Theorem isphtpyd

Description: Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Hypotheses	isphtpy.2	⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) )
	isphtpy.3	⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) )
	isphtpyd.1	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) )
	isphtpyd.2	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) )
	isphtpyd.3	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) )
Assertion	isphtpyd	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		isphtpy.2	⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) )
2		isphtpy.3	⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) )
3		isphtpyd.1	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) )
4		isphtpyd.2	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) )
5		isphtpyd.3	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) )
6	4 5	jca	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) ) )
7	6	ralrimiva	⊢ ( 𝜑 → ∀ 𝑠 ∈ ( 0 [,] 1 ) ( ( 0 𝐻 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝐻 𝑠 ) = ( 𝐹 ‘ 1 ) ) )
8	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 ) ) ) ) )
9	3 7 8	mpbir2and	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) )