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Theorem fourierdlem74 46195
Description: Given a piecewise smooth function 𝐹, the derived function 𝐻 has a limit at the upper bound of each interval of the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem74.xre (𝜑𝑋 ∈ ℝ)
fourierdlem74.p 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem74.f (𝜑𝐹:ℝ⟶ℝ)
fourierdlem74.x (𝜑𝑋 ∈ ran 𝑉)
fourierdlem74.y (𝜑𝑌 ∈ ℝ)
fourierdlem74.w (𝜑𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))
fourierdlem74.h 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem74.m (𝜑𝑀 ∈ ℕ)
fourierdlem74.v (𝜑𝑉 ∈ (𝑃𝑀))
fourierdlem74.r ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))
fourierdlem74.q 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
fourierdlem74.o 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem74.g 𝐺 = (ℝ D 𝐹)
fourierdlem74.gcn ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
fourierdlem74.e (𝜑𝐸 ∈ ((𝐺 ↾ (-∞(,)𝑋)) lim 𝑋))
fourierdlem74.a 𝐴 = if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1))))
Assertion
Ref Expression
fourierdlem74 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
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 fourierdlem74
Dummy variables 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfzofz 13715 . . . . . 6 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
2 pire 26500 . . . . . . . . . . . 12 π ∈ ℝ
32renegcli 11570 . . . . . . . . . . 11 -π ∈ ℝ
43a1i 11 . . . . . . . . . 10 (𝜑 → -π ∈ ℝ)
5 fourierdlem74.xre . . . . . . . . . 10 (𝜑𝑋 ∈ ℝ)
64, 5readdcld 11290 . . . . . . . . 9 (𝜑 → (-π + 𝑋) ∈ ℝ)
72a1i 11 . . . . . . . . . 10 (𝜑 → π ∈ ℝ)
87, 5readdcld 11290 . . . . . . . . 9 (𝜑 → (π + 𝑋) ∈ ℝ)
96, 8iccssred 13474 . . . . . . . 8 (𝜑 → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
109adantr 480 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
11 fourierdlem74.p . . . . . . . . 9 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
12 fourierdlem74.m . . . . . . . . 9 (𝜑𝑀 ∈ ℕ)
13 fourierdlem74.v . . . . . . . . 9 (𝜑𝑉 ∈ (𝑃𝑀))
1411, 12, 13fourierdlem15 46137 . . . . . . . 8 (𝜑𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)))
1514ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)))
1610, 15sseldd 3984 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ℝ)
171, 16sylan2 593 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ)
1817adantr 480 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑉𝑖) ∈ ℝ)
195ad2antrr 726 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝑋 ∈ ℝ)
2011fourierdlem2 46124 . . . . . . . . . 10 (𝑀 ∈ ℕ → (𝑉 ∈ (𝑃𝑀) ↔ (𝑉 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))))
2112, 20syl 17 . . . . . . . . 9 (𝜑 → (𝑉 ∈ (𝑃𝑀) ↔ (𝑉 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))))
2213, 21mpbid 232 . . . . . . . 8 (𝜑 → (𝑉 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))))
2322simprrd 774 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))
2423r19.21bi 3251 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
2524adantr 480 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
26 eqcom 2744 . . . . . . 7 ((𝑉‘(𝑖 + 1)) = 𝑋𝑋 = (𝑉‘(𝑖 + 1)))
2726biimpi 216 . . . . . 6 ((𝑉‘(𝑖 + 1)) = 𝑋𝑋 = (𝑉‘(𝑖 + 1)))
2827adantl 481 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝑋 = (𝑉‘(𝑖 + 1)))
2925, 28breqtrrd 5171 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑉𝑖) < 𝑋)
30 fourierdlem74.f . . . . . 6 (𝜑𝐹:ℝ⟶ℝ)
31 ioossre 13448 . . . . . . 7 ((𝑉𝑖)(,)𝑋) ⊆ ℝ
3231a1i 11 . . . . . 6 (𝜑 → ((𝑉𝑖)(,)𝑋) ⊆ ℝ)
3330, 32fssresd 6775 . . . . 5 (𝜑 → (𝐹 ↾ ((𝑉𝑖)(,)𝑋)):((𝑉𝑖)(,)𝑋)⟶ℝ)
3433ad2antrr 726 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝐹 ↾ ((𝑉𝑖)(,)𝑋)):((𝑉𝑖)(,)𝑋)⟶ℝ)
35 limcresi 25920 . . . . . . . 8 ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ (((𝐹 ↾ (-∞(,)𝑋)) ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋)
36 fourierdlem74.w . . . . . . . 8 (𝜑𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))
3735, 36sselid 3981 . . . . . . 7 (𝜑𝑊 ∈ (((𝐹 ↾ (-∞(,)𝑋)) ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋))
3837adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ (((𝐹 ↾ (-∞(,)𝑋)) ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋))
39 mnfxr 11318 . . . . . . . . . 10 -∞ ∈ ℝ*
4039a1i 11 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → -∞ ∈ ℝ*)
4117rexrd 11311 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ*)
4217mnfltd 13166 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → -∞ < (𝑉𝑖))
4340, 41, 42xrltled 13192 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → -∞ ≤ (𝑉𝑖))
44 iooss1 13422 . . . . . . . . 9 ((-∞ ∈ ℝ* ∧ -∞ ≤ (𝑉𝑖)) → ((𝑉𝑖)(,)𝑋) ⊆ (-∞(,)𝑋))
4540, 43, 44syl2anc 584 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖)(,)𝑋) ⊆ (-∞(,)𝑋))
4645resabs1d 6026 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-∞(,)𝑋)) ↾ ((𝑉𝑖)(,)𝑋)) = (𝐹 ↾ ((𝑉𝑖)(,)𝑋)))
4746oveq1d 7446 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ (-∞(,)𝑋)) ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋) = ((𝐹 ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋))
4838, 47eleqtrd 2843 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋))
4948adantr 480 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝑊 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋))
50 eqid 2737 . . . 4 (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) = (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋)))
51 ax-resscn 11212 . . . . . . . . . 10 ℝ ⊆ ℂ
5251a1i 11 . . . . . . . . 9 (𝜑 → ℝ ⊆ ℂ)
5330, 52fssd 6753 . . . . . . . . 9 (𝜑𝐹:ℝ⟶ℂ)
54 ssid 4006 . . . . . . . . . 10 ℝ ⊆ ℝ
5554a1i 11 . . . . . . . . 9 (𝜑 → ℝ ⊆ ℝ)
56 eqid 2737 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
57 tgioo4 24826 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
5856, 57dvres 25946 . . . . . . . . 9 (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ ((𝑉𝑖)(,)𝑋) ⊆ ℝ)) → (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑉𝑖)(,)𝑋))))
5952, 53, 55, 32, 58syl22anc 839 . . . . . . . 8 (𝜑 → (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑉𝑖)(,)𝑋))))
60 fourierdlem74.g . . . . . . . . . . 11 𝐺 = (ℝ D 𝐹)
6160eqcomi 2746 . . . . . . . . . 10 (ℝ D 𝐹) = 𝐺
62 ioontr 45524 . . . . . . . . . 10 ((int‘(topGen‘ran (,)))‘((𝑉𝑖)(,)𝑋)) = ((𝑉𝑖)(,)𝑋)
6361, 62reseq12i 5995 . . . . . . . . 9 ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑉𝑖)(,)𝑋))) = (𝐺 ↾ ((𝑉𝑖)(,)𝑋))
6463a1i 11 . . . . . . . 8 (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑉𝑖)(,)𝑋))) = (𝐺 ↾ ((𝑉𝑖)(,)𝑋)))
6559, 64eqtrd 2777 . . . . . . 7 (𝜑 → (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) = (𝐺 ↾ ((𝑉𝑖)(,)𝑋)))
6665dmeqd 5916 . . . . . 6 (𝜑 → dom (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) = dom (𝐺 ↾ ((𝑉𝑖)(,)𝑋)))
6766ad2antrr 726 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → dom (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) = dom (𝐺 ↾ ((𝑉𝑖)(,)𝑋)))
68 fourierdlem74.gcn . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
6968adantr 480 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
70 oveq2 7439 . . . . . . . . . 10 ((𝑉‘(𝑖 + 1)) = 𝑋 → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) = ((𝑉𝑖)(,)𝑋))
7170reseq2d 5997 . . . . . . . . 9 ((𝑉‘(𝑖 + 1)) = 𝑋 → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = (𝐺 ↾ ((𝑉𝑖)(,)𝑋)))
7271, 70feq12d 6724 . . . . . . . 8 ((𝑉‘(𝑖 + 1)) = 𝑋 → ((𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ ↔ (𝐺 ↾ ((𝑉𝑖)(,)𝑋)):((𝑉𝑖)(,)𝑋)⟶ℝ))
7372adantl 481 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → ((𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ ↔ (𝐺 ↾ ((𝑉𝑖)(,)𝑋)):((𝑉𝑖)(,)𝑋)⟶ℝ))
7469, 73mpbid 232 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝐺 ↾ ((𝑉𝑖)(,)𝑋)):((𝑉𝑖)(,)𝑋)⟶ℝ)
75 fdm 6745 . . . . . 6 ((𝐺 ↾ ((𝑉𝑖)(,)𝑋)):((𝑉𝑖)(,)𝑋)⟶ℝ → dom (𝐺 ↾ ((𝑉𝑖)(,)𝑋)) = ((𝑉𝑖)(,)𝑋))
7674, 75syl 17 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → dom (𝐺 ↾ ((𝑉𝑖)(,)𝑋)) = ((𝑉𝑖)(,)𝑋))
7767, 76eqtrd 2777 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → dom (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) = ((𝑉𝑖)(,)𝑋))
78 limcresi 25920 . . . . . . . 8 ((𝐺 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ (((𝐺 ↾ (-∞(,)𝑋)) ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋)
7945resabs1d 6026 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ (-∞(,)𝑋)) ↾ ((𝑉𝑖)(,)𝑋)) = (𝐺 ↾ ((𝑉𝑖)(,)𝑋)))
8079oveq1d 7446 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐺 ↾ (-∞(,)𝑋)) ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋) = ((𝐺 ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋))
8178, 80sseqtrid 4026 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ ((𝐺 ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋))
82 fourierdlem74.e . . . . . . . 8 (𝜑𝐸 ∈ ((𝐺 ↾ (-∞(,)𝑋)) lim 𝑋))
8382adantr 480 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ ((𝐺 ↾ (-∞(,)𝑋)) lim 𝑋))
8481, 83sseldd 3984 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ ((𝐺 ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋))
8559, 64eqtr2d 2778 . . . . . . . 8 (𝜑 → (𝐺 ↾ ((𝑉𝑖)(,)𝑋)) = (ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))))
8685oveq1d 7446 . . . . . . 7 (𝜑 → ((𝐺 ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋) = ((ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) lim 𝑋))
8786adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ ((𝑉𝑖)(,)𝑋)) lim 𝑋) = ((ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) lim 𝑋))
8884, 87eleqtrd 2843 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ ((ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) lim 𝑋))
8988adantr 480 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝐸 ∈ ((ℝ D (𝐹 ↾ ((𝑉𝑖)(,)𝑋))) lim 𝑋))
90 eqid 2737 . . . 4 (𝑠 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ ((((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊) / 𝑠)) = (𝑠 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ ((((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊) / 𝑠))
91 oveq2 7439 . . . . . . 7 (𝑥 = 𝑠 → (𝑋 + 𝑥) = (𝑋 + 𝑠))
9291fveq2d 6910 . . . . . 6 (𝑥 = 𝑠 → ((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑥)) = ((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)))
9392oveq1d 7446 . . . . 5 (𝑥 = 𝑠 → (((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑥)) − 𝑊) = (((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊))
9493cbvmptv 5255 . . . 4 (𝑥 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ (((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑥)) − 𝑊)) = (𝑠 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ (((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊))
95 id 22 . . . . 5 (𝑥 = 𝑠𝑥 = 𝑠)
9695cbvmptv 5255 . . . 4 (𝑥 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ 𝑥) = (𝑠 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ 𝑠)
9718, 19, 29, 34, 49, 50, 77, 89, 90, 94, 96fourierdlem60 46181 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝐸 ∈ ((𝑠 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ ((((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊) / 𝑠)) lim 0))
98 fourierdlem74.a . . . . 5 𝐴 = if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1))))
99 iftrue 4531 . . . . 5 ((𝑉‘(𝑖 + 1)) = 𝑋 → if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) = 𝐸)
10098, 99eqtrid 2789 . . . 4 ((𝑉‘(𝑖 + 1)) = 𝑋𝐴 = 𝐸)
101100adantl 481 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝐴 = 𝐸)
102 fourierdlem74.h . . . . . . 7 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
103102reseq1i 5993 . . . . . 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))))
104103a1i 11 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 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)))))
105 ioossicc 13473 . . . . . . . 8 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
1063rexri 11319 . . . . . . . . . 10 -π ∈ ℝ*
107106a1i 11 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ*)
1082rexri 11319 . . . . . . . . . 10 π ∈ ℝ*
109108a1i 11 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → π ∈ ℝ*)
1103a1i 11 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → -π ∈ ℝ)
1112a1i 11 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → π ∈ ℝ)
1125adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
11316, 112resubcld 11691 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
1144recnd 11289 . . . . . . . . . . . . . . . 16 (𝜑 → -π ∈ ℂ)
1155recnd 11289 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ ℂ)
116114, 115pncand 11621 . . . . . . . . . . . . . . 15 (𝜑 → ((-π + 𝑋) − 𝑋) = -π)
117116eqcomd 2743 . . . . . . . . . . . . . 14 (𝜑 → -π = ((-π + 𝑋) − 𝑋))
118117adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → -π = ((-π + 𝑋) − 𝑋))
1196adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → (-π + 𝑋) ∈ ℝ)
1208adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0...𝑀)) → (π + 𝑋) ∈ ℝ)
121 elicc2 13452 . . . . . . . . . . . . . . . . 17 (((-π + 𝑋) ∈ ℝ ∧ (π + 𝑋) ∈ ℝ) → ((𝑉𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)) ↔ ((𝑉𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉𝑖) ∧ (𝑉𝑖) ≤ (π + 𝑋))))
122119, 120, 121syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)) ↔ ((𝑉𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉𝑖) ∧ (𝑉𝑖) ≤ (π + 𝑋))))
12315, 122mpbid 232 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉𝑖) ∧ (𝑉𝑖) ≤ (π + 𝑋)))
124123simp2d 1144 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → (-π + 𝑋) ≤ (𝑉𝑖))
125119, 16, 112, 124lesub1dd 11879 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → ((-π + 𝑋) − 𝑋) ≤ ((𝑉𝑖) − 𝑋))
126118, 125eqbrtrd 5165 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → -π ≤ ((𝑉𝑖) − 𝑋))
127123simp3d 1145 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ≤ (π + 𝑋))
12816, 120, 112, 127lesub1dd 11879 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ≤ ((π + 𝑋) − 𝑋))
129111recnd 11289 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → π ∈ ℂ)
130115adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℂ)
131129, 130pncand 11621 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0...𝑀)) → ((π + 𝑋) − 𝑋) = π)
132128, 131breqtrd 5169 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ≤ π)
133110, 111, 113, 126, 132eliccd 45517 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ (-π[,]π))
134 fourierdlem74.q . . . . . . . . . . 11 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
135133, 134fmptd 7134 . . . . . . . . . 10 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
136135adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
137 simpr 484 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
138107, 109, 136, 137fourierdlem8 46130 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
139105, 138sstrid 3995 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
140139resmptd 6058 . . . . . 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))))
141140adantr 480 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 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))))
1421adantl 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
1431, 113sylan2 593 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
144134fvmpt2 7027 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ ℝ) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
145142, 143, 144syl2anc 584 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
146145adantr 480 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
147 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑉𝑖) = (𝑉𝑗))
148147oveq1d 7446 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑗) − 𝑋))
149148cbvmptv 5255 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
150134, 149eqtri 2765 . . . . . . . . . . 11 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
151150a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋)))
152 fveq2 6906 . . . . . . . . . . . 12 (𝑗 = (𝑖 + 1) → (𝑉𝑗) = (𝑉‘(𝑖 + 1)))
153152oveq1d 7446 . . . . . . . . . . 11 (𝑗 = (𝑖 + 1) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
154153adantl 481 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
155 fzofzp1 13803 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
156155adantl 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
15722simpld 494 . . . . . . . . . . . . . 14 (𝜑𝑉 ∈ (ℝ ↑m (0...𝑀)))
158 elmapi 8889 . . . . . . . . . . . . . 14 (𝑉 ∈ (ℝ ↑m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
159157, 158syl 17 . . . . . . . . . . . . 13 (𝜑𝑉:(0...𝑀)⟶ℝ)
160159adantr 480 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
161160, 156ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
1625adantr 480 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
163161, 162resubcld 11691 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
164151, 154, 156, 163fvmptd 7023 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
165164adantr 480 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
166 oveq1 7438 . . . . . . . . 9 ((𝑉‘(𝑖 + 1)) = 𝑋 → ((𝑉‘(𝑖 + 1)) − 𝑋) = (𝑋𝑋))
167166adantl 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → ((𝑉‘(𝑖 + 1)) − 𝑋) = (𝑋𝑋))
168115ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝑋 ∈ ℂ)
169168subidd 11608 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑋𝑋) = 0)
1701, 169sylanl2 681 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑋𝑋) = 0)
171165, 167, 1703eqtrd 2781 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑄‘(𝑖 + 1)) = 0)
172146, 171oveq12d 7449 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (((𝑉𝑖) − 𝑋)(,)0))
173 simplr 769 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 0) → 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
174 fourierdlem74.o . . . . . . . . . . . . 13 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
17512adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑠 = 0) → 𝑀 ∈ ℕ)
1764, 7, 5, 11, 174, 12, 13, 134fourierdlem14 46136 . . . . . . . . . . . . . 14 (𝜑𝑄 ∈ (𝑂𝑀))
177176adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑠 = 0) → 𝑄 ∈ (𝑂𝑀))
178 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑠 = 0) → 𝑠 = 0)
179 fourierdlem74.x . . . . . . . . . . . . . . . . . 18 (𝜑𝑋 ∈ ran 𝑉)
180 ffn 6736 . . . . . . . . . . . . . . . . . . 19 (𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)) → 𝑉 Fn (0...𝑀))
181 fvelrnb 6969 . . . . . . . . . . . . . . . . . . 19 (𝑉 Fn (0...𝑀) → (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (0...𝑀)(𝑉𝑖) = 𝑋))
18214, 180, 1813syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (0...𝑀)(𝑉𝑖) = 𝑋))
183179, 182mpbid 232 . . . . . . . . . . . . . . . . 17 (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑉𝑖) = 𝑋)
184 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
185134fvmpt2 7027 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ (-π[,]π)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
186184, 133, 185syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
187186adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
188 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉𝑖) = 𝑋 → ((𝑉𝑖) − 𝑋) = (𝑋𝑋))
189188adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → ((𝑉𝑖) − 𝑋) = (𝑋𝑋))
190115subidd 11608 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑋𝑋) = 0)
191190ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑋𝑋) = 0)
192187, 189, 1913eqtrd 2781 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0...𝑀)) ∧ (𝑉𝑖) = 𝑋) → (𝑄𝑖) = 0)
193192ex 412 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) = 𝑋 → (𝑄𝑖) = 0))
194193reximdva 3168 . . . . . . . . . . . . . . . . 17 (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑉𝑖) = 𝑋 → ∃𝑖 ∈ (0...𝑀)(𝑄𝑖) = 0))
195183, 194mpd 15 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄𝑖) = 0)
196113, 134fmptd 7134 . . . . . . . . . . . . . . . . 17 (𝜑𝑄:(0...𝑀)⟶ℝ)
197 ffn 6736 . . . . . . . . . . . . . . . . 17 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
198 fvelrnb 6969 . . . . . . . . . . . . . . . . 17 (𝑄 Fn (0...𝑀) → (0 ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄𝑖) = 0))
199196, 197, 1983syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (0 ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄𝑖) = 0))
200195, 199mpbird 257 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ ran 𝑄)
201200adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠 = 0) → 0 ∈ ran 𝑄)
202178, 201eqeltrd 2841 . . . . . . . . . . . . 13 ((𝜑𝑠 = 0) → 𝑠 ∈ ran 𝑄)
203174, 175, 177, 202fourierdlem12 46134 . . . . . . . . . . . 12 (((𝜑𝑠 = 0) ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
204203an32s 652 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = 0) → ¬ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
205204adantlr 715 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 0) → ¬ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
206173, 205pm2.65da 817 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
207206adantlr 715 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
208207iffalsed 4536 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 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))
209 elioore 13417 . . . . . . . . . . . 12 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
210209adantl 481 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
211 0red 11264 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℝ)
212 elioo3g 13416 . . . . . . . . . . . . . . 15 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑠 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑠𝑠 < (𝑄‘(𝑖 + 1)))))
213212biimpi 216 . . . . . . . . . . . . . 14 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (((𝑄𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ*𝑠 ∈ ℝ*) ∧ ((𝑄𝑖) < 𝑠𝑠 < (𝑄‘(𝑖 + 1)))))
214213simprrd 774 . . . . . . . . . . . . 13 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 < (𝑄‘(𝑖 + 1)))
215214adantl 481 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
216171adantr 480 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) = 0)
217215, 216breqtrd 5169 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < 0)
218210, 211, 217ltnsymd 11410 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 0 < 𝑠)
219218iffalsed 4536 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑊)
220219oveq2d 7447 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑊))
221220oveq1d 7446 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) = (((𝐹‘(𝑋 + 𝑠)) − 𝑊) / 𝑠))
22241ad2antrr 726 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉𝑖) ∈ ℝ*)
2235rexrd 11311 . . . . . . . . . . . 12 (𝜑𝑋 ∈ ℝ*)
224223ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ*)
225162ad2antrr 726 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
226225, 210readdcld 11290 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
227115adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
228 iccssre 13469 . . . . . . . . . . . . . . . . . . 19 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
2293, 2, 228mp2an 692 . . . . . . . . . . . . . . . . . 18 (-π[,]π) ⊆ ℝ
230229, 51sstri 3993 . . . . . . . . . . . . . . . . 17 (-π[,]π) ⊆ ℂ
231186, 133eqeltrd 2841 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ (-π[,]π))
2321, 231sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ (-π[,]π))
233230, 232sselid 3981 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℂ)
234227, 233addcomd 11463 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄𝑖)) = ((𝑄𝑖) + 𝑋))
235145oveq1d 7446 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) + 𝑋) = (((𝑉𝑖) − 𝑋) + 𝑋))
23617recnd 11289 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℂ)
237236, 227npcand 11624 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝑉𝑖) − 𝑋) + 𝑋) = (𝑉𝑖))
238234, 235, 2373eqtrrd 2782 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) = (𝑋 + (𝑄𝑖)))
239238adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉𝑖) = (𝑋 + (𝑄𝑖)))
240145, 143eqeltrd 2841 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
241240adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) ∈ ℝ)
242209adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
2435ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
244213simprld 772 . . . . . . . . . . . . . . 15 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑄𝑖) < 𝑠)
245244adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) < 𝑠)
246241, 242, 243, 245ltadd2dd 11420 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄𝑖)) < (𝑋 + 𝑠))
247239, 246eqbrtrd 5165 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉𝑖) < (𝑋 + 𝑠))
248247adantlr 715 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉𝑖) < (𝑋 + 𝑠))
249 ltaddneg 11477 . . . . . . . . . . . . 13 ((𝑠 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (𝑠 < 0 ↔ (𝑋 + 𝑠) < 𝑋))
250210, 225, 249syl2anc 584 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑠 < 0 ↔ (𝑋 + 𝑠) < 𝑋))
251217, 250mpbid 232 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < 𝑋)
252222, 224, 226, 248, 251eliood 45511 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉𝑖)(,)𝑋))
253 fvres 6925 . . . . . . . . . . 11 ((𝑋 + 𝑠) ∈ ((𝑉𝑖)(,)𝑋) → ((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
254253eqcomd 2743 . . . . . . . . . 10 ((𝑋 + 𝑠) ∈ ((𝑉𝑖)(,)𝑋) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)))
255252, 254syl 17 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)))
256255oveq1d 7446 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − 𝑊) = (((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊))
257256oveq1d 7446 . . . . . . 7 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − 𝑊) / 𝑠) = ((((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊) / 𝑠))
258208, 221, 2573eqtrd 2781 . . . . . 6 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = ((((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊) / 𝑠))
259172, 258mpteq12dva 5231 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) = (𝑠 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ ((((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊) / 𝑠)))
260104, 141, 2593eqtrd 2781 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ ((((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊) / 𝑠)))
261260, 171oveq12d 7449 . . 3 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ (((𝑉𝑖) − 𝑋)(,)0) ↦ ((((𝐹 ↾ ((𝑉𝑖)(,)𝑋))‘(𝑋 + 𝑠)) − 𝑊) / 𝑠)) lim 0))
26297, 101, 2613eltr4d 2856 . 2 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
263 eqid 2737 . . . . 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)))
264 eqid 2737 . . . . 5 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠)
265 eqid 2737 . . . . 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))
26630adantr 480 . . . . . . . . . 10 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℝ)
2675adantr 480 . . . . . . . . . . 11 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
268209adantl 481 . . . . . . . . . . 11 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
269267, 268readdcld 11290 . . . . . . . . . 10 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
270266, 269ffvelcdmd 7105 . . . . . . . . 9 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
271270recnd 11289 . . . . . . . 8 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
272271adantlr 715 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
2732723adantl3 1169 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
274 fourierdlem74.y . . . . . . . . . 10 (𝜑𝑌 ∈ ℝ)
275274recnd 11289 . . . . . . . . 9 (𝜑𝑌 ∈ ℂ)
276 limccl 25910 . . . . . . . . . 10 ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ ℂ
277276, 36sselid 3981 . . . . . . . . 9 (𝜑𝑊 ∈ ℂ)
278275, 277ifcld 4572 . . . . . . . 8 (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
279278adantr 480 . . . . . . 7 ((𝜑𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
2802793ad2antl1 1186 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
281273, 280subcld 11620 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
282209recnd 11289 . . . . . . 7 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℂ)
283282adantl 481 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
284 velsn 4642 . . . . . . . 8 (𝑠 ∈ {0} ↔ 𝑠 = 0)
285206, 284sylnibr 329 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 ∈ {0})
2862853adantl3 1169 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 ∈ {0})
287283, 286eldifd 3962 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (ℂ ∖ {0}))
288 eqid 2737 . . . . . . . . . 10 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠)))
289 eqid 2737 . . . . . . . . . 10 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊)
290 eqid 2737 . . . . . . . . . 10 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊))
291277ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑊 ∈ ℂ)
29230adantr 480 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ)
293 ioossre 13448 . . . . . . . . . . . 12 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
294293a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
29541adantr 480 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉𝑖) ∈ ℝ*)
296161rexrd 11311 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ*)
297296adantr 480 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈ ℝ*)
298269adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
299196adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
300299, 156ffvelcdmd 7105 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
301300adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
302214adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
303242, 301, 243, 302ltadd2dd 11420 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑋 + (𝑄‘(𝑖 + 1))))
304164oveq2d 7447 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)))
305161recnd 11289 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
306227, 305pncan3d 11623 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)) = (𝑉‘(𝑖 + 1)))
307304, 306eqtrd 2777 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
308307adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
309303, 308breqtrd 5169 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1)))
310295, 297, 298, 247, 309eliood 45511 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
311 ioossre 13448 . . . . . . . . . . . 12 ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ
312311a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)
313242, 302ltned 11397 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ (𝑄‘(𝑖 + 1)))
314 fourierdlem74.r . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))
315307eqcomd 2743 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) = (𝑋 + (𝑄‘(𝑖 + 1))))
316315oveq2d 7447 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑋 + (𝑄‘(𝑖 + 1)))))
317314, 316eleqtrd 2843 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑋 + (𝑄‘(𝑖 + 1)))))
318300recnd 11289 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
319292, 162, 294, 288, 310, 312, 313, 317, 318fourierdlem53 46174 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) lim (𝑄‘(𝑖 + 1))))
320 ioosscn 13449 . . . . . . . . . . . 12 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
321320a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
322277adantr 480 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ℂ)
323289, 321, 322, 318constlimc 45639 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) lim (𝑄‘(𝑖 + 1))))
324288, 289, 290, 272, 291, 319, 323sublimc 45667 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑅𝑊) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim (𝑄‘(𝑖 + 1))))
325324adantr 480 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑅𝑊) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim (𝑄‘(𝑖 + 1))))
326 iftrue 4531 . . . . . . . . . 10 ((𝑉‘(𝑖 + 1)) < 𝑋 → if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌) = 𝑊)
327326oveq2d 7447 . . . . . . . . 9 ((𝑉‘(𝑖 + 1)) < 𝑋 → (𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) = (𝑅𝑊))
328327adantl 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) = (𝑅𝑊))
329209adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
330 0red 11264 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℝ)
331300ad2antrr 726 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
332214adantl 481 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
333164adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
334161adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
3355ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → 𝑋 ∈ ℝ)
336 simpr 484 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑉‘(𝑖 + 1)) < 𝑋)
337334, 335, 336ltled 11409 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑉‘(𝑖 + 1)) ≤ 𝑋)
338334, 335suble0d 11854 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (((𝑉‘(𝑖 + 1)) − 𝑋) ≤ 0 ↔ (𝑉‘(𝑖 + 1)) ≤ 𝑋))
339337, 338mpbird 257 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → ((𝑉‘(𝑖 + 1)) − 𝑋) ≤ 0)
340333, 339eqbrtrd 5165 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑄‘(𝑖 + 1)) ≤ 0)
341340adantr 480 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ≤ 0)
342329, 331, 330, 332, 341ltletrd 11421 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < 0)
343329, 330, 342ltnsymd 11410 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 0 < 𝑠)
344343iffalsed 4536 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑊)
345344oveq2d 7447 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑊))
346345mpteq2dva 5242 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)))
347346oveq1d 7446 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim (𝑄‘(𝑖 + 1))))
348325, 328, 3473eltr4d 2856 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄‘(𝑖 + 1))))
3493483adantl3 1169 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄‘(𝑖 + 1))))
350 simpl1 1192 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → 𝜑)
351 simpl2 1193 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → 𝑖 ∈ (0..^𝑀))
3525ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → 𝑋 ∈ ℝ)
3533523adantl3 1169 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → 𝑋 ∈ ℝ)
354161adantr 480 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
3553543adantl3 1169 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
356 neqne 2948 . . . . . . . . . . 11 (¬ (𝑉‘(𝑖 + 1)) = 𝑋 → (𝑉‘(𝑖 + 1)) ≠ 𝑋)
357356necomd 2996 . . . . . . . . . 10 (¬ (𝑉‘(𝑖 + 1)) = 𝑋𝑋 ≠ (𝑉‘(𝑖 + 1)))
358357adantr 480 . . . . . . . . 9 ((¬ (𝑉‘(𝑖 + 1)) = 𝑋 ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → 𝑋 ≠ (𝑉‘(𝑖 + 1)))
3593583ad2antl3 1188 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → 𝑋 ≠ (𝑉‘(𝑖 + 1)))
360 simpr 484 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → ¬ (𝑉‘(𝑖 + 1)) < 𝑋)
361353, 355, 359, 360lttri5d 45311 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1)))
362 eqid 2737 . . . . . . . 8 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(0 < 𝑠, 𝑌, 𝑊)) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(0 < 𝑠, 𝑌, 𝑊))
363272adantlr 715 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
364278ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
365319adantr 480 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → 𝑅 ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) lim (𝑄‘(𝑖 + 1))))
366 eqid 2737 . . . . . . . . . . 11 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌)
367275adantr 480 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ℂ)
368366, 321, 367, 318constlimc 45639 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) lim (𝑄‘(𝑖 + 1))))
369368adantr 480 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → 𝑌 ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) lim (𝑄‘(𝑖 + 1))))
3705ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → 𝑋 ∈ ℝ)
371161adantr 480 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
372 simpr 484 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → 𝑋 < (𝑉‘(𝑖 + 1)))
373370, 371, 372ltnsymd 11410 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → ¬ (𝑉‘(𝑖 + 1)) < 𝑋)
374373iffalsed 4536 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌) = 𝑌)
375 0red 11264 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℝ)
376240ad2antrr 726 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) ∈ ℝ)
377209adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
378190eqcomd 2743 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 = (𝑋𝑋))
379378ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → 0 = (𝑋𝑋))
38017adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → (𝑉𝑖) ∈ ℝ)
38141ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → (𝑉𝑖) ∈ ℝ*)
382296ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → (𝑉‘(𝑖 + 1)) ∈ ℝ*)
383162ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → 𝑋 ∈ ℝ)
384 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → ¬ 𝑋 ≤ (𝑉𝑖))
38517adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → (𝑉𝑖) ∈ ℝ)
3865ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → 𝑋 ∈ ℝ)
387385, 386ltnled 11408 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → ((𝑉𝑖) < 𝑋 ↔ ¬ 𝑋 ≤ (𝑉𝑖)))
388384, 387mpbird 257 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → (𝑉𝑖) < 𝑋)
389388adantlr 715 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → (𝑉𝑖) < 𝑋)
390 simplr 769 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → 𝑋 < (𝑉‘(𝑖 + 1)))
391381, 382, 383, 389, 390eliood 45511 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → 𝑋 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
39211, 12, 13, 179fourierdlem12 46134 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
393392ad2antrr 726 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ ¬ 𝑋 ≤ (𝑉𝑖)) → ¬ 𝑋 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
394391, 393condan 818 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → 𝑋 ≤ (𝑉𝑖))
395370, 380, 370, 394lesub1dd 11879 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → (𝑋𝑋) ≤ ((𝑉𝑖) − 𝑋))
396379, 395eqbrtrd 5165 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → 0 ≤ ((𝑉𝑖) − 𝑋))
397145eqcomd 2743 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) = (𝑄𝑖))
398397adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → ((𝑉𝑖) − 𝑋) = (𝑄𝑖))
399396, 398breqtrd 5169 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → 0 ≤ (𝑄𝑖))
400399adantr 480 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ≤ (𝑄𝑖))
401244adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄𝑖) < 𝑠)
402375, 376, 377, 400, 401lelttrd 11419 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 < 𝑠)
403402iftrued 4533 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌)
404403mpteq2dva 5242 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(0 < 𝑠, 𝑌, 𝑊)) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌))
405404oveq1d 7446 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(0 < 𝑠, 𝑌, 𝑊)) lim (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) lim (𝑄‘(𝑖 + 1))))
406369, 374, 4053eltr4d 2856 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(0 < 𝑠, 𝑌, 𝑊)) lim (𝑄‘(𝑖 + 1))))
407288, 362, 263, 363, 364, 365, 406sublimc 45667 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑋 < (𝑉‘(𝑖 + 1))) → (𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄‘(𝑖 + 1))))
408350, 351, 361, 407syl21anc 838 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) < 𝑋) → (𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄‘(𝑖 + 1))))
409349, 408pm2.61dan 813 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim (𝑄‘(𝑖 + 1))))
410321, 264, 318idlimc 45641 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) lim (𝑄‘(𝑖 + 1))))
4114103adant3 1133 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑄‘(𝑖 + 1)) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) lim (𝑄‘(𝑖 + 1))))
4121643adant3 1133 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
4133053adant3 1133 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
4142273adant3 1133 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝑋 ∈ ℂ)
4153563ad2ant3 1136 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑉‘(𝑖 + 1)) ≠ 𝑋)
416413, 414, 415subne0d 11629 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → ((𝑉‘(𝑖 + 1)) − 𝑋) ≠ 0)
417412, 416eqnetrd 3008 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝑄‘(𝑖 + 1)) ≠ 0)
4182063adantl3 1169 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
419418neqned 2947 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ 0)
420263, 264, 265, 281, 287, 409, 411, 417, 419divlimc 45671 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) lim (𝑄‘(𝑖 + 1))))
421 iffalse 4534 . . . . . 6 (¬ (𝑉‘(𝑖 + 1)) = 𝑋 → if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) = ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1))))
42298, 421eqtrid 2789 . . . . 5 (¬ (𝑉‘(𝑖 + 1)) = 𝑋𝐴 = ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1))))
4234223ad2ant3 1136 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝐴 = ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1))))
424 ioossre 13448 . . . . . . . . . . . . 13 (-∞(,)𝑋) ⊆ ℝ
425424a1i 11 . . . . . . . . . . . 12 (𝜑 → (-∞(,)𝑋) ⊆ ℝ)
42630, 425fssresd 6775 . . . . . . . . . . 11 (𝜑 → (𝐹 ↾ (-∞(,)𝑋)):(-∞(,)𝑋)⟶ℝ)
427424, 52sstrid 3995 . . . . . . . . . . 11 (𝜑 → (-∞(,)𝑋) ⊆ ℂ)
42839a1i 11 . . . . . . . . . . . 12 (𝜑 → -∞ ∈ ℝ*)
4295mnfltd 13166 . . . . . . . . . . . 12 (𝜑 → -∞ < 𝑋)
43056, 428, 5, 429lptioo2cn 45660 . . . . . . . . . . 11 (𝜑𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘(-∞(,)𝑋)))
431426, 427, 430, 36limcrecl 45644 . . . . . . . . . 10 (𝜑𝑊 ∈ ℝ)
43230, 5, 274, 431, 102fourierdlem9 46131 . . . . . . . . 9 (𝜑𝐻:(-π[,]π)⟶ℝ)
433432adantr 480 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐻:(-π[,]π)⟶ℝ)
434433, 139feqresmpt 6978 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻𝑠)))
435139sselda 3983 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (-π[,]π))
436 0cnd 11254 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℂ)
437278ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
438272, 437subcld 11620 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
439282adantl 481 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
440206neqned 2947 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ 0)
441438, 439, 440divcld 12043 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℂ)
442436, 441ifcld 4572 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℂ)
443102fvmpt2 7027 . . . . . . . . . 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)))
444435, 442, 443syl2anc 584 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
445206iffalsed 4536 . . . . . . . . 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))
446444, 445eqtrd 2777 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
447446mpteq2dva 5242 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻𝑠)) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
448434, 447eqtrd 2777 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
4494483adant3 1133 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → (𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
450449oveq1d 7446 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) lim (𝑄‘(𝑖 + 1))))
451420, 423, 4503eltr4d 2856 . . 3 ((𝜑𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
4524513expa 1119 . 2 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
453262, 452pm2.61dan 813 1 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  wss 3951  ifcif 4525  {csn 4626   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  cres 5687   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  -∞cmnf 11293  *cxr 11294   < clt 11295  cle 11296  cmin 11492  -cneg 11493   / cdiv 11920  cn 12266  (,)cioo 13387  [,]cicc 13390  ...cfz 13547  ..^cfzo 13694  πcpi 16102  TopOpenctopn 17466  topGenctg 17482  fldccnfld 21364  intcnt 23025   lim climc 25897   D cdv 25898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723  df-ef 16103  df-sin 16105  df-cos 16106  df-pi 16108  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902
This theorem is referenced by:  fourierdlem88  46209  fourierdlem103  46224  fourierdlem104  46225
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