Theorem phtpy01

Description: Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Hypotheses	isphtpy.2	⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) )
	isphtpy.3	⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) )
	phtpyi.1	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) )
Assertion	phtpy01	⊢ ( 𝜑 → ( ( 𝐹 ‘ 0 ) = ( 𝐺 ‘ 0 ) ∧ ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		isphtpy.2	⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) )
2		isphtpy.3	⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) )
3		phtpyi.1	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) )
4		1elunit	⊢ 1 ∈ ( 0 [,] 1 )
5	1 2 3	phtpyi	⊢ ( ( 𝜑 ∧ 1 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝐻 1 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 1 ) = ( 𝐹 ‘ 1 ) ) )
6	4 5	mpan2	⊢ ( 𝜑 → ( ( 0 𝐻 1 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 1 ) = ( 𝐹 ‘ 1 ) ) )
7	6	simpld	⊢ ( 𝜑 → ( 0 𝐻 1 ) = ( 𝐹 ‘ 0 ) )
8		0elunit	⊢ 0 ∈ ( 0 [,] 1 )
9		iitopon	⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) )
10	9	a1i	⊢ ( 𝜑 → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) )
11	1 2	phtpyhtpy	⊢ ( 𝜑 → ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ⊆ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) )
12	11 3	sseldd	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) )
13	10 1 2 12	htpyi	⊢ ( ( 𝜑 ∧ 0 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝐻 0 ) = ( 𝐹 ‘ 0 ) ∧ ( 0 𝐻 1 ) = ( 𝐺 ‘ 0 ) ) )
14	8 13	mpan2	⊢ ( 𝜑 → ( ( 0 𝐻 0 ) = ( 𝐹 ‘ 0 ) ∧ ( 0 𝐻 1 ) = ( 𝐺 ‘ 0 ) ) )
15	14	simprd	⊢ ( 𝜑 → ( 0 𝐻 1 ) = ( 𝐺 ‘ 0 ) )
16	7 15	eqtr3d	⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) = ( 𝐺 ‘ 0 ) )
17	6	simprd	⊢ ( 𝜑 → ( 1 𝐻 1 ) = ( 𝐹 ‘ 1 ) )
18	10 1 2 12	htpyi	⊢ ( ( 𝜑 ∧ 1 ∈ ( 0 [,] 1 ) ) → ( ( 1 𝐻 0 ) = ( 𝐹 ‘ 1 ) ∧ ( 1 𝐻 1 ) = ( 𝐺 ‘ 1 ) ) )
19	4 18	mpan2	⊢ ( 𝜑 → ( ( 1 𝐻 0 ) = ( 𝐹 ‘ 1 ) ∧ ( 1 𝐻 1 ) = ( 𝐺 ‘ 1 ) ) )
20	19	simprd	⊢ ( 𝜑 → ( 1 𝐻 1 ) = ( 𝐺 ‘ 1 ) )
21	17 20	eqtr3d	⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 1 ) )
22	16 21	jca	⊢ ( 𝜑 → ( ( 𝐹 ‘ 0 ) = ( 𝐺 ‘ 0 ) ∧ ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 1 ) ) )