fourierdlem75 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem75 43722

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	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem75.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem75.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem75.x	⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
fourierdlem75.y	⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem75.w	⊢ (𝜑 → 𝑊 ∈ ℝ)
fourierdlem75.h	⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem75.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem75.v	⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
fourierdlem75.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
fourierdlem75.q	⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
fourierdlem75.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem75.g	⊢ 𝐺 = (ℝ D 𝐹)
fourierdlem75.gcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
fourierdlem75.e	⊢ (𝜑 → 𝐸 ∈ ((𝐺 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem75.a	⊢ 𝐴 = if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)))

**Assertion**
Ref	Expression
fourierdlem75	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))

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.

Step	Hyp	Ref	Expression
1		fourierdlem75.xre	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ)
2	1	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑋 ∈ ℝ)
3		fourierdlem75.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
4		fourierdlem75.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
5		fourierdlem75.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
6	5	fourierdlem2 43650	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))))
7	4, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))))
8	3, 7	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))
9	8	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_m (0...𝑀)))
10		elmapi 8637	. . . . . . . 8 ⊢ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ)
12	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
13		fzofzp1 13484	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
14	13	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
15	12, 14	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
16	15	adantr 481	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
17		eqcom 2745	. . . . . . 7 ⊢ ((𝑉‘𝑖) = 𝑋 ↔ 𝑋 = (𝑉‘𝑖))
18	17	biimpi 215	. . . . . 6 ⊢ ((𝑉‘𝑖) = 𝑋 → 𝑋 = (𝑉‘𝑖))
19	18	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑋 = (𝑉‘𝑖))
20	8	simprrd 771	. . . . . . 7 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
21	20	r19.21bi 3134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
22	21	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
23	19, 22	eqbrtrd 5096	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1)))
24		fourierdlem75.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
25	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ)
26		ioossre 13140	. . . . . . 7 ⊢ (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ
27	26	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)
28	25, 27	fssresd 6641	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
29	28	adantr 481	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
30		limcresi 25049	. . . . . . . 8 ⊢ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ⊆ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋)
31		fourierdlem75.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
32	30, 31	sselid 3919	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋))
33	32	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋))
34		pnfxr 11029	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
35	34	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → +∞ ∈ ℝ^*)
36	15	rexrd 11025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ^*)
37	15	ltpnfd 12857	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) < +∞)
38	36, 35, 37	xrltled 12884	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ≤ +∞)
39		iooss2 13115	. . . . . . . . 9 ⊢ ((+∞ ∈ ℝ^* ∧ (𝑉‘(𝑖 + 1)) ≤ +∞) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ (𝑋(,)+∞))
40	35, 38, 39	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ (𝑋(,)+∞))
41	40	resabs1d 5922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
42	41	oveq1d 7290	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋))
43	33, 42	eleqtrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋))
44	43	adantr 481	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋))
45		eqid 2738	. . . 4 ⊢ (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
46		ax-resscn 10928	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
47	46	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
48	24, 47	fssd 6618	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
49		ssid 3943	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ
50	49	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℝ)
51	26	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)
52		eqid 2738	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
53	52	tgioo2 23966	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
54	52, 53	dvres 25075	. . . . . . . . . 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))))))
55	47, 48, 50, 51, 54	syl22anc 836	. . . . . . . . 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)
57	56	eqcomi 2747	. . . . . . . . . 10 ⊢ (ℝ D 𝐹) = 𝐺
58		ioontr 43049	. . . . . . . . . 10 ⊢ ((int‘(topGen‘ran (,)))‘(𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1)))
59	57, 58	reseq12i 5889	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))
60	55, 59	eqtrdi 2794	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
61	60	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑉‘𝑖) = 𝑋) → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
62	61	dmeqd 5814	. . . . . 6 ⊢ ((𝜑 ∧ (𝑉‘𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
63	62	adantlr 712	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
64		fourierdlem75.gcn	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
65	64	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
66		oveq1 7282	. . . . . . . . . . 11 ⊢ ((𝑉‘𝑖) = 𝑋 → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
67	66	reseq2d 5891	. . . . . . . . . 10 ⊢ ((𝑉‘𝑖) = 𝑋 → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
68	67	feq1d 6585	. . . . . . . . 9 ⊢ ((𝑉‘𝑖) = 𝑋 → ((𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ))
69	68	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ))
70	65, 69	mpbid 231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
71	66	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
72	71	feq2d 6586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ))
73	70, 72	mpbid 231	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
74		fdm 6609	. . . . . 6 ⊢ ((𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ → dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
75	73, 74	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
76	63, 75	eqtrd 2778	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝑋(,)(𝑉‘(𝑖 + 1))))
77		limcresi 25049	. . . . . . . 8 ⊢ ((𝐺 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ⊆ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋)
78		fourierdlem75.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ((𝐺 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
79	77, 78	sselid 3919	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋))
80	79	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋))
81	40	resabs1d 5922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
82	60	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))
83	81, 82	eqtr4d 2781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))))
84	83	oveq1d 7290	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) lim_ℂ 𝑋) = ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) lim_ℂ 𝑋))
85	80, 84	eleqtrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) lim_ℂ 𝑋))
86	85	adantr 481	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝐸 ∈ ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) lim_ℂ 𝑋))
87		eqid 2738	. . . 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 7283	. . . . . . 7 ⊢ (𝑥 = 𝑠 → (𝑋 + 𝑥) = (𝑋 + 𝑠))
89	88	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = 𝑠 → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑥)) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)))
90	89	oveq1d 7290	. . . . 5 ⊢ (𝑥 = 𝑠 → (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑥)) − 𝑌) = (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌))
91	90	cbvmptv 5187	. . . 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 = 𝑠 → 𝑥 = 𝑠)
93	92	cbvmptv 5187	. . . 4 ⊢ (𝑥 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ 𝑥) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ 𝑠)
94	2, 16, 23, 29, 44, 45, 76, 86, 87, 91, 93	fourierdlem61 43708	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝐸 ∈ ((𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) lim_ℂ 0))
95		fourierdlem75.a	. . . . 5 ⊢ 𝐴 = if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)))
96		iftrue 4465	. . . . 5 ⊢ ((𝑉‘𝑖) = 𝑋 → if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) = 𝐸)
97	95, 96	eqtrid 2790	. . . 4 ⊢ ((𝑉‘𝑖) = 𝑋 → 𝐴 = 𝐸)
98	97	adantl 482	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝐴 = 𝐸)
99		fourierdlem75.h	. . . . . . 7 ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
100	99	reseq1i 5887	. . . . . 6 ⊢ (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
101	100	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
102		ioossicc 13165	. . . . . . . 8 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
103		pire 25615	. . . . . . . . . . . 12 ⊢ π ∈ ℝ
104	103	renegcli 11282	. . . . . . . . . . 11 ⊢ -π ∈ ℝ
105	104	rexri 11033	. . . . . . . . . 10 ⊢ -π ∈ ℝ^*
106	105	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
107	103	rexri 11033	. . . . . . . . . 10 ⊢ π ∈ ℝ^*
108	107	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
109	104	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → -π ∈ ℝ)
110	103	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → π ∈ ℝ)
111	104	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → -π ∈ ℝ)
112	111, 1	readdcld 11004	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (-π + 𝑋) ∈ ℝ)
113	103	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → π ∈ ℝ)
114	113, 1	readdcld 11004	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (π + 𝑋) ∈ ℝ)
115	112, 114	iccssred 13166	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
116	115	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
117	5, 4, 3	fourierdlem15 43663	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)))
118	117	ffvelrnda 6961	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑉‘𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)))
119	116, 118	sseldd 3922	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑉‘𝑖) ∈ ℝ)
120	1	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
121	119, 120	resubcld 11403	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
122	111	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → -π ∈ ℂ)
123	1	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ ℂ)
124	122, 123	pncand 11333	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((-π + 𝑋) − 𝑋) = -π)
125	124	eqcomd 2744	. . . . . . . . . . . . . 14 ⊢ (𝜑 → -π = ((-π + 𝑋) − 𝑋))
126	125	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → -π = ((-π + 𝑋) − 𝑋))
127	112	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (-π + 𝑋) ∈ ℝ)
128	114	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (π + 𝑋) ∈ ℝ)
129		elicc2 13144	. . . . . . . . . . . . . . . . 17 ⊢ (((-π + 𝑋) ∈ ℝ ∧ (π + 𝑋) ∈ ℝ) → ((𝑉‘𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)) ↔ ((𝑉‘𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉‘𝑖) ∧ (𝑉‘𝑖) ≤ (π + 𝑋))))
130	127, 128, 129	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)) ↔ ((𝑉‘𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉‘𝑖) ∧ (𝑉‘𝑖) ≤ (π + 𝑋))))
131	118, 130	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉‘𝑖) ∧ (𝑉‘𝑖) ≤ (π + 𝑋)))
132	131	simp2d 1142	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (-π + 𝑋) ≤ (𝑉‘𝑖))
133	127, 119, 120, 132	lesub1dd 11591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((-π + 𝑋) − 𝑋) ≤ ((𝑉‘𝑖) − 𝑋))
134	126, 133	eqbrtrd 5096	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → -π ≤ ((𝑉‘𝑖) − 𝑋))
135	131	simp3d 1143	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑉‘𝑖) ≤ (π + 𝑋))
136	119, 128, 120, 135	lesub1dd 11591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ≤ ((π + 𝑋) − 𝑋))
137	110	recnd 11003	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → π ∈ ℂ)
138	123	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℂ)
139	137, 138	pncand 11333	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((π + 𝑋) − 𝑋) = π)
140	136, 139	breqtrd 5100	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ≤ π)
141	109, 110, 121, 134, 140	eliccd 43042	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ (-π[,]π))
142		fourierdlem75.q	. . . . . . . . . . 11 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
143	141, 142	fmptd 6988	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
144	143	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
145		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
146	106, 108, 144, 145	fourierdlem8 43656	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
147	102, 146	sstrid 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
148	147	resmptd 5948	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))))
149	148	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))))
150		elfzofz 13403	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
151		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
152	142	fvmpt2 6886	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ (-π[,]π)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
153	151, 141, 152	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
154	153	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
155		oveq1 7282	. . . . . . . . . 10 ⊢ ((𝑉‘𝑖) = 𝑋 → ((𝑉‘𝑖) − 𝑋) = (𝑋 − 𝑋))
156	155	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝑉‘𝑖) − 𝑋) = (𝑋 − 𝑋))
157	123	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑋 ∈ ℂ)
158	157	subidd 11320	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑋 − 𝑋) = 0)
159	154, 156, 158	3eqtrd 2782	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) = 0)
160	150, 159	sylanl2 678	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) = 0)
161		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
162	161	oveq1d 7290	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
163	162	cbvmptv 5187	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
164	142, 163	eqtri 2766	. . . . . . . . . 10 ⊢ 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
165	164	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)))
166		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
167	166	oveq1d 7290	. . . . . . . . . 10 ⊢ (𝑗 = (𝑖 + 1) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
168	167	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
169	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
170	15, 169	resubcld 11403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
171	165, 168, 14, 170	fvmptd 6882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
172	171	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
173	160, 172	oveq12d 7293	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)))
174		simplr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 0) → 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
175		fourierdlem75.o	. . . . . . . . . . . . 13 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
176	4	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 = 0) → 𝑀 ∈ ℕ)
177	111, 113, 1, 5, 175, 4, 3, 142	fourierdlem14 43662	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))
178	177	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 = 0) → 𝑄 ∈ (𝑂‘𝑀))
179		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 = 0) → 𝑠 = 0)
180		fourierdlem75.x	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
181		ffn 6600	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)) → 𝑉 Fn (0...𝑀))
182		fvelrnb 6830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 Fn (0...𝑀) → (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (0...𝑀)(𝑉‘𝑖) = 𝑋))
183	117, 181, 182	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (0...𝑀)(𝑉‘𝑖) = 𝑋))
184	180, 183	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑉‘𝑖) = 𝑋)
185	159	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) = 𝑋 → (𝑄‘𝑖) = 0))
186	185	reximdva 3203	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑉‘𝑖) = 𝑋 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = 0))
187	184, 186	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = 0)
188	121, 142	fmptd 6988	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
189		ffn 6600	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
190		fvelrnb 6830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 Fn (0...𝑀) → (0 ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = 0))
191	188, 189, 190	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = 0))
192	187, 191	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ran 𝑄)
193	192	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 = 0) → 0 ∈ ran 𝑄)
194	179, 193	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 = 0) → 𝑠 ∈ ran 𝑄)
195	175, 176, 178, 194	fourierdlem12 43660	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 = 0) ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
196	195	an32s 649	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = 0) → ¬ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
197	196	adantlr 712	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 0) → ¬ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
198	174, 197	pm2.65da 814	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
199	198	adantlr 712	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
200	199	iffalsed 4470	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(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))
201	160	eqcomd 2744	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 0 = (𝑄‘𝑖))
202	201	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 = (𝑄‘𝑖))
203		elioo3g 13108	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑠 ∈ ℝ^*) ∧ ((𝑄‘𝑖) < 𝑠 ∧ 𝑠 < (𝑄‘(𝑖 + 1)))))
204	203	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑠 ∈ ℝ^*) ∧ ((𝑄‘𝑖) < 𝑠 ∧ 𝑠 < (𝑄‘(𝑖 + 1)))))
205	204	simprld 769	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) < 𝑠)
206	205	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠)
207	202, 206	eqbrtrd 5096	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 < 𝑠)
208	207	iftrued 4467	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌)
209	208	oveq2d 7291	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑌))
210	209	oveq1d 7290	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) = (((𝐹‘(𝑋 + 𝑠)) − 𝑌) / 𝑠))
211	1	rexrd 11025	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
212	211	ad3antrrr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ^*)
213	36	ad2antrr 723	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈ ℝ^*)
214	169	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
215		elioore 13109	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
216	215	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
217	214, 216	readdcld 11004	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
218	216, 207	elrpd 12769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ⁺)
219	214, 218	ltaddrpd 12805	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 < (𝑋 + 𝑠))
220	215	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
221	188	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
222	221, 14	ffvelrnd 6962	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
223	222	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
224	1	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
225	204	simprrd 771	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 < (𝑄‘(𝑖 + 1)))
226	225	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
227	220, 223, 224, 226	ltadd2dd 11134	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑋 + (𝑄‘(𝑖 + 1))))
228	171	oveq2d 7291	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)))
229	123	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
230	15	recnd 11003	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
231	229, 230	pncan3d 11335	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)) = (𝑉‘(𝑖 + 1)))
232	228, 231	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
233	232	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
234	227, 233	breqtrd 5100	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1)))
235	234	adantlr 712	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1)))
236	212, 213, 217, 219, 235	eliood 43036	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ (𝑋(,)(𝑉‘(𝑖 + 1))))
237		fvres 6793	. . . . . . . . . . 11 ⊢ ((𝑋 + 𝑠) ∈ (𝑋(,)(𝑉‘(𝑖 + 1))) → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
238	236, 237	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
239	238	eqcomd 2744	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)))
240	239	oveq1d 7290	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − 𝑌) = (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌))
241	240	oveq1d 7290	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − 𝑌) / 𝑠) = ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠))
242	200, 210, 241	3eqtrd 2782	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(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))
243	173, 242	mpteq12dva 5163	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑠 ∈ ((𝑄‘𝑖)(,)(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)))
244	101, 149, 243	3eqtrd 2782	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)))
245	244, 160	oveq12d 7293	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) lim_ℂ 0))
246	94, 98, 245	3eltr4d 2854	. 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
247		eqid 2738	. . . . 5 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)))
248		eqid 2738	. . . . 5 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠)
249		eqid 2738	. . . . 5 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
250	24	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℝ)
251	1	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
252	215	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
253	251, 252	readdcld 11004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
254	250, 253	ffvelrnd 6962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
255	254	recnd 11003	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
256	255	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
257	256	3adantl3 1167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
258		limccl 25039	. . . . . . . . . 10 ⊢ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ⊆ ℂ
259	258, 31	sselid 3919	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℂ)
260		fourierdlem75.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ ℝ)
261	260	recnd 11003	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℂ)
262	259, 261	ifcld 4505	. . . . . . . 8 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
263	262	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
264	263	3ad2antl1 1184	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
265	257, 264	subcld 11332	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
266	215	recnd 11003	. . . . . . 7 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℂ)
267	266	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
268		velsn 4577	. . . . . . . 8 ⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0)
269	198, 268	sylnibr 329	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 ∈ {0})
270	269	3adantl3 1167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 ∈ {0})
271	267, 270	eldifd 3898	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (ℂ ∖ {0}))
272		eqid 2738	. . . . . . . . . 10 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠)))
273		eqid 2738	. . . . . . . . . 10 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊)
274		eqid 2738	. . . . . . . . . 10 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊))
275	261	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑊 ∈ ℂ)
276		ioossre 13140	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
277	276	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
278	150, 119	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
279	278	rexrd 11025	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ^*)
280	279	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) ∈ ℝ^*)
281	36	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈ ℝ^*)
282	253	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
283		iccssre 13161	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
284	104, 103, 283	mp2an 689	. . . . . . . . . . . . . . . . . 18 ⊢ (-π[,]π) ⊆ ℝ
285	284, 46	sstri 3930	. . . . . . . . . . . . . . . . 17 ⊢ (-π[,]π) ⊆ ℂ
286	153, 141	eqeltrd 2839	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π))
287	150, 286	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π))
288	285, 287	sselid 3919	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
289	229, 288	addcomd 11177	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘𝑖)) = ((𝑄‘𝑖) + 𝑋))
290	150	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
291	150, 121	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
292	142	fvmpt2 6886	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
293	290, 291, 292	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
294	293	oveq1d 7290	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋))
295	278	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ)
296	295, 229	npcand 11336	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖))
297	289, 294, 296	3eqtrrd 2783	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖)))
298	297	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖)))
299	293, 291	eqeltrd 2839	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
300	299	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
301	205	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠)
302	300, 220, 224, 301	ltadd2dd 11134	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘𝑖)) < (𝑋 + 𝑠))
303	298, 302	eqbrtrd 5096	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) < (𝑋 + 𝑠))
304	280, 281, 282, 303, 234	eliood 43036	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
305		ioossre 13140	. . . . . . . . . . . 12 ⊢ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ
306	305	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)
307	300, 301	gtned 11110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ (𝑄‘𝑖))
308		fourierdlem75.r	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
309	297	oveq2d 7291	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)) = ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑋 + (𝑄‘𝑖))))
310	308, 309	eleqtrd 2841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑋 + (𝑄‘𝑖))))
311	25, 169, 277, 272, 304, 306, 307, 310, 288	fourierdlem53 43700	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) lim_ℂ (𝑄‘𝑖)))
312		ioosscn 13141	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
313	312	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
314	261	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ℂ)
315	273, 313, 314, 288	constlimc 43165	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) lim_ℂ (𝑄‘𝑖)))
316	272, 273, 274, 256, 275, 311, 315	sublimc 43193	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 − 𝑊) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim_ℂ (𝑄‘𝑖)))
317	316	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑅 − 𝑊) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim_ℂ (𝑄‘𝑖)))
318		iftrue 4465	. . . . . . . . . 10 ⊢ ((𝑉‘𝑖) < 𝑋 → if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌) = 𝑊)
319	318	oveq2d 7291	. . . . . . . . 9 ⊢ ((𝑉‘𝑖) < 𝑋 → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅 − 𝑊))
320	319	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅 − 𝑊))
321	215	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
322		0red 10978	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℝ)
323	222	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
324	225	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
325	171	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
326	279	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘𝑖) ∈ ℝ^*)
327	36	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ^*)
328	169	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ℝ)
329		simplr 766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘𝑖) < 𝑋)
330		simpr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋)
331	1	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ℝ)
332	15	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
333	331, 332	ltnled 11122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑋 < (𝑉‘(𝑖 + 1)) ↔ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋))
334	330, 333	mpbird 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1)))
335	334	adantlr 712	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1)))
336	326, 327, 328, 329, 335	eliood 43036	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
337	5, 4, 3, 180	fourierdlem12 43660	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
338	337	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → ¬ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
339	336, 338	condan 815	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑉‘(𝑖 + 1)) ≤ 𝑋)
340	15	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
341	1	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → 𝑋 ∈ ℝ)
342	340, 341	suble0d 11566	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (((𝑉‘(𝑖 + 1)) − 𝑋) ≤ 0 ↔ (𝑉‘(𝑖 + 1)) ≤ 𝑋))
343	339, 342	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → ((𝑉‘(𝑖 + 1)) − 𝑋) ≤ 0)
344	325, 343	eqbrtrd 5096	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑄‘(𝑖 + 1)) ≤ 0)
345	344	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ≤ 0)
346	321, 323, 322, 324, 345	ltletrd 11135	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < 0)
347	321, 322, 346	ltnsymd 11124	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 0 < 𝑠)
348	347	iffalsed 4470	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑊)
349	348	oveq2d 7291	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑊))
350	349	mpteq2dva 5174	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)))
351	350	oveq1d 7290	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) lim_ℂ (𝑄‘𝑖)))
352	317, 320, 351	3eltr4d 2854	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim_ℂ (𝑄‘𝑖)))
353	352	3adantl3 1167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim_ℂ (𝑄‘𝑖)))
354		eqid 2738	. . . . . . . . . 10 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌)
355		eqid 2738	. . . . . . . . . 10 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌))
356	259	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑌 ∈ ℂ)
357	259	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ℂ)
358	354, 313, 357, 288	constlimc 43165	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) lim_ℂ (𝑄‘𝑖)))
359	272, 354, 355, 256, 356, 311, 358	sublimc 43193	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 − 𝑌) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) lim_ℂ (𝑄‘𝑖)))
360	359	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑅 − 𝑌) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) lim_ℂ (𝑄‘𝑖)))
361		iffalse 4468	. . . . . . . . . 10 ⊢ (¬ (𝑉‘𝑖) < 𝑋 → if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌) = 𝑌)
362	361	oveq2d 7291	. . . . . . . . 9 ⊢ (¬ (𝑉‘𝑖) < 𝑋 → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅 − 𝑌))
363	362	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅 − 𝑌))
364		0red 10978	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℝ)
365	299	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
366	215	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
367	1	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → 𝑋 ∈ ℝ)
368	278	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑉‘𝑖) ∈ ℝ)
369		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → ¬ (𝑉‘𝑖) < 𝑋)
370	367, 368, 369	nltled 11125	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → 𝑋 ≤ (𝑉‘𝑖))
371	368, 367	subge0d 11565	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (0 ≤ ((𝑉‘𝑖) − 𝑋) ↔ 𝑋 ≤ (𝑉‘𝑖)))
372	370, 371	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → 0 ≤ ((𝑉‘𝑖) − 𝑋))
373	293	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) = (𝑄‘𝑖))
374	373	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → ((𝑉‘𝑖) − 𝑋) = (𝑄‘𝑖))
375	372, 374	breqtrd 5100	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → 0 ≤ (𝑄‘𝑖))
376	375	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ≤ (𝑄‘𝑖))
377	205	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠)
378	364, 365, 366, 376, 377	lelttrd 11133	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 < 𝑠)
379	378	iftrued 4467	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌)
380	379	oveq2d 7291	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑌))
381	380	mpteq2dva 5174	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)))
382	381	oveq1d 7290	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) lim_ℂ (𝑄‘𝑖)))
383	360, 363, 382	3eltr4d 2854	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim_ℂ (𝑄‘𝑖)))
384	383	3adantl3 1167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim_ℂ (𝑄‘𝑖)))
385	353, 384	pm2.61dan 810	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) lim_ℂ (𝑄‘𝑖)))
386	313, 248, 288	idlimc 43167	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) lim_ℂ (𝑄‘𝑖)))
387	386	3adant3 1131	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) lim_ℂ (𝑄‘𝑖)))
388	293	3adant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
389	295	3adant3 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑉‘𝑖) ∈ ℂ)
390	229	3adant3 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → 𝑋 ∈ ℂ)
391		neqne 2951	. . . . . . . 8 ⊢ (¬ (𝑉‘𝑖) = 𝑋 → (𝑉‘𝑖) ≠ 𝑋)
392	391	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑉‘𝑖) ≠ 𝑋)
393	389, 390, 392	subne0d 11341	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → ((𝑉‘𝑖) − 𝑋) ≠ 0)
394	388, 393	eqnetrd 3011	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) ≠ 0)
395	198	3adantl3 1167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0)
396	395	neqned 2950	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ 0)
397	247, 248, 249, 265, 271, 385, 387, 394, 396	divlimc 43197	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) lim_ℂ (𝑄‘𝑖)))
398		iffalse 4468	. . . . . 6 ⊢ (¬ (𝑉‘𝑖) = 𝑋 → if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) = ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)))
399	95, 398	eqtrid 2790	. . . . 5 ⊢ (¬ (𝑉‘𝑖) = 𝑋 → 𝐴 = ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)))
400	399	3ad2ant3 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → 𝐴 = ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)))
401		ioossre 13140	. . . . . . . . . . . . 13 ⊢ (𝑋(,)+∞) ⊆ ℝ
402	401	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
403	24, 402	fssresd 6641	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
404	401, 47	sstrid 3932	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
405	34	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → +∞ ∈ ℝ^*)
406	1	ltpnfd 12857	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 < +∞)
407	52, 405, 1, 406	lptioo1cn 43187	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)+∞)))
408	403, 404, 407, 31	limcrecl 43170	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ)
409	24, 1, 408, 260, 99	fourierdlem9 43657	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
410	409	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐻:(-π[,]π)⟶ℝ)
411	410, 147	feqresmpt 6838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)))
412	147	sselda 3921	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (-π[,]π))
413		0cnd 10968	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈ ℂ)
414	262	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
415	256, 414	subcld 11332	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
416	266	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
417	198	neqned 2950	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ 0)
418	415, 416, 417	divcld 11751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℂ)
419	413, 418	ifcld 4505	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℂ)
420	99	fvmpt2 6886	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (-π[,]π) ∧ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℂ) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
421	412, 419, 420	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
422	198	iffalsed 4470	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(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))
423	421, 422	eqtrd 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
424	423	mpteq2dva 5174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
425	411, 424	eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
426	425	3adant3 1131	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
427	426	oveq1d 7290	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) lim_ℂ (𝑄‘𝑖)))
428	397, 400, 427	3eltr4d 2854	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
429	428	3expa 1117	. 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
430	246, 429	pm2.61dan 810	1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 ⊆ wss 3887 ifcif 4459 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 ↾ cres 5591 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 +∞cpnf 11006 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕcn 11973 (,)cioo 13079 [,]cicc 13082 ...cfz 13239 ..^cfzo 13382 πcpi 15776 TopOpenctopn 17132 topGenctg 17148 ℂ_fldccnfld 20597 intcnt 22168 lim_ℂ climc 25026 D cdv 25027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031

This theorem is referenced by: fourierdlem85 43732 fourierdlem88 43735 fourierdlem103 43750 fourierdlem104 43751

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