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Metamath Proof Explorer

Theorem fourierdlem67

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

Ref Expression
Hypotheses fourierdlem67.f ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ )
fourierdlem67.x ⊢ ( 𝜑 → 𝑋 ∈ ℝ )
fourierdlem67.y ⊢ ( 𝜑 → 𝑌 ∈ ℝ )
fourierdlem67.w ⊢ ( 𝜑 → 𝑊 ∈ ℝ )
fourierdlem67.h ⊢ 𝐻 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 0 , ( ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) − if ( 0 < 𝑠 , 𝑌 , 𝑊 ) ) / 𝑠 ) ) )
fourierdlem67.k ⊢ 𝐾 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 1 , ( 𝑠 / ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ) ) )
fourierdlem67.u ⊢ 𝑈 = ( 𝑠 ∈ ( - π [,] π ) ↦ ( ( 𝐻 ‘ 𝑠 ) · ( 𝐾 ‘ 𝑠 ) ) )
fourierdlem67.n ⊢ ( 𝜑 → 𝑁 ∈ ℝ )
fourierdlem67.s ⊢ 𝑆 = ( 𝑠 ∈ ( - π [,] π ) ↦ ( sin ‘ ( ( 𝑁 + ( 1 / 2 ) ) · 𝑠 ) ) )
fourierdlem67.g ⊢ 𝐺 = ( 𝑠 ∈ ( - π [,] π ) ↦ ( ( 𝑈 ‘ 𝑠 ) · ( 𝑆 ‘ 𝑠 ) ) )
Assertion fourierdlem67 ( 𝜑 → 𝐺 : ( - π [,] π ) ⟶ ℝ )

Proof

Step Hyp Ref Expression
1 fourierdlem67.f ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ )
2 fourierdlem67.x ⊢ ( 𝜑 → 𝑋 ∈ ℝ )
3 fourierdlem67.y ⊢ ( 𝜑 → 𝑌 ∈ ℝ )
4 fourierdlem67.w ⊢ ( 𝜑 → 𝑊 ∈ ℝ )
5 fourierdlem67.h ⊢ 𝐻 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 0 , ( ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) − if ( 0 < 𝑠 , 𝑌 , 𝑊 ) ) / 𝑠 ) ) )
6 fourierdlem67.k ⊢ 𝐾 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 1 , ( 𝑠 / ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ) ) )
7 fourierdlem67.u ⊢ 𝑈 = ( 𝑠 ∈ ( - π [,] π ) ↦ ( ( 𝐻 ‘ 𝑠 ) · ( 𝐾 ‘ 𝑠 ) ) )
8 fourierdlem67.n ⊢ ( 𝜑 → 𝑁 ∈ ℝ )
9 fourierdlem67.s ⊢ 𝑆 = ( 𝑠 ∈ ( - π [,] π ) ↦ ( sin ‘ ( ( 𝑁 + ( 1 / 2 ) ) · 𝑠 ) ) )
10 fourierdlem67.g ⊢ 𝐺 = ( 𝑠 ∈ ( - π [,] π ) ↦ ( ( 𝑈 ‘ 𝑠 ) · ( 𝑆 ‘ 𝑠 ) ) )
11 1 2 3 4 5 6 7 fourierdlem55 ⊢ ( 𝜑 → 𝑈 : ( - π [,] π ) ⟶ ℝ )
12 11 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( - π [,] π ) ) → ( 𝑈 ‘ 𝑠 ) ∈ ℝ )
13 9 fourierdlem5 ⊢ ( 𝑁 ∈ ℝ → 𝑆 : ( - π [,] π ) ⟶ ℝ )
14 8 13 syl ⊢ ( 𝜑 → 𝑆 : ( - π [,] π ) ⟶ ℝ )
15 14 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( - π [,] π ) ) → ( 𝑆 ‘ 𝑠 ) ∈ ℝ )
16 12 15 remulcld ⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( - π [,] π ) ) → ( ( 𝑈 ‘ 𝑠 ) · ( 𝑆 ‘ 𝑠 ) ) ∈ ℝ )
17 16 10 fmptd ⊢ ( 𝜑 → 𝐺 : ( - π [,] π ) ⟶ ℝ )