Theorem fourierdlem105

Description: A piecewise continuous function is integrable on any closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fourierdlem105.p	⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
	fourierdlem105.t	⊢ 𝑇 = ( 𝐵 − 𝐴 )
	fourierdlem105.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
	fourierdlem105.q	⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) )
	fourierdlem105.f	⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ )
	fourierdlem105.6	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
	fourierdlem105.fcn	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
	fourierdlem105.r	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
	fourierdlem105.l	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
	fourierdlem105.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	fourierdlem105.d	⊢ ( 𝜑 → 𝐷 ∈ ( 𝐶 (,) +∞ ) )
Assertion	fourierdlem105	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )

Proof

Step	Hyp	Ref	Expression
1		fourierdlem105.p	⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
2		fourierdlem105.t	⊢ 𝑇 = ( 𝐵 − 𝐴 )
3		fourierdlem105.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
4		fourierdlem105.q	⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) )
5		fourierdlem105.f	⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ )
6		fourierdlem105.6	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
7		fourierdlem105.fcn	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
8		fourierdlem105.r	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
9		fourierdlem105.l	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
10		fourierdlem105.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
11		fourierdlem105.d	⊢ ( 𝜑 → 𝐷 ∈ ( 𝐶 (,) +∞ ) )
12		eqid	⊢ ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
13		eqid	⊢ ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 )
14		oveq1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 + ( 𝑗 · 𝑇 ) ) = ( 𝑦 + ( 𝑗 · 𝑇 ) ) )
15	14	eleq1d	⊢ ( 𝑤 = 𝑦 → ( ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ) )
16	15	rexbidv	⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ℤ ( 𝑦 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ) )
17		oveq1	⊢ ( 𝑗 = 𝑘 → ( 𝑗 · 𝑇 ) = ( 𝑘 · 𝑇 ) )
18	17	oveq2d	⊢ ( 𝑗 = 𝑘 → ( 𝑦 + ( 𝑗 · 𝑇 ) ) = ( 𝑦 + ( 𝑘 · 𝑇 ) ) )
19	18	eleq1d	⊢ ( 𝑗 = 𝑘 → ( ( 𝑦 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) )
20	19	cbvrexvw	⊢ ( ∃ 𝑗 ∈ ℤ ( 𝑦 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 )
21	16 20	bitrdi	⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) )
22	21	cbvrabv	⊢ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 }
23	22	uneq2i	⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } )
24		isoeq1	⊢ ( 𝑔 = 𝑓 → ( 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ↔ 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) )
25	24	cbviotavw	⊢ ( ℩ 𝑔 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) )
26		id	⊢ ( 𝑤 = 𝑥 → 𝑤 = 𝑥 )
27		oveq2	⊢ ( 𝑤 = 𝑥 → ( 𝐵 − 𝑤 ) = ( 𝐵 − 𝑥 ) )
28	27	oveq1d	⊢ ( 𝑤 = 𝑥 → ( ( 𝐵 − 𝑤 ) / 𝑇 ) = ( ( 𝐵 − 𝑥 ) / 𝑇 ) )
29	28	fveq2d	⊢ ( 𝑤 = 𝑥 → ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) )
30	29	oveq1d	⊢ ( 𝑤 = 𝑥 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) )
31	26 30	oveq12d	⊢ ( 𝑤 = 𝑥 → ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) )
32	31	cbvmptv	⊢ ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) )
33		eqeq1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 = 𝐵 ↔ 𝑦 = 𝐵 ) )
34		id	⊢ ( 𝑤 = 𝑦 → 𝑤 = 𝑦 )
35	33 34	ifbieq2d	⊢ ( 𝑤 = 𝑦 → if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) = if ( 𝑦 = 𝐵 , 𝐴 , 𝑦 ) )
36	35	cbvmptv	⊢ ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) = ( 𝑦 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑦 = 𝐵 , 𝐴 , 𝑦 ) )
37		fveq2	⊢ ( 𝑧 = 𝑥 → ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) = ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) )
38	37	fveq2d	⊢ ( 𝑧 = 𝑥 → ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) = ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) )
39	38	breq2d	⊢ ( 𝑧 = 𝑥 → ( ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) ↔ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ) )
40	39	rabbidv	⊢ ( 𝑧 = 𝑥 → { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) } = { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } )
41		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 𝑖 ) )
42	41	breq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ↔ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ) )
43	42	cbvrabv	⊢ { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } = { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) }
44	40 43	eqtrdi	⊢ ( 𝑧 = 𝑥 → { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) } = { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } )
45	44	supeq1d	⊢ ( 𝑧 = 𝑥 → sup ( { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) } , ℝ , < ) = sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } , ℝ , < ) )
46	45	cbvmptv	⊢ ( 𝑧 ∈ ℝ ↦ sup ( { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) } , ℝ , < ) ) = ( 𝑥 ∈ ℝ ↦ sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } , ℝ , < ) )
47	1 2 3 4 5 6 7 8 9 10 11 12 13 23 25 32 36 46	fourierdlem100	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )