Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem75 Structured version   Visualization version   GIF version

Theorem fourierdlem75 39074
 Description: Given a piecewise smooth function 𝐹, the derived function 𝐻 has a limit at the lower bound of each interval of the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem75.xre (𝜑𝑋 ∈ ℝ)
fourierdlem75.p 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem75.f (𝜑𝐹:ℝ⟶ℝ)
fourierdlem75.x (𝜑𝑋 ∈ ran 𝑉)
fourierdlem75.y (𝜑𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))
fourierdlem75.w (𝜑𝑊 ∈ ℝ)
fourierdlem75.h 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem75.m (𝜑𝑀 ∈ ℕ)
fourierdlem75.v (𝜑𝑉 ∈ (𝑃𝑀))
fourierdlem75.r ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))
fourierdlem75.q 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
fourierdlem75.o 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem75.g 𝐺 = (ℝ D 𝐹)
fourierdlem75.gcn ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
fourierdlem75.e (𝜑𝐸 ∈ ((𝐺 ↾ (𝑋(,)+∞)) lim 𝑋))
fourierdlem75.a 𝐴 = if((𝑉𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖)))
Assertion
Ref Expression
fourierdlem75 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
Distinct variable groups:   𝐸,𝑠   𝐹,𝑠   𝐻,𝑠   𝑖,𝑀,𝑚,𝑝   𝑀,𝑠,𝑖   𝑄,𝑖,𝑝   𝑄,𝑠   𝑅,𝑠   𝑖,𝑉,𝑝   𝑉,𝑠   𝑊,𝑠   𝑖,𝑋,𝑚,𝑝   𝑋,𝑠   𝑌,𝑠   𝜑,𝑖,𝑠
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝐴(𝑖,𝑚,𝑠,𝑝)   𝑃(𝑖,𝑚,𝑠,𝑝)   𝑄(𝑚)   𝑅(𝑖,𝑚,𝑝)   𝐸(𝑖,𝑚,𝑝)   𝐹(𝑖,𝑚,𝑝)   𝐺(𝑖,𝑚,𝑠,𝑝)   𝐻(𝑖,𝑚,𝑝)   𝑂(𝑖,𝑚,𝑠,𝑝)   𝑉(𝑚)   𝑊(𝑖,𝑚,𝑝)   𝑌(𝑖,𝑚,𝑝)

Proof of Theorem fourierdlem75
Dummy variables 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem75.xre . . . . 5 (𝜑𝑋 ∈ ℝ)
21ad2antrr 758 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝑋 ∈ ℝ)
3 fourierdlem75.v . . . . . . . . . 10 (𝜑𝑉 ∈ (𝑃𝑀))
4 fourierdlem75.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℕ)
5 fourierdlem75.p . . . . . . . . . . . 12 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
65fourierdlem2 39002 . . . . . . . . . . 11 (𝑀 ∈ ℕ → (𝑉 ∈ (𝑃𝑀) ↔ (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))))
74, 6syl 17 . . . . . . . . . 10 (𝜑 → (𝑉 ∈ (𝑃𝑀) ↔ (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))))
83, 7mpbid 221 . . . . . . . . 9 (𝜑 → (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))))
98simpld 474 . . . . . . . 8 (𝜑𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)))
10 elmapi 7765 . . . . . . . 8 (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
119, 10syl 17 . . . . . . 7 (𝜑𝑉:(0...𝑀)⟶ℝ)
1211adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
13 fzofzp1 12431 . . . . . . 7 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
1413adantl 481 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
1512, 14ffvelrnd 6268 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
1615adantr 480 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
17 eqcom 2617 . . . . . . 7 ((𝑉𝑖) = 𝑋𝑋 = (𝑉𝑖))
1817biimpi 205 . . . . . 6 ((𝑉𝑖) = 𝑋𝑋 = (𝑉𝑖))
1918adantl 481 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝑋 = (𝑉𝑖))
208simprrd 793 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))
2120r19.21bi 2916 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
2221adantr 480 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
2319, 22eqbrtrd 4605 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1)))
24 fourierdlem75.f . . . . . . 7 (𝜑𝐹:ℝ⟶ℝ)
2524adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ)
26 ioossre 12106 . . . . . . 7 (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ
2726a1i 11 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)
2825, 27fssresd 5984 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
2928adantr 480 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
30 limcresi 23455 . . . . . . . 8 ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ⊆ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋)
31 fourierdlem75.y . . . . . . . 8 (𝜑𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))
3230, 31sseldi 3566 . . . . . . 7 (𝜑𝑌 ∈ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋))
3332adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋))
34 pnfxr 9971 . . . . . . . . . 10 +∞ ∈ ℝ*
3534a1i 11 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → +∞ ∈ ℝ*)
3615rexrd 9968 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ*)
3715ltpnfd 11831 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) < +∞)
3836, 35, 37xrltled 38427 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ≤ +∞)
39 iooss2 12082 . . . . . . . . 9 ((+∞ ∈ ℝ* ∧ (𝑉‘(𝑖 + 1)) ≤ +∞) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ (𝑋(,)+∞))
4035, 38, 39syl2anc 691 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ (𝑋(,)+∞))
4140resabs1d 5348 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
4241oveq1d 6564 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋))
4333, 42eleqtrd 2690 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋))
4443adantr 480 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋))
45 eqid 2610 . . . 4 (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
46 ax-resscn 9872 . . . . . . . . . . 11 ℝ ⊆ ℂ
4746a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ⊆ ℂ)
4824, 47fssd 5970 . . . . . . . . . 10 (𝜑𝐹:ℝ⟶ℂ)
49 ssid 3587 . . . . . . . . . . 11 ℝ ⊆ ℝ
5049a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ⊆ ℝ)
5126a1i 11 . . . . . . . . . 10 (𝜑 → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)
52 eqid 2610 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
5352tgioo2 22414 . . . . . . . . . . 11 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
5452, 53dvres 23481 . . . . . . . . . 10 (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑋(,)(𝑉‘(𝑖 + 1))))))
5547, 48, 50, 51, 54syl22anc 1319 . . . . . . . . 9 (𝜑 → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑋(,)(𝑉‘(𝑖 + 1))))))
56 fourierdlem75.g . . . . . . . . . . 11 𝐺 = (ℝ D 𝐹)
5756eqcomi 2619 . . . . . . . . . 10 (ℝ D 𝐹) = 𝐺
58 ioontr 38583 . . . . . . . . . 10 ((int‘(topGen‘ran (,)))‘(𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1)))
5957, 58reseq12i 5315 . . . . . . . . 9 ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))
6055, 59syl6eq 2660 . . . . . . . 8 (𝜑 → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
6160adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑉𝑖) = 𝑋) → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
6261dmeqd 5248 . . . . . 6 ((𝜑 ∧ (𝑉𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
6362adantlr 747 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
64 fourierdlem75.gcn . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
6564adantr 480 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
66 oveq1 6556 . . . . . . . . . . 11 ((𝑉𝑖) = 𝑋 → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
6766reseq2d 5317 . . . . . . . . . 10 ((𝑉𝑖) = 𝑋 → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
6867feq1d 5943 . . . . . . . . 9 ((𝑉𝑖) = 𝑋 → ((𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ))
6968adantl 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → ((𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ))
7065, 69mpbid 221 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
7166adantl 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
7271feq2d 5944 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → ((𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ))
7370, 72mpbid 221 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
74 fdm 5964 . . . . . 6 ((𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ → dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
7573, 74syl 17 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
7663, 75eqtrd 2644 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
77 limcresi 23455 . . . . . . . 8 ((𝐺 ↾ (𝑋(,)+∞)) lim 𝑋) ⊆ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋)
78 fourierdlem75.e . . . . . . . 8 (𝜑𝐸 ∈ ((𝐺 ↾ (𝑋(,)+∞)) lim 𝑋))
7977, 78sseldi 3566 . . . . . . 7 (𝜑𝐸 ∈ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋))
8079adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋))
8140resabs1d 5348 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
8260adantr 480 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
8381, 82eqtr4d 2647 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))))
8483oveq1d 6564 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim 𝑋) = ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) lim 𝑋))
8580, 84eleqtrd 2690 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) lim 𝑋))
8685adantr 480 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝐸 ∈ ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) lim 𝑋))
87 eqid 2610 . . . 4 (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠))
88 oveq2 6557 . . . . . . 7 (𝑥 = 𝑠 → (𝑋 + 𝑥) = (𝑋 + 𝑠))
8988fveq2d 6107 . . . . . 6 (𝑥 = 𝑠 → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑥)) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)))
9089oveq1d 6564 . . . . 5 (𝑥 = 𝑠 → (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑥)) − 𝑌) = (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌))
9190cbvmptv 4678 . . . 4 (𝑥 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑥)) − 𝑌)) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌))
92 id 22 . . . . 5 (𝑥 = 𝑠𝑥 = 𝑠)
9392cbvmptv 4678 . . . 4 (𝑥 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ 𝑥) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ 𝑠)
942, 16, 23, 29, 44, 45, 76, 86, 87, 91, 93fourierdlem61 39060 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝐸 ∈ ((𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) lim 0))
95 fourierdlem75.a . . . . 5 𝐴 = if((𝑉𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖)))
96 iftrue 4042 . . . . 5 ((𝑉𝑖) = 𝑋 → if((𝑉𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖))) = 𝐸)
9795, 96syl5eq 2656 . . . 4 ((𝑉𝑖) = 𝑋𝐴 = 𝐸)
9897adantl 481 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝐴 = 𝐸)
99 fourierdlem75.h . . . . . . 7 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
10099reseq1i 5313 . . . . . 6 (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
101100a1i 11 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
102 ioossicc 12130 . . . . . . . 8 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
103 pire 24014 . . . . . . . . . . . 12 π ∈ ℝ
104103renegcli 10221 . . . . . . . . . . 11 -π ∈ ℝ
105104rexri 9976 . . . . . . . . . 10 -π ∈ ℝ*
106105a1i 11 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ*)
107103rexri 9976 . . . . . . . . . 10 π ∈ ℝ*
108107a1i 11 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → π ∈ ℝ*)
109104a1i 11 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → -π ∈ ℝ)
110103a1i 11 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → π ∈ ℝ)
111104a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → -π ∈ ℝ)
112111, 1readdcld 9948 . . . . . . . . . . . . . . . 16 (𝜑 → (-π + 𝑋) ∈ ℝ)
113103a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → π ∈ ℝ)
114113, 1readdcld 9948 . . . . . . . . . . . . . . . 16 (𝜑 → (π + 𝑋) ∈ ℝ)
115112, 114iccssred 38574 . . . . . . . . . . . . . . 15 (𝜑 → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
116115adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
1175, 4, 3fourierdlem15 39015 . . . . . . . . . . . . . . 15 (𝜑𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)))
118117ffvelrnda 6267 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)))
119116, 118sseldd 3569 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ℝ)
1201adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
121119, 120resubcld 10337 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
122111recnd 9947 . . . . . . . . . . . . . . . 16 (𝜑 → -π ∈ ℂ)
1231recnd 9947 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ ℂ)
124122, 123pncand 10272 . . . . . . . . . . . . . . 15 (𝜑 → ((-π + 𝑋) − 𝑋) = -π)
125124eqcomd 2616 . . . . . . . . . . . . . 14 (𝜑 → -π = ((-π + 𝑋) − 𝑋))
126125adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → -π = ((-π + 𝑋) − 𝑋))
127112adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → (-π + 𝑋) ∈ ℝ)
128114adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0...𝑀)) → (π + 𝑋) ∈ ℝ)
129 elicc2 12109 . . . . . . . . . . . . . . . . 17 (((-π + 𝑋) ∈ ℝ ∧ (π + 𝑋) ∈ ℝ) → ((𝑉𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)) ↔ ((𝑉𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉𝑖) ∧ (𝑉𝑖) ≤ (π + 𝑋))))
130127, 128, 129syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)) ↔ ((𝑉𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉𝑖) ∧ (𝑉𝑖) ≤ (π + 𝑋))))
131118, 130mpbid 221 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉𝑖) ∧ (𝑉𝑖) ≤ (π + 𝑋)))
132131simp2d 1067 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → (-π + 𝑋) ≤ (𝑉𝑖))
133127, 119, 120, 132lesub1dd 10522 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → ((-π + 𝑋) − 𝑋) ≤ ((𝑉𝑖) − 𝑋))
134126, 133eqbrtrd 4605 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → -π ≤ ((𝑉𝑖) − 𝑋))
135131simp3d 1068 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ≤ (π + 𝑋))
136119, 128, 120, 135lesub1dd 10522 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ≤ ((π + 𝑋) − 𝑋))
137110recnd 9947 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → π ∈ ℂ)
138123adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℂ)
139137, 138pncand 10272 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → ((π + 𝑋) − 𝑋) = π)
140136, 139breqtrd 4609 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ≤ π)
141109, 110, 121, 134, 140eliccd 38573 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ (-π[,]π))
142 fourierdlem75.q . . . . . . . . . . 11 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
143141, 142fmptd 6292 . . . . . . . . . 10 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
144143adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
145 simpr 476 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
146106, 108, 144, 145fourierdlem8 39008 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
147102, 146syl5ss 3579 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
148147resmptd 5371 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))))
149148adantr 480 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))))
150 elfzofz 12354 . . . . . . . 8 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
151 simpr 476 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
152142fvmpt2 6200 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ (-π[,]π)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
153151, 141, 152syl2anc 691 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
154153adantr 480 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
155 oveq1 6556 . . . . . . . . . 10 ((𝑉𝑖) = 𝑋 → ((𝑉𝑖) − 𝑋) = (𝑋𝑋))
156155adantl 481 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → ((𝑉𝑖) − 𝑋) = (𝑋𝑋))
157123ad2antrr 758 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝑋 ∈ ℂ)
158157subidd 10259 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑋𝑋) = 0)
159154, 156, 1583eqtrd 2648 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑄𝑖) = 0)
160150, 159sylanl2 681 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑄𝑖) = 0)
161 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑉𝑖) = (𝑉𝑗))
162161oveq1d 6564 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑗) − 𝑋))
163162cbvmptv 4678 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
164142, 163eqtri 2632 . . . . . . . . . 10 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
165164a1i 11 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋)))
166 fveq2 6103 . . . . . . . . . . 11 (𝑗 = (𝑖 + 1) → (𝑉𝑗) = (𝑉‘(𝑖 + 1)))
167166oveq1d 6564 . . . . . . . . . 10 (𝑗 = (𝑖 + 1) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
168167adantl 481 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
1691adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
17015, 169resubcld 10337 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
171165, 168, 14, 170fvmptd 6197 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
172171adantr 480 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
173160, 172oveq12d 6567 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)))
174 simplr 788 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 0) → 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
175 fourierdlem75.o . . . . . . . . . . . . 13 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
1764adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑠 = 0) → 𝑀 ∈ ℕ)
177111, 113, 1, 5, 175, 4, 3, 142fourierdlem14 39014 . . . . . . . . . . . . . 14 (𝜑𝑄 ∈ (𝑂𝑀))
178177adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑠 = 0) → 𝑄 ∈ (𝑂𝑀))
179 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑠 = 0) → 𝑠 = 0)
180 fourierdlem75.x . . . . . . . . . . . . . . . . . 18 (𝜑𝑋 ∈ ran 𝑉)
181 ffn 5958 . . . . . . . . . . . . . . . . . . 19 (𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)) → 𝑉 Fn (0...𝑀))
182 fvelrnb 6153 . . . . . . . . . . . . . . . . . . 19 (𝑉 Fn (0...𝑀) → (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (0...𝑀)(𝑉𝑖) = 𝑋))
183117, 181, 1823syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (0...𝑀)(𝑉𝑖) = 𝑋))
184180, 183mpbid 221 . . . . . . . . . . . . . . . . 17 (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑉𝑖) = 𝑋)
185159ex 449 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) = 𝑋 → (𝑄𝑖) = 0))
186185reximdva 3000 . . . . . . . . . . . . . . . . 17 (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑉𝑖) = 𝑋 → ∃𝑖 ∈ (0...𝑀)(𝑄𝑖) = 0))
187184, 186mpd 15 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄𝑖) = 0)
188121, 142fmptd 6292 . . . . . . . . . . . . . . . . 17 (𝜑𝑄:(0...𝑀)⟶ℝ)
189 ffn 5958 . . . . . . . . . . . . . . . . 17 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
190 fvelrnb 6153 . . . . . . . . . . . . . . . . 17 (𝑄 Fn (0...𝑀) → (0 ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄𝑖) = 0))
191188, 189, 1903syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (0 ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄𝑖) = 0))
192187, 191mpbird 246 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ ran 𝑄)
193192adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠 = 0) → 0 ∈ ran 𝑄)
194179, 193eqeltrd 2688 . . . . . . . . . . . . 13 ((𝜑𝑠 = 0) → 𝑠 ∈ ran 𝑄)
195175, 176, 178, 194fourierdlem12 39012 . . . . . . . . . . . 12 (((𝜑𝑠 = 0) ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
196195an32s 842 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = 0) → ¬ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
197196adantlr 747 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 0) → ¬ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
198174, 197pm2.65da 598 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
199198adantlr 747 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
200199iffalsed 4047 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
201160eqcomd 2616 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 0 = (𝑄𝑖))
202201adantr 480 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 = (𝑄𝑖))
203 elioo3g 12075 . . . . . . . . . . . . . 14 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑠 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑠𝑠 < (𝑄‘(𝑖 + 1)))))
204203biimpi 205 . . . . . . . . . . . . 13 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑠 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑠𝑠 < (𝑄‘(𝑖 + 1)))))
205204simprld 791 . . . . . . . . . . . 12 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑄𝑖) < 𝑠)
206205adantl 481 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) < 𝑠)
207202, 206eqbrtrd 4605 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 < 𝑠)
208207iftrued 4044 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌)
209208oveq2d 6565 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑌))
210209oveq1d 6564 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) = (((𝐹‘(𝑋 + 𝑠)) − 𝑌) / 𝑠))
2111rexrd 9968 . . . . . . . . . . . . 13 (𝜑𝑋 ∈ ℝ*)
212211ad3antrrr 762 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ*)
21336ad2antrr 758 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈ ℝ*)
214169ad2antrr 758 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
215 elioore 12076 . . . . . . . . . . . . . 14 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
216215adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
217214, 216readdcld 9948 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
218216, 207elrpd 11745 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ+)
219214, 218ltaddrpd 11781 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 < (𝑋 + 𝑠))
220215adantl 481 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
221188adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
222221, 14ffvelrnd 6268 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
223222adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
2241ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
225204simprrd 793 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 < (𝑄‘(𝑖 + 1)))
226225adantl 481 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
227220, 223, 224, 226ltadd2dd 10075 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑋 + (𝑄‘(𝑖 + 1))))
228171oveq2d 6565 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)))
229123adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
23015recnd 9947 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
231229, 230pncan3d 10274 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)) = (𝑉‘(𝑖 + 1)))
232228, 231eqtrd 2644 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
233232adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
234227, 233breqtrd 4609 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1)))
235234adantlr 747 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1)))
236212, 213, 217, 219, 235eliood 38567 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ (𝑋(,)(𝑉‘(𝑖 + 1))))
237 fvres 6117 . . . . . . . . . . 11 ((𝑋 + 𝑠) ∈ (𝑋(,)(𝑉‘(𝑖 + 1))) → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
238236, 237syl 17 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
239238eqcomd 2616 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)))
240239oveq1d 6564 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − 𝑌) = (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌))
241240oveq1d 6564 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − 𝑌) / 𝑠) = ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠))
242200, 210, 2413eqtrd 2648 . . . . . 6 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠))
243173, 242mpteq12dva 4662 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)))
244101, 149, 2433eqtrd 2648 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)))
245244, 160oveq12d 6567 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) lim 0))
24694, 98, 2453eltr4d 2703 . 2 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
247 eqid 2610 . . . . 5 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)))
248 eqid 2610 . . . . 5 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠)
249 eqid 2610 . . . . 5 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
25024adantr 480 . . . . . . . . . 10 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℝ)
2511adantr 480 . . . . . . . . . . 11 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
252215adantl 481 . . . . . . . . . . 11 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
253251, 252readdcld 9948 . . . . . . . . . 10 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
254250, 253ffvelrnd 6268 . . . . . . . . 9 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
255254recnd 9947 . . . . . . . 8 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
256255adantlr 747 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
2572563adantl3 1212 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
258 limccl 23445 . . . . . . . . . 10 ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ⊆ ℂ
259258, 31sseldi 3566 . . . . . . . . 9 (𝜑𝑌 ∈ ℂ)
260 fourierdlem75.w . . . . . . . . . 10 (𝜑𝑊 ∈ ℝ)
261260recnd 9947 . . . . . . . . 9 (𝜑𝑊 ∈ ℂ)
262259, 261ifcld 4081 . . . . . . . 8 (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
263262adantr 480 . . . . . . 7 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
2642633ad2antl1 1216 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
265257, 264subcld 10271 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
266215recnd 9947 . . . . . . 7 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℂ)
267266adantl 481 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
268 velsn 4141 . . . . . . . 8 (𝑠 ∈ {0} ↔ 𝑠 = 0)
269198, 268sylnibr 318 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 ∈ {0})
2702693adantl3 1212 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 ∈ {0})
271267, 270eldifd 3551 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (ℂ ∖ {0}))
272 eqid 2610 . . . . . . . . . 10 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠)))
273 eqid 2610 . . . . . . . . . 10 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊)
274 eqid 2610 . . . . . . . . . 10 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊))
275261ad2antrr 758 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑊 ∈ ℂ)
276 ioossre 12106 . . . . . . . . . . . 12 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
277276a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
278150, 119sylan2 490 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ)
279278rexrd 9968 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ*)
280279adantr 480 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉𝑖) ∈ ℝ*)
28136adantr 480 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈ ℝ*)
282253adantlr 747 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
283 iccssre 12126 . . . . . . . . . . . . . . . . . . 19 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
284104, 103, 283mp2an 704 . . . . . . . . . . . . . . . . . 18 (-π[,]π) ⊆ ℝ
285284, 46sstri 3577 . . . . . . . . . . . . . . . . 17 (-π[,]π) ⊆ ℂ
286153, 141eqeltrd 2688 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ (-π[,]π))
287150, 286sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ (-π[,]π))
288285, 287sseldi 3566 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℂ)
289229, 288addcomd 10117 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄𝑖)) = ((𝑄𝑖) + 𝑋))
290150adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
291150, 121sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
292142fvmpt2 6200 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ ℝ) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
293290, 291, 292syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
294293oveq1d 6564 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) + 𝑋) = (((𝑉𝑖) − 𝑋) + 𝑋))
295278recnd 9947 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℂ)
296295, 229npcand 10275 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝑉𝑖) − 𝑋) + 𝑋) = (𝑉𝑖))
297289, 294, 2963eqtrrd 2649 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) = (𝑋 + (𝑄𝑖)))
298297adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉𝑖) = (𝑋 + (𝑄𝑖)))
299293, 291eqeltrd 2688 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
300299adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) ∈ ℝ)
301205adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) < 𝑠)
302300, 220, 224, 301ltadd2dd 10075 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄𝑖)) < (𝑋 + 𝑠))
303298, 302eqbrtrd 4605 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉𝑖) < (𝑋 + 𝑠))
304280, 281, 282, 303, 234eliood 38567 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
305 ioossre 12106 . . . . . . . . . . . 12 ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ
306305a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)
307300, 301gtned 10051 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ (𝑄𝑖))
308 fourierdlem75.r . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))
309297oveq2d 6565 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)) = ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑋 + (𝑄𝑖))))
310308, 309eleqtrd 2690 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑋 + (𝑄𝑖))))
31125, 169, 277, 272, 304, 306, 307, 310, 288fourierdlem53 39052 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) lim (𝑄𝑖)))
312 ioosscn 38563 . . . . . . . . . . . 12 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
313312a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
314261adantr 480 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ℂ)
315273, 313, 314, 288constlimc 38691 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) lim (𝑄𝑖)))
316272, 273, 274, 256, 275, 311, 315sublimc 38719 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑅𝑊) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim (𝑄𝑖)))
317316adantr 480 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (𝑅𝑊) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim (𝑄𝑖)))
318 iftrue 4042 . . . . . . . . . 10 ((𝑉𝑖) < 𝑋 → if((𝑉𝑖) < 𝑋, 𝑊, 𝑌) = 𝑊)
319318oveq2d 6565 . . . . . . . . 9 ((𝑉𝑖) < 𝑋 → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅𝑊))
320319adantl 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅𝑊))
321215adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
322 0red 9920 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℝ)
323222ad2antrr 758 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
324225adantl 481 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
325171adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
326279ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉𝑖) ∈ ℝ*)
32736ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ*)
328169ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ℝ)
329 simplr 788 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉𝑖) < 𝑋)
330 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋)
3311ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ℝ)
33215adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
333331, 332ltnled 10063 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑋 < (𝑉‘(𝑖 + 1)) ↔ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋))
334330, 333mpbird 246 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1)))
335334adantlr 747 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1)))
336326, 327, 328, 329, 335eliood 38567 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
3375, 4, 3, 180fourierdlem12 39012 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
338337ad2antrr 758 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → ¬ 𝑋 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
339336, 338condan 831 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (𝑉‘(𝑖 + 1)) ≤ 𝑋)
34015adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
3411ad2antrr 758 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → 𝑋 ∈ ℝ)
342340, 341suble0d 10497 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (((𝑉‘(𝑖 + 1)) − 𝑋) ≤ 0 ↔ (𝑉‘(𝑖 + 1)) ≤ 𝑋))
343339, 342mpbird 246 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → ((𝑉‘(𝑖 + 1)) − 𝑋) ≤ 0)
344325, 343eqbrtrd 4605 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (𝑄‘(𝑖 + 1)) ≤ 0)
345344adantr 480 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ≤ 0)
346321, 323, 322, 324, 345ltletrd 10076 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < 0)
347321, 322, 346ltnsymd 10065 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 0 < 𝑠)
348347iffalsed 4047 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑊)
349348oveq2d 6565 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑊))
350349mpteq2dva 4672 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)))
351350oveq1d 6564 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄𝑖)) = ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim (𝑄𝑖)))
352317, 320, 3513eltr4d 2703 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉𝑖) < 𝑋) → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄𝑖)))
3533523adantl3 1212 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ (𝑉𝑖) < 𝑋) → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄𝑖)))
354 eqid 2610 . . . . . . . . . 10 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌)
355 eqid 2610 . . . . . . . . . 10 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌))
356259ad2antrr 758 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑌 ∈ ℂ)
357259adantr 480 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ℂ)
358354, 313, 357, 288constlimc 38691 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) lim (𝑄𝑖)))
359272, 354, 355, 256, 356, 311, 358sublimc 38719 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑅𝑌) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) lim (𝑄𝑖)))
360359adantr 480 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → (𝑅𝑌) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) lim (𝑄𝑖)))
361 iffalse 4045 . . . . . . . . . 10 (¬ (𝑉𝑖) < 𝑋 → if((𝑉𝑖) < 𝑋, 𝑊, 𝑌) = 𝑌)
362361oveq2d 6565 . . . . . . . . 9 (¬ (𝑉𝑖) < 𝑋 → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅𝑌))
363362adantl 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅𝑌))
364 0red 9920 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℝ)
365299ad2antrr 758 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) ∈ ℝ)
366215adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
3671ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → 𝑋 ∈ ℝ)
368278adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → (𝑉𝑖) ∈ ℝ)
369 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → ¬ (𝑉𝑖) < 𝑋)
370367, 368, 369nltled 10066 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → 𝑋 ≤ (𝑉𝑖))
371368, 367subge0d 10496 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → (0 ≤ ((𝑉𝑖) − 𝑋) ↔ 𝑋 ≤ (𝑉𝑖)))
372370, 371mpbird 246 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → 0 ≤ ((𝑉𝑖) − 𝑋))
373293eqcomd 2616 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) = (𝑄𝑖))
374373adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → ((𝑉𝑖) − 𝑋) = (𝑄𝑖))
375372, 374breqtrd 4609 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → 0 ≤ (𝑄𝑖))
376375adantr 480 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ≤ (𝑄𝑖))
377205adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) < 𝑠)
378364, 365, 366, 376, 377lelttrd 10074 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 < 𝑠)
379378iftrued 4044 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌)
380379oveq2d 6565 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑌))
381380mpteq2dva 4672 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)))
382381oveq1d 6564 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄𝑖)) = ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) lim (𝑄𝑖)))
383360, 363, 3823eltr4d 2703 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) < 𝑋) → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄𝑖)))
3843833adantl3 1212 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ ¬ (𝑉𝑖) < 𝑋) → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄𝑖)))
385353, 384pm2.61dan 828 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → (𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄𝑖)))
386313, 248, 288idlimc 38693 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) lim (𝑄𝑖)))
3873863adant3 1074 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → (𝑄𝑖) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) lim (𝑄𝑖)))
3882933adant3 1074 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
3892953adant3 1074 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → (𝑉𝑖) ∈ ℂ)
3902293adant3 1074 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → 𝑋 ∈ ℂ)
391 neqne 2790 . . . . . . . 8 (¬ (𝑉𝑖) = 𝑋 → (𝑉𝑖) ≠ 𝑋)
3923913ad2ant3 1077 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → (𝑉𝑖) ≠ 𝑋)
393389, 390, 392subne0d 10280 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → ((𝑉𝑖) − 𝑋) ≠ 0)
394388, 393eqnetrd 2849 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → (𝑄𝑖) ≠ 0)
3951983adantl3 1212 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
396395neqned 2789 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ 0)
397247, 248, 249, 265, 271, 385, 387, 394, 396divlimc 38723 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) lim (𝑄𝑖)))
398 iffalse 4045 . . . . . 6 (¬ (𝑉𝑖) = 𝑋 → if((𝑉𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖))) = ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖)))
39995, 398syl5eq 2656 . . . . 5 (¬ (𝑉𝑖) = 𝑋𝐴 = ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖)))
4003993ad2ant3 1077 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → 𝐴 = ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖)))
401 ioossre 12106 . . . . . . . . . . . . 13 (𝑋(,)+∞) ⊆ ℝ
402401a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
40324, 402fssresd 5984 . . . . . . . . . . 11 (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
404401, 47syl5ss 3579 . . . . . . . . . . 11 (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
40534a1i 11 . . . . . . . . . . . 12 (𝜑 → +∞ ∈ ℝ*)
4061ltpnfd 11831 . . . . . . . . . . . 12 (𝜑𝑋 < +∞)
40752, 405, 1, 406lptioo1cn 38713 . . . . . . . . . . 11 (𝜑𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝑋(,)+∞)))
408403, 404, 407, 31limcrecl 38696 . . . . . . . . . 10 (𝜑𝑌 ∈ ℝ)
40924, 1, 408, 260, 99fourierdlem9 39009 . . . . . . . . 9 (𝜑𝐻:(-π[,]π)⟶ℝ)
410409adantr 480 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐻:(-π[,]π)⟶ℝ)
411410, 147feqresmpt 6160 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻𝑠)))
412147sselda 3568 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (-π[,]π))
413 0cnd 9912 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℂ)
414262ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
415256, 414subcld 10271 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
416266adantl 481 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
417198neqned 2789 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ 0)
418415, 416, 417divcld 10680 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℂ)
419413, 418ifcld 4081 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℂ)
42099fvmpt2 6200 . . . . . . . . . 10 ((𝑠 ∈ (-π[,]π) ∧ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℂ) → (𝐻𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
421412, 419, 420syl2anc 691 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
422198iffalsed 4047 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
423421, 422eqtrd 2644 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
424423mpteq2dva 4672 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻𝑠)) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
425411, 424eqtrd 2644 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
4264253adant3 1074 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
427426oveq1d 6564 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) lim (𝑄𝑖)))
428397, 400, 4273eltr4d 2703 . . 3 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
4294283expa 1257 . 2 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
430246, 429pm2.61dan 828 1 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  ℕcn 10897  (,)cioo 12046  [,]cicc 12049  ...cfz 12197  ..^cfzo 12334  πcpi 14636  TopOpenctopn 15905  topGenctg 15921  ℂfldccnfld 19567  intcnt 20631   limℂ climc 23432   D cdv 23433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437 This theorem is referenced by:  fourierdlem85  39084  fourierdlem88  39087  fourierdlem103  39102  fourierdlem104  39103
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