Theorem phtpyi

Description: Membership in the class of path homotopies between two continuous functions. (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	phtpyi	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝐻 𝐴 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝐴 ) = ( 𝐹 ‘ 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	1 2	isphtpy	⊢ ( 𝜑 → ( 𝐻 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ↔ ( 𝐻 ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) ∧ ∀ 𝑠 ∈ ( 0 [,] 1 ) ( ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) ) ) ) )
5	3 4	mpbid	⊢ ( 𝜑 → ( 𝐻 ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) ∧ ∀ 𝑠 ∈ ( 0 [,] 1 ) ( ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) ) ) )
6	5	simprd	⊢ ( 𝜑 → ∀ 𝑠 ∈ ( 0 [,] 1 ) ( ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) ) )
7		oveq2	⊢ ( 𝑠 = 𝐴 → ( 0 𝐻 𝑠 ) = ( 0 𝐻 𝐴 ) )
8	7	eqeq1d	⊢ ( 𝑠 = 𝐴 → ( ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) ↔ ( 0 𝐻 𝐴 ) = ( 𝐹 ‘ 0 ) ) )
9		oveq2	⊢ ( 𝑠 = 𝐴 → ( 1 𝐻 𝑠 ) = ( 1 𝐻 𝐴 ) )
10	9	eqeq1d	⊢ ( 𝑠 = 𝐴 → ( ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) ↔ ( 1 𝐻 𝐴 ) = ( 𝐹 ‘ 1 ) ) )
11	8 10	anbi12d	⊢ ( 𝑠 = 𝐴 → ( ( ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) ) ↔ ( ( 0 𝐻 𝐴 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝐴 ) = ( 𝐹 ‘ 1 ) ) ) )
12	11	rspccva	⊢ ( ( ∀ 𝑠 ∈ ( 0 [,] 1 ) ( ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) ) ∧ 𝐴 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝐻 𝐴 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝐴 ) = ( 𝐹 ‘ 1 ) ) )
13	6 12	sylan	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝐻 𝐴 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝐴 ) = ( 𝐹 ‘ 1 ) ) )