Metamath Proof Explorer

Theorem fourierdlem106

Description: For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses fourierdlem106.f ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ )
fourierdlem106.t ⊢ 𝑇 = ( 2 · π )
fourierdlem106.per ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
fourierdlem106.g ⊢ 𝐺 = ( ( ℝ D 𝐹 ) ↾ ( - π (,) π ) )
fourierdlem106.dmdv ⊢ ( 𝜑 → ( ( - π (,) π ) ∖ dom 𝐺 ) ∈ Fin )
fourierdlem106.dvcn ⊢ ( 𝜑 → 𝐺 ∈ ( dom 𝐺 –cn→ ℂ ) )
fourierdlem106.rlim ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( - π [,) π ) ∖ dom 𝐺 ) ) → ( ( 𝐺 ↾ ( 𝑥 (,) +∞ ) ) limℂ 𝑥 ) ≠ ∅ )
fourierdlem106.llim ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( - π (,] π ) ∖ dom 𝐺 ) ) → ( ( 𝐺 ↾ ( -∞ (,) 𝑥 ) ) limℂ 𝑥 ) ≠ ∅ )
fourierdlem106.x ⊢ ( 𝜑 → 𝑋 ∈ ℝ )
Assertion fourierdlem106 ( 𝜑 → ( ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) ≠ ∅ ∧ ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) )

Proof

Step Hyp Ref Expression
1 fourierdlem106.f ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ )
2 fourierdlem106.t ⊢ 𝑇 = ( 2 · π )
3 fourierdlem106.per ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
4 fourierdlem106.g ⊢ 𝐺 = ( ( ℝ D 𝐹 ) ↾ ( - π (,) π ) )
5 fourierdlem106.dmdv ⊢ ( 𝜑 → ( ( - π (,) π ) ∖ dom 𝐺 ) ∈ Fin )
6 fourierdlem106.dvcn ⊢ ( 𝜑 → 𝐺 ∈ ( dom 𝐺 –cn→ ℂ ) )
7 fourierdlem106.rlim ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( - π [,) π ) ∖ dom 𝐺 ) ) → ( ( 𝐺 ↾ ( 𝑥 (,) +∞ ) ) limℂ 𝑥 ) ≠ ∅ )
8 fourierdlem106.llim ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( - π (,] π ) ∖ dom 𝐺 ) ) → ( ( 𝐺 ↾ ( -∞ (,) 𝑥 ) ) limℂ 𝑥 ) ≠ ∅ )
9 fourierdlem106.x ⊢ ( 𝜑 → 𝑋 ∈ ℝ )
10 eqid ⊢ ( 𝑘 ∈ ℕ ↦ { 𝑤 ∈ ( ℝ ↑m ( 0 ... 𝑘 ) ) ∣ ( ( ( 𝑤 ‘ 0 ) = - π ∧ ( 𝑤 ‘ 𝑘 ) = π ) ∧ ∀ 𝑧 ∈ ( 0 ..^ 𝑘 ) ( 𝑤 ‘ 𝑧 ) < ( 𝑤 ‘ ( 𝑧 + 1 ) ) ) } ) = ( 𝑘 ∈ ℕ ↦ { 𝑤 ∈ ( ℝ ↑m ( 0 ... 𝑘 ) ) ∣ ( ( ( 𝑤 ‘ 0 ) = - π ∧ ( 𝑤 ‘ 𝑘 ) = π ) ∧ ∀ 𝑧 ∈ ( 0 ..^ 𝑘 ) ( 𝑤 ‘ 𝑧 ) < ( 𝑤 ‘ ( 𝑧 + 1 ) ) ) } )
11 id ⊢ ( 𝑦 = 𝑥 → 𝑦 = 𝑥 )
12 oveq2 ⊢ ( 𝑦 = 𝑥 → ( π − 𝑦 ) = ( π − 𝑥 ) )
13 12 oveq1d ⊢ ( 𝑦 = 𝑥 → ( ( π − 𝑦 ) / 𝑇 ) = ( ( π − 𝑥 ) / 𝑇 ) )
14 13 fveq2d ⊢ ( 𝑦 = 𝑥 → ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( π − 𝑥 ) / 𝑇 ) ) )
15 14 oveq1d ⊢ ( 𝑦 = 𝑥 → ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( π − 𝑥 ) / 𝑇 ) ) · 𝑇 ) )
16 11 15 oveq12d ⊢ ( 𝑦 = 𝑥 → ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( π − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) )
17 16 cbvmptv ⊢ ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( π − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) )
18 eqid ⊢ ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) = ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) )
19 eqid ⊢ ( ( ♯ ‘ ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) − 1 ) = ( ( ♯ ‘ ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) − 1 )
20 isoeq1 ⊢ ( 𝑔 = 𝑓 → ( 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) − 1 ) ) , ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) ↔ 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) − 1 ) ) , ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) ) )
21 20 cbviotavw ⊢ ( ℩ 𝑔 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) − 1 ) ) , ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) ) = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) − 1 ) ) , ( { - π , π , ( ( 𝑦 ∈ ℝ ↦ ( 𝑦 + ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑋 ) } ∪ ( ( - π [,] π ) ∖ dom 𝐺 ) ) ) )
22 1 2 3 4 5 6 7 8 9 10 17 18 19 21 fourierdlem102 ⊢ ( 𝜑 → ( ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) ≠ ∅ ∧ ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) )