Description: Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | sconnpht2.1 | |
|
sconnpht2.2 | |
||
sconnpht2.3 | |
||
sconnpht2.4 | |
||
sconnpht2.5 | |
||
Assertion | sconnpht2 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sconnpht2.1 | |
|
2 | sconnpht2.2 | |
|
3 | sconnpht2.3 | |
|
4 | sconnpht2.4 | |
|
5 | sconnpht2.5 | |
|
6 | eqid | |
|
7 | 6 | pcorevcl | |
8 | 3 7 | syl | |
9 | 8 | simp1d | |
10 | 8 | simp2d | |
11 | 5 10 | eqtr4d | |
12 | 2 9 11 | pcocn | |
13 | 2 9 | pco0 | |
14 | 2 9 | pco1 | |
15 | 8 | simp3d | |
16 | 4 15 | eqtr4d | |
17 | 14 16 | eqtr4d | |
18 | 13 17 | eqtr4d | |
19 | sconnpht | |
|
20 | 1 12 18 19 | syl3anc | |
21 | 13 | sneqd | |
22 | 21 | xpeq2d | |
23 | 20 22 | breqtrd | |
24 | eqid | |
|
25 | 6 24 2 3 4 5 | pcophtb | |
26 | 23 25 | mpbid | |