Metamath Proof Explorer

Theorem fourierdlem69

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

Ref Expression
Hypotheses fourierdlem69.p ⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
fourierdlem69.m ⊢ ( 𝜑 → 𝑀 ∈ ℕ )
fourierdlem69.q ⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) )
fourierdlem69.f ⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ )
fourierdlem69.fcn ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
fourierdlem69.r ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
fourierdlem69.l ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
Assertion fourierdlem69 ( 𝜑 → 𝐹 ∈ 𝐿1 )

Proof

Step Hyp Ref Expression
1 fourierdlem69.p ⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
2 fourierdlem69.m ⊢ ( 𝜑 → 𝑀 ∈ ℕ )
3 fourierdlem69.q ⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) )
4 fourierdlem69.f ⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ )
5 fourierdlem69.fcn ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
6 fourierdlem69.r ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
7 fourierdlem69.l ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
8 1 fourierdlem2 ⊢ ( 𝑀 ∈ ℕ → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
9 2 8 syl ⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
10 3 9 mpbid ⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
11 10 simprd ⊢ ( 𝜑 → ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
12 11 simpld ⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) )
13 12 simpld ⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = 𝐴 )
14 12 simprd ⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = 𝐵 )
15 13 14 oveq12d ⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) = ( 𝐴 [,] 𝐵 ) )
16 15 feq2d ⊢ ( 𝜑 → ( 𝐹 : ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ⟶ ℂ ↔ 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) )
17 4 16 mpbird ⊢ ( 𝜑 → 𝐹 : ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ⟶ ℂ )
18 17 feqmptd ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) )
19 nfv ⊢ Ⅎ 𝑥 𝜑
20 0zd ⊢ ( 𝜑 → 0 ∈ ℤ )
21 nnuz ⊢ ℕ = ( ℤ≥ ‘ 1 )
22 1e0p1 ⊢ 1 = ( 0 + 1 )
23 22 fveq2i ⊢ ( ℤ≥ ‘ 1 ) = ( ℤ≥ ‘ ( 0 + 1 ) )
24 21 23 eqtri ⊢ ℕ = ( ℤ≥ ‘ ( 0 + 1 ) )
25 2 24 eleqtrdi ⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ ( 0 + 1 ) ) )
26 10 simpld ⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) )
27 elmapi ⊢ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ )
28 26 27 syl ⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ )
29 28 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
30 11 simprd ⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
31 30 r19.21bi ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
32 4 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ )
33 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) )
34 13 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( 𝑄 ‘ 0 ) = 𝐴 )
35 14 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( 𝑄 ‘ 𝑀 ) = 𝐵 )
36 34 35 oveq12d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) = ( 𝐴 [,] 𝐵 ) )
37 33 36 eleqtrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
38 32 37 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ )
39 28 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ )
40 elfzofz ⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) )
41 40 adantl ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) )
42 39 41 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
43 fzofzp1 ⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) )
44 43 adantl ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) )
45 39 44 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
46 4 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ )
47 ioossicc ⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
48 1 2 3 fourierdlem11 ⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) )
49 48 simp1d ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
50 49 rexrd ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
51 50 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐴 ∈ ℝ* )
52 48 simp2d ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
53 52 rexrd ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
54 53 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐵 ∈ ℝ* )
55 1 2 3 fourierdlem15 ⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
56 55 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
57 simpr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) )
58 51 54 56 57 fourierdlem8 ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
59 47 58 sstrid ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
60 46 59 feqresmpt ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) )
61 60 5 eqeltrrd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
62 60 oveq1d ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
63 7 62 eleqtrd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
64 60 oveq1d ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
65 6 64 eleqtrd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
66 42 45 61 63 65 iblcncfioo ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )
67 46 adantr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ )
68 58 sselda ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
69 67 68 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ )
70 42 45 66 69 ibliooicc ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )
71 19 20 25 29 31 38 70 iblspltprt ⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )
72 18 71 eqeltrd ⊢ ( 𝜑 → 𝐹 ∈ 𝐿1 )