Theorem fourierdlem55

Description: U is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fourierdlem55.f	⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ )
	fourierdlem55.x	⊢ ( 𝜑 → 𝑋 ∈ ℝ )
	fourierdlem55.r	⊢ ( 𝜑 → 𝑌 ∈ ℝ )
	fourierdlem55.w	⊢ ( 𝜑 → 𝑊 ∈ ℝ )
	fourierdlem55.h	⊢ 𝐻 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 0 , ( ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) − if ( 0 < 𝑠 , 𝑌 , 𝑊 ) ) / 𝑠 ) ) )
	fourierdlem55.k	⊢ 𝐾 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 1 , ( 𝑠 / ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ) ) )
	fourierdlem55.u	⊢ 𝑈 = ( 𝑠 ∈ ( - π [,] π ) ↦ ( ( 𝐻 ‘ 𝑠 ) · ( 𝐾 ‘ 𝑠 ) ) )
Assertion	fourierdlem55	⊢ ( 𝜑 → 𝑈 : ( - π [,] π ) ⟶ ℝ )

Proof

Step	Hyp	Ref	Expression
1		fourierdlem55.f	⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ )
2		fourierdlem55.x	⊢ ( 𝜑 → 𝑋 ∈ ℝ )
3		fourierdlem55.r	⊢ ( 𝜑 → 𝑌 ∈ ℝ )
4		fourierdlem55.w	⊢ ( 𝜑 → 𝑊 ∈ ℝ )
5		fourierdlem55.h	⊢ 𝐻 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 0 , ( ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) − if ( 0 < 𝑠 , 𝑌 , 𝑊 ) ) / 𝑠 ) ) )
6		fourierdlem55.k	⊢ 𝐾 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 1 , ( 𝑠 / ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ) ) )
7		fourierdlem55.u	⊢ 𝑈 = ( 𝑠 ∈ ( - π [,] π ) ↦ ( ( 𝐻 ‘ 𝑠 ) · ( 𝐾 ‘ 𝑠 ) ) )
8	1 2 3 4 5	fourierdlem9	⊢ ( 𝜑 → 𝐻 : ( - π [,] π ) ⟶ ℝ )
9	8	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( - π [,] π ) ) → ( 𝐻 ‘ 𝑠 ) ∈ ℝ )
10	6	fourierdlem43	⊢ 𝐾 : ( - π [,] π ) ⟶ ℝ
11	10	ffvelrni	⊢ ( 𝑠 ∈ ( - π [,] π ) → ( 𝐾 ‘ 𝑠 ) ∈ ℝ )
12	11	adantl	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( - π [,] π ) ) → ( 𝐾 ‘ 𝑠 ) ∈ ℝ )
13	9 12	remulcld	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( - π [,] π ) ) → ( ( 𝐻 ‘ 𝑠 ) · ( 𝐾 ‘ 𝑠 ) ) ∈ ℝ )
14	13 7	fmptd	⊢ ( 𝜑 → 𝑈 : ( - π [,] π ) ⟶ ℝ )