑅 |
| 10 |
9
|
toptopon |
⊢ ( 𝑅 ∈ Top ↔ 𝑅 ∈ ( TopOn ‘ ∪ 𝑅 ) ) |
| 11 |
8 10
|
sylib |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑅 ∈ ( TopOn ‘ ∪ 𝑅 ) ) |
| 12 |
|
sconntop |
⊢ ( 𝑆 ∈ SConn → 𝑆 ∈ Top ) |
| 13 |
12
|
ad2antlr |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑆 ∈ Top ) |
| 14 |
|
eqid |
⊢ ∪ 𝑆 = ∪ 𝑆 |
| 15 |
14
|
toptopon |
⊢ ( 𝑆 ∈ Top ↔ 𝑆 ∈ ( TopOn ‘ ∪ 𝑆 ) ) |
| 16 |
13 15
|
sylib |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑆 ∈ ( TopOn ‘ ∪ 𝑆 ) ) |
| 17 |
|
tx1cn |
⊢ ( ( 𝑅 ∈ ( TopOn ‘ ∪ 𝑅 ) ∧ 𝑆 ∈ ( TopOn ‘ ∪ 𝑆 ) ) → ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∈ ( ( 𝑅 ×t 𝑆 ) Cn 𝑅 ) ) |
| 18 |
11 16 17
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∈ ( ( 𝑅 ×t 𝑆 ) Cn 𝑅 ) ) |
| 19 |
|
cnco |
⊢ ( ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∈ ( ( 𝑅 ×t 𝑆 ) Cn 𝑅 ) ) → ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ∈ ( II Cn 𝑅 ) ) |
| 20 |
6 18 19
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ∈ ( II Cn 𝑅 ) ) |
| 21 |
|
simprr |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) |
| 22 |
21
|
fveq2d |
⊢ ( ( ( 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 |
⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) |
| 24 |
23
|
a1i |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ) |
| 25 |
|
txtopon |
⊢ ( ( 𝑅 ∈ ( TopOn ‘ ∪ 𝑅 ) ∧ 𝑆 ∈ ( TopOn ‘ ∪ 𝑆 ) ) → ( 𝑅 ×t 𝑆 ) ∈ ( TopOn ‘ ( ∪ 𝑅 × ∪ 𝑆 ) ) ) |
| 26 |
11 16 25
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( 𝑅 ×t 𝑆 ) ∈ ( TopOn ‘ ( ∪ 𝑅 × ∪ 𝑆 ) ) ) |
| 27 |
|
cnf2 |
⊢ ( ( II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ∧ ( 𝑅 ×t 𝑆 ) ∈ ( TopOn ‘ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∧ 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ) → 𝑓 : ( 0 [,] 1 ) ⟶ ( ∪ 𝑅 × ∪ 𝑆 ) ) |
| 28 |
24 26 6 27
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑓 : ( 0 [,] 1 ) ⟶ ( ∪ 𝑅 × ∪ 𝑆 ) ) |
| 29 |
|
0elunit |
⊢ 0 ∈ ( 0 [,] 1 ) |
| 30 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 0 [,] 1 ) ⟶ ( ∪ 𝑅 × ∪ 𝑆 ) ∧ 0 ∈ ( 0 [,] 1 ) ) → ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) = ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ‘ ( 𝑓 ‘ 0 ) ) ) |
| 31 |
28 29 30
|
sylancl |
⊢ ( ( ( 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 |
⊢ 1 ∈ ( 0 [,] 1 ) |
| 33 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 0 [,] 1 ) ⟶ ( ∪ 𝑅 × ∪ 𝑆 ) ∧ 1 ∈ ( 0 [,] 1 ) ) → ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 1 ) = ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ‘ ( 𝑓 ‘ 1 ) ) ) |
| 34 |
28 32 33
|
sylancl |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 1 ) = ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ‘ ( 𝑓 ‘ 1 ) ) ) |
| 35 |
22 31 34
|
3eqtr4d |
⊢ ( ( ( 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 |
⊢ ( ( 𝑅 ∈ SConn ∧ ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ∈ ( II Cn 𝑅 ) ∧ ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) = ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 1 ) ) → ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( ≃ph ‘ 𝑅 ) ( ( 0 [,] 1 ) × { ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) |
| 37 |
5 20 35 36
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( ≃ph ‘ 𝑅 ) ( ( 0 [,] 1 ) × { ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) |
| 38 |
|
isphtpc |
⊢ ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( ≃ph ‘ 𝑅 ) ( ( 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 ) } ) ) ≠ ∅ ) ) |
| 39 |
37 38
|
sylib |
⊢ ( ( ( 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 ) } ) ) ≠ ∅ ) ) |
| 40 |
39
|
simp3d |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 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 ) } ) ) ) |
| 42 |
40 41
|
sylib |
⊢ ( ( ( 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 |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑆 ∈ SConn ) |
| 44 |
|
tx2cn |
⊢ ( ( 𝑅 ∈ ( TopOn ‘ ∪ 𝑅 ) ∧ 𝑆 ∈ ( TopOn ‘ ∪ 𝑆 ) ) → ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∈ ( ( 𝑅 ×t 𝑆 ) Cn 𝑆 ) ) |
| 45 |
11 16 44
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∈ ( ( 𝑅 ×t 𝑆 ) Cn 𝑆 ) ) |
| 46 |
|
cnco |
⊢ ( ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∈ ( ( 𝑅 ×t 𝑆 ) Cn 𝑆 ) ) → ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ∈ ( II Cn 𝑆 ) ) |
| 47 |
6 45 46
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ∈ ( II Cn 𝑆 ) ) |
| 48 |
21
|
fveq2d |
⊢ ( ( ( 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 |
⊢ ( ( 𝑓 : ( 0 [,] 1 ) ⟶ ( ∪ 𝑅 × ∪ 𝑆 ) ∧ 0 ∈ ( 0 [,] 1 ) ) → ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) = ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ‘ ( 𝑓 ‘ 0 ) ) ) |
| 50 |
28 29 49
|
sylancl |
⊢ ( ( ( 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 |
⊢ ( ( 𝑓 : ( 0 [,] 1 ) ⟶ ( ∪ 𝑅 × ∪ 𝑆 ) ∧ 1 ∈ ( 0 [,] 1 ) ) → ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 1 ) = ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ‘ ( 𝑓 ‘ 1 ) ) ) |
| 52 |
28 32 51
|
sylancl |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 1 ) = ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ‘ ( 𝑓 ‘ 1 ) ) ) |
| 53 |
48 50 52
|
3eqtr4d |
⊢ ( ( ( 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 |
⊢ ( ( 𝑆 ∈ SConn ∧ ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ∈ ( II Cn 𝑆 ) ∧ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) = ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 1 ) ) → ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( ≃ph ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) |
| 55 |
43 47 53 54
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( ≃ph ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) |
| 56 |
|
isphtpc |
⊢ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( ≃ph ‘ 𝑆 ) ( ( 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 ) } ) ) ≠ ∅ ) ) |
| 57 |
55 56
|
sylib |
⊢ ( ( ( 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 ) } ) ) ≠ ∅ ) ) |
| 58 |
57
|
simp3d |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ≠ ∅ ↔ ∃ ℎ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) |
| 60 |
58 59
|
sylib |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ∃ ℎ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) |
| 61 |
|
exdistrv |
⊢ ( ∃ 𝑔 ∃ ℎ ( 𝑔 ∈ ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑅 ) ( ( 0 [,] 1 ) × { ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ∧ ℎ ∈ ( ( ( 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 ) } ) ) ∧ ∃ ℎ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) ) |
| 62 |
8
|
adantr |
⊢ ( ( ( ( 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 ) } ) ) ∧ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) ) → 𝑅 ∈ Top ) |
| 63 |
13
|
adantr |
⊢ ( ( ( ( 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 ) } ) ) ∧ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) ) → 𝑆 ∈ Top ) |
| 64 |
6
|
adantr |
⊢ ( ( ( ( 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 ) } ) ) ∧ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) ) → 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ) |
| 65 |
|
eqid |
⊢ ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) = ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) |
| 66 |
|
eqid |
⊢ ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) = ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) |
| 67 |
|
simprl |
⊢ ( ( ( ( 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 ) } ) ) ∧ ℎ ∈ ( ( ( 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 |
⊢ ( ( ( ( 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 ) } ) ) ∧ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) ) → ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) |
| 69 |
62 63 64 65 66 67 68
|
txsconnlem |
⊢ ( ( ( ( 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 ) } ) ) ∧ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) ) → 𝑓 ( ≃ph ‘ ( 𝑅 ×t 𝑆 ) ) ( ( 0 [,] 1 ) × { ( 𝑓 ‘ 0 ) } ) ) |
| 70 |
69
|
ex |
⊢ ( ( ( 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 ) } ) ) ∧ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) → 𝑓 ( ≃ph ‘ ( 𝑅 ×t 𝑆 ) ) ( ( 0 [,] 1 ) × { ( 𝑓 ‘ 0 ) } ) ) ) |
| 71 |
70
|
exlimdvv |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → ( ∃ 𝑔 ∃ ℎ ( 𝑔 ∈ ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑅 ) ( ( 0 [,] 1 ) × { ( ( ( 1st ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ∧ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) → 𝑓 ( ≃ph ‘ ( 𝑅 ×t 𝑆 ) ) ( ( 0 [,] 1 ) × { ( 𝑓 ‘ 0 ) } ) ) ) |
| 72 |
61 71
|
biimtrrid |
⊢ ( ( ( 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 ) } ) ) ∧ ∃ ℎ ℎ ∈ ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ( PHtpy ‘ 𝑆 ) ( ( 0 [,] 1 ) × { ( ( ( 2nd ↾ ( ∪ 𝑅 × ∪ 𝑆 ) ) ∘ 𝑓 ) ‘ 0 ) } ) ) ) → 𝑓 ( ≃ph ‘ ( 𝑅 ×t 𝑆 ) ) ( ( 0 [,] 1 ) × { ( 𝑓 ‘ 0 ) } ) ) ) |
| 73 |
42 60 72
|
mp2and |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ ( 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) ) ) → 𝑓 ( ≃ph ‘ ( 𝑅 ×t 𝑆 ) ) ( ( 0 [,] 1 ) × { ( 𝑓 ‘ 0 ) } ) ) |
| 74 |
73
|
expr |
⊢ ( ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) ∧ 𝑓 ∈ ( II Cn ( 𝑅 ×t 𝑆 ) ) ) → ( ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) → 𝑓 ( ≃ph ‘ ( 𝑅 ×t 𝑆 ) ) ( ( 0 [,] 1 ) × { ( 𝑓 ‘ 0 ) } ) ) ) |
| 75 |
74
|
ralrimiva |
⊢ ( ( 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 |
⊢ ( ( 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 ) } ) ) ) ) |
| 77 |
4 75 76
|
sylanbrc |
⊢ ( ( 𝑅 ∈ SConn ∧ 𝑆 ∈ SConn ) → ( 𝑅 ×t 𝑆 ) ∈ SConn ) |