Metamath Proof Explorer

Theorem fourierdlem94

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 fourierdlem94.f ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ )
fourierdlem94.t ⊢ 𝑇 = ( 2 · π )
fourierdlem94.per ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
fourierdlem94.x ⊢ ( 𝜑 → 𝑋 ∈ ℝ )
fourierdlem94.p ⊢ 𝑃 = ( 𝑛 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑛 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = - π ∧ ( 𝑝 ‘ 𝑛 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑛 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
fourierdlem94.m ⊢ ( 𝜑 → 𝑀 ∈ ℕ )
fourierdlem94.q ⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) )
fourierdlem94.dvcn ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
fourierdlem94.dvlb ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ≠ ∅ )
fourierdlem94.dvub ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ≠ ∅ )
Assertion fourierdlem94 ( 𝜑 → ( ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) ≠ ∅ ∧ ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) )

Proof

Step Hyp Ref Expression
1 fourierdlem94.f ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ )
2 fourierdlem94.t ⊢ 𝑇 = ( 2 · π )
3 fourierdlem94.per ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
4 fourierdlem94.x ⊢ ( 𝜑 → 𝑋 ∈ ℝ )
5 fourierdlem94.p ⊢ 𝑃 = ( 𝑛 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑛 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = - π ∧ ( 𝑝 ‘ 𝑛 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑛 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
6 fourierdlem94.m ⊢ ( 𝜑 → 𝑀 ∈ ℕ )
7 fourierdlem94.q ⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) )
8 fourierdlem94.dvcn ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
9 fourierdlem94.dvlb ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ≠ ∅ )
10 fourierdlem94.dvub ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ≠ ∅ )
11 pire ⊢ π ∈ ℝ
12 11 renegcli ⊢ - π ∈ ℝ
13 12 a1i ⊢ ( 𝜑 → - π ∈ ℝ )
14 11 a1i ⊢ ( 𝜑 → π ∈ ℝ )
15 negpilt0 ⊢ - π < 0
16 pipos ⊢ 0 < π
17 0re ⊢ 0 ∈ ℝ
18 12 17 11 lttri ⊢ ( ( - π < 0 ∧ 0 < π ) → - π < π )
19 15 16 18 mp2an ⊢ - π < π
20 19 a1i ⊢ ( 𝜑 → - π < π )
21 picn ⊢ π ∈ ℂ
22 21 2timesi ⊢ ( 2 · π ) = ( π + π )
23 21 21 subnegi ⊢ ( π − - π ) = ( π + π )
24 22 2 23 3eqtr4i ⊢ 𝑇 = ( π − - π )
25 ssid ⊢ ℝ ⊆ ℝ
26 25 a1i ⊢ ( 𝜑 → ℝ ⊆ ℝ )
27 simp2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℤ ) → 𝑥 ∈ ℝ )
28 zre ⊢ ( 𝑘 ∈ ℤ → 𝑘 ∈ ℝ )
29 28 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℤ ) → 𝑘 ∈ ℝ )
30 2re ⊢ 2 ∈ ℝ
31 30 11 remulcli ⊢ ( 2 · π ) ∈ ℝ
32 31 a1i ⊢ ( 𝜑 → ( 2 · π ) ∈ ℝ )
33 2 32 eqeltrid ⊢ ( 𝜑 → 𝑇 ∈ ℝ )
34 33 adantr ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → 𝑇 ∈ ℝ )
35 34 3adant2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℤ ) → 𝑇 ∈ ℝ )
36 29 35 remulcld ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℤ ) → ( 𝑘 · 𝑇 ) ∈ ℝ )
37 27 36 readdcld ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℤ ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ℝ )
38 simp1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℤ ) → 𝜑 )
39 simp3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℤ ) → 𝑘 ∈ ℤ )
40 ax-resscn ⊢ ℝ ⊆ ℂ
41 40 a1i ⊢ ( 𝜑 → ℝ ⊆ ℂ )
42 1 41 fssd ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ )
43 42 adantr ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → 𝐹 : ℝ ⟶ ℂ )
44 43 adantr ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑥 ∈ ℝ ) → 𝐹 : ℝ ⟶ ℂ )
45 34 adantr ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑥 ∈ ℝ ) → 𝑇 ∈ ℝ )
46 simplr ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑥 ∈ ℝ ) → 𝑘 ∈ ℤ )
47 simpr ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑥 ∈ ℝ ) → 𝑥 ∈ ℝ )
48 eleq1w ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ℝ ↔ 𝑦 ∈ ℝ ) )
49 48 anbi2d ⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ↔ ( 𝜑 ∧ 𝑦 ∈ ℝ ) ) )
50 oveq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 + 𝑇 ) = ( 𝑦 + 𝑇 ) )
51 50 fveq2d ⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) )
52 fveq2 ⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) )
53 51 52 eqeq12d ⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) ) )
54 49 53 imbi12d ⊢ ( 𝑥 = 𝑦 → ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) ) ) )
55 54 3 chvarvv ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) )
56 55 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) )
57 44 45 46 47 56 fperiodmul ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) )
58 38 39 27 57 syl21anc ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) )
59 40 a1i ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ℝ ⊆ ℂ )
60 ioossre ⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ
61 60 a1i ⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ )
62 1 61 fssresd ⊢ ( 𝜑 → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℝ )
63 62 41 fssd ⊢ ( 𝜑 → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
64 63 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
65 60 a1i ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ )
66 42 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 : ℝ ⟶ ℂ )
67 25 a1i ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ℝ ⊆ ℝ )
68 eqid ⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
69 tgioo4 ⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ )
70 68 69 dvres ⊢ ( ( ( ℝ ⊆ ℂ ∧ 𝐹 : ℝ ⟶ ℂ ) ∧ ( ℝ ⊆ ℝ ∧ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) ) → ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
71 59 66 67 65 70 syl22anc ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
72 71 dmeqd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = dom ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
73 ioontr ⊢ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
74 73 reseq2i ⊢ ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
75 74 dmeqi ⊢ dom ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = dom ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
76 75 a1i ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = dom ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
77 cncff ⊢ ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) → ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
78 fdm ⊢ ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ → dom ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
79 8 77 78 3syl ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
80 72 76 79 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
81 dvcn ⊢ ( ( ( ℝ ⊆ ℂ ∧ ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ∧ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) ∧ dom ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
82 59 64 65 80 81 syl31anc ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
83 65 40 sstrdi ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ )
84 5 fourierdlem2 ⊢ ( 𝑀 ∈ ℕ → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = - π ∧ ( 𝑄 ‘ 𝑀 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
85 6 84 syl ⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = - π ∧ ( 𝑄 ‘ 𝑀 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
86 7 85 mpbid ⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = - π ∧ ( 𝑄 ‘ 𝑀 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
87 86 simpld ⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) )
88 elmapi ⊢ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ )
89 87 88 syl ⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ )
90 89 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ )
91 elfzofz ⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) )
92 91 adantl ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) )
93 90 92 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
94 93 rexrd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* )
95 fzofzp1 ⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) )
96 95 adantl ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) )
97 90 96 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
98 86 simprrd ⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
99 98 r19.21bi ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
100 68 94 97 99 lptioo2cn ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
101 62 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℝ )
102 41 42 26 dvbss ⊢ ( 𝜑 → dom ( ℝ D 𝐹 ) ⊆ ℝ )
103 dvfre ⊢ ( ( 𝐹 : ℝ ⟶ ℝ ∧ ℝ ⊆ ℝ ) → ( ℝ D 𝐹 ) : dom ( ℝ D 𝐹 ) ⟶ ℝ )
104 1 26 103 syl2anc ⊢ ( 𝜑 → ( ℝ D 𝐹 ) : dom ( ℝ D 𝐹 ) ⟶ ℝ )
105 86 simprd ⊢ ( 𝜑 → ( ( ( 𝑄 ‘ 0 ) = - π ∧ ( 𝑄 ‘ 𝑀 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
106 105 simplld ⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = - π )
107 105 simplrd ⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = π )
108 8 77 syl ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
109 97 rexrd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
110 68 109 93 99 lptioo1cn ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
111 108 83 110 9 68 ellimciota ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℩ 𝑥 𝑥 ∈ ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) ∈ ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
112 108 83 100 10 68 ellimciota ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℩ 𝑥 𝑥 ∈ ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
113 28 adantl ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → 𝑘 ∈ ℝ )
114 113 34 remulcld ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → ( 𝑘 · 𝑇 ) ∈ ℝ )
115 43 adantr ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑡 ∈ ℝ ) → 𝐹 : ℝ ⟶ ℂ )
116 34 adantr ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑡 ∈ ℝ ) → 𝑇 ∈ ℝ )
117 simplr ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑡 ∈ ℝ ) → 𝑘 ∈ ℤ )
118 simpr ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑡 ∈ ℝ ) → 𝑡 ∈ ℝ )
119 3 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑡 ∈ ℝ ) ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
120 115 116 117 118 119 fperiodmul ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑡 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑡 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑡 ) )
121 eqid ⊢ ( ℝ D 𝐹 ) = ( ℝ D 𝐹 )
122 43 114 120 121 fperdvper ⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑡 ∈ dom ( ℝ D 𝐹 ) ) → ( ( 𝑡 + ( 𝑘 · 𝑇 ) ) ∈ dom ( ℝ D 𝐹 ) ∧ ( ( ℝ D 𝐹 ) ‘ ( 𝑡 + ( 𝑘 · 𝑇 ) ) ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) )
123 122 an32s ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ dom ( ℝ D 𝐹 ) ) ∧ 𝑘 ∈ ℤ ) → ( ( 𝑡 + ( 𝑘 · 𝑇 ) ) ∈ dom ( ℝ D 𝐹 ) ∧ ( ( ℝ D 𝐹 ) ‘ ( 𝑡 + ( 𝑘 · 𝑇 ) ) ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) )
124 123 simpld ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ dom ( ℝ D 𝐹 ) ) ∧ 𝑘 ∈ ℤ ) → ( 𝑡 + ( 𝑘 · 𝑇 ) ) ∈ dom ( ℝ D 𝐹 ) )
125 123 simprd ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ dom ( ℝ D 𝐹 ) ) ∧ 𝑘 ∈ ℤ ) → ( ( ℝ D 𝐹 ) ‘ ( 𝑡 + ( 𝑘 · 𝑇 ) ) ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) )
126 fveq2 ⊢ ( 𝑗 = 𝑖 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 𝑖 ) )
127 oveq1 ⊢ ( 𝑗 = 𝑖 → ( 𝑗 + 1 ) = ( 𝑖 + 1 ) )
128 127 fveq2d ⊢ ( 𝑗 = 𝑖 → ( 𝑄 ‘ ( 𝑗 + 1 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
129 126 128 oveq12d ⊢ ( 𝑗 = 𝑖 → ( ( 𝑄 ‘ 𝑗 ) (,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
130 129 cbvmptv ⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑗 ) (,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) = ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
131 eqid ⊢ ( 𝑡 ∈ ℝ ↦ ( 𝑡 + ( ( ⌊ ‘ ( ( π − 𝑡 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝑡 ∈ ℝ ↦ ( 𝑡 + ( ( ⌊ ‘ ( ( π − 𝑡 ) / 𝑇 ) ) · 𝑇 ) ) )
132 102 104 13 14 20 24 6 89 106 107 8 111 112 124 125 130 131 fourierdlem71 ⊢ ( 𝜑 → ∃ 𝑧 ∈ ℝ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 )
133 132 adantr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∃ 𝑧 ∈ ℝ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 )
134 nfv ⊢ Ⅎ 𝑡 ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) )
135 nfra1 ⊢ Ⅎ 𝑡 ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧
136 134 135 nfan ⊢ Ⅎ 𝑡 ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 )
137 71 74 eqtrdi ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
138 137 fveq1d ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) = ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) )
139 fvres ⊢ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) )
140 138 139 sylan9eq ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) )
141 140 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) ) = ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) )
142 141 adantlr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) ) = ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) )
143 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 )
144 ssdmres ⊢ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom ( ℝ D 𝐹 ) ↔ dom ( ( ℝ D 𝐹 ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
145 79 144 sylibr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom ( ℝ D 𝐹 ) )
146 145 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom ( ℝ D 𝐹 ) )
147 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
148 146 147 sseldd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ dom ( ℝ D 𝐹 ) )
149 rspa ⊢ ( ( ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ∧ 𝑡 ∈ dom ( ℝ D 𝐹 ) ) → ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 )
150 143 148 149 syl2anc ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 )
151 142 150 eqbrtrd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) ) ≤ 𝑧 )
152 151 ex ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( abs ‘ ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) ) ≤ 𝑧 ) )
153 136 152 ralrimi ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 ) → ∀ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) ) ≤ 𝑧 )
154 153 ex ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 → ∀ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) ) ≤ 𝑧 ) )
155 154 reximdv ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∃ 𝑧 ∈ ℝ ∀ 𝑡 ∈ dom ( ℝ D 𝐹 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ≤ 𝑧 → ∃ 𝑧 ∈ ℝ ∀ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) ) ≤ 𝑧 ) )
156 133 155 mpd ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∃ 𝑧 ∈ ℝ ∀ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ‘ 𝑡 ) ) ≤ 𝑧 )
157 93 97 101 80 156 ioodvbdlimc2 ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ≠ ∅ )
158 64 83 100 157 68 ellimciota ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℩ 𝑦 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
159 oveq2 ⊢ ( 𝑦 = 𝑥 → ( π − 𝑦 ) = ( π − 𝑥 ) )
160 159 oveq1d ⊢ ( 𝑦 = 𝑥 → ( ( π − 𝑦 ) / 𝑇 ) = ( ( π − 𝑥 ) / 𝑇 ) )
161 160 fveq2d ⊢ ( 𝑦 = 𝑥 → ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( π − 𝑥 ) / 𝑇 ) ) )
162 161 oveq1d ⊢ ( 𝑦 = 𝑥 → ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( π − 𝑥 ) / 𝑇 ) ) · 𝑇 ) )
163 162 cbvmptv ⊢ ( 𝑦 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) = ( 𝑥 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( π − 𝑥 ) / 𝑇 ) ) · 𝑇 ) )
164 id ⊢ ( 𝑧 = 𝑥 → 𝑧 = 𝑥 )
165 fveq2 ⊢ ( 𝑧 = 𝑥 → ( ( 𝑦 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ‘ 𝑧 ) = ( ( 𝑦 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ‘ 𝑥 ) )
166 164 165 oveq12d ⊢ ( 𝑧 = 𝑥 → ( 𝑧 + ( ( 𝑦 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ‘ 𝑧 ) ) = ( 𝑥 + ( ( 𝑦 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ‘ 𝑥 ) ) )
167 166 cbvmptv ⊢ ( 𝑧 ∈ ℝ ↦ ( 𝑧 + ( ( 𝑦 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ‘ 𝑧 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( 𝑦 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( π − 𝑦 ) / 𝑇 ) ) · 𝑇 ) ) ‘ 𝑥 ) ) )
168 13 14 20 5 24 6 7 26 1 37 58 82 158 4 163 167 fourierdlem49 ⊢ ( 𝜑 → ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) ≠ ∅ )
169 93 97 101 80 156 ioodvbdlimc1 ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ≠ ∅ )
170 64 83 110 169 68 ellimciota ⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℩ 𝑦 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
171 biid ⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) )
172 13 14 20 5 24 6 7 1 37 58 82 170 4 163 167 171 fourierdlem48 ⊢ ( 𝜑 → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ )
173 168 172 jca ⊢ ( 𝜑 → ( ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) ≠ ∅ ∧ ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) )