fourierdlem81 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem81		Structured version Visualization version GIF version

Theorem fourierdlem81 44418

Description: The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by its period 𝑇. In this lemma, 𝑇 is assumed to be strictly positive. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem81.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem81.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem81.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem81.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem81.t	⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
fourierdlem81.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem81.fper	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem81.s	⊢ 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇))
fourierdlem81.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
fourierdlem81.cncf	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem81.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
fourierdlem81.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
fourierdlem81.g	⊢ 𝐺 = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))))
fourierdlem81.h	⊢ 𝐻 = (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ (𝐺‘(𝑥 − 𝑇)))

**Assertion**
Ref	Expression
fourierdlem81	⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Distinct variable groups: 𝐴,𝑖,𝑚,𝑝 𝑥,𝐴,𝑖 𝐵,𝑖,𝑚,𝑝 𝑥,𝐵 𝑖,𝐹,𝑥 𝑥,𝐺 𝑥,𝐿 𝑖,𝑀,𝑚,𝑝 𝑥,𝑀 𝑄,𝑖,𝑝 𝑥,𝑄 𝑥,𝑅 𝑆,𝑖,𝑥 𝑇,𝑖,𝑥 𝜑,𝑖,𝑥 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝑃(𝑥,𝑖,𝑚,𝑝) 𝑄(𝑚) 𝑅(𝑖,𝑚,𝑝) 𝑆(𝑚,𝑝) 𝑇(𝑚,𝑝) 𝐹(𝑚,𝑝) 𝐺(𝑖,𝑚,𝑝) 𝐻(𝑥,𝑖,𝑚,𝑝) 𝐿(𝑖,𝑚,𝑝)

Proof of Theorem fourierdlem81 Dummy variables 𝑦 𝑤 𝑧 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem81.q	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
2		fourierdlem81.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem81.p	. . . . . . . . . . 11 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
4	3	fourierdlem2 44340	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
5	2, 4	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
6	1, 5	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simprd 496	. . . . . . 7 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
8	7	simpld 495	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
9	8	simpld 495	. . . . 5 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
10	9	eqcomd 2742	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
11	8	simprd 496	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
12	11	eqcomd 2742	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
13	10, 12	oveq12d 7375	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
14	13	itgeq1d 44188	. 2 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝑄‘0)[,](𝑄‘𝑀))(𝐹‘𝑥) d𝑥)
15		0zd 12511	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
16		nnuz 12806	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
17		0p1e1 12275	. . . . . 6 ⊢ (0 + 1) = 1
18	17	fveq2i 6845	. . . . 5 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
19	16, 18	eqtr4i 2767	. . . 4 ⊢ ℕ = (ℤ_≥‘(0 + 1))
20	2, 19	eleqtrdi 2848	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(0 + 1)))
21	6	simpld 495	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
22		reex 11142	. . . . . 6 ⊢ ℝ ∈ V
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
24		ovex 7390	. . . . . 6 ⊢ (0...𝑀) ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
26	23, 25	elmapd 8779	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
27	21, 26	mpbid 231	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
28	7	simprd 496	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
29	28	r19.21bi 3234	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
30		fourierdlem81.f	. . . . 5 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
31	30	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → 𝐹:ℝ⟶ℂ)
32		fourierdlem81.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
33	9, 32	eqeltrd 2838	. . . . . 6 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
34		fourierdlem81.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
35	11, 34	eqeltrd 2838	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
36	33, 35	iccssred 13351	. . . . 5 ⊢ (𝜑 → ((𝑄‘0)[,](𝑄‘𝑀)) ⊆ ℝ)
37	36	sselda 3944	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → 𝑥 ∈ ℝ)
38	31, 37	ffvelcdmd 7036	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → (𝐹‘𝑥) ∈ ℂ)
39	27	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
40		elfzofz 13588	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
41	40	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
42	39, 41	ffvelcdmd 7036	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
43		fzofzp1 13669	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
44	43	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
45	39, 44	ffvelcdmd 7036	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
46	30	feqmptd 6910	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
47	46	reseq1d 5936	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
48	47	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
49		ioossre 13325	. . . . . . . 8 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
50	49	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
51	50	resmptd 5994	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
52	48, 51	eqtr2d 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
53		fourierdlem81.cncf	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
54		fourierdlem81.l	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
55		fourierdlem81.r	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
56	42, 45, 53, 54, 55	iblcncfioo 44209	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ 𝐿¹)
57	52, 56	eqeltrd 2838	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
58	30	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℂ)
59	42, 45	iccssred 13351	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ ℝ)
60	59	sselda 3944	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
61	58, 60	ffvelcdmd 7036	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹‘𝑥) ∈ ℂ)
62	42, 45, 57, 61	ibliooicc 44202	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
63	15, 20, 27, 29, 38, 62	itgspltprt 44210	. 2 ⊢ (𝜑 → ∫((𝑄‘0)[,](𝑄‘𝑀))(𝐹‘𝑥) d𝑥 = Σ𝑖 ∈ (0..^𝑀)∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
64		fourierdlem81.s	. . . . . . . 8 ⊢ 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇))
65	64	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇)))
66		fveq2 6842	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
67	66	oveq1d 7372	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
68	67	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
69	2	nnnn0d 12473	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70		nn0uz 12805	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
71	69, 70	eleqtrdi 2848	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
72		eluzfz1 13448	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
73	71, 72	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0...𝑀))
74		fourierdlem81.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
75	74	rpred 12957	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℝ)
76	33, 75	readdcld 11184	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) ∈ ℝ)
77	65, 68, 73, 76	fvmptd 6955	. . . . . 6 ⊢ (𝜑 → (𝑆‘0) = ((𝑄‘0) + 𝑇))
78	9	oveq1d 7372	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) = (𝐴 + 𝑇))
79	77, 78	eqtr2d 2777	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑇) = (𝑆‘0))
80		fveq2 6842	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
81	80	oveq1d 7372	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
82	81	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
83		eluzfz2 13449	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
84	71, 83	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
85	35, 75	readdcld 11184	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) ∈ ℝ)
86	65, 82, 84, 85	fvmptd 6955	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑀) = ((𝑄‘𝑀) + 𝑇))
87	11	oveq1d 7372	. . . . . 6 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) = (𝐵 + 𝑇))
88	86, 87	eqtr2d 2777	. . . . 5 ⊢ (𝜑 → (𝐵 + 𝑇) = (𝑆‘𝑀))
89	79, 88	oveq12d 7375	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = ((𝑆‘0)[,](𝑆‘𝑀)))
90	89	itgeq1d 44188	. . 3 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝑆‘0)[,](𝑆‘𝑀))(𝐹‘𝑥) d𝑥)
91	27	ffvelcdmda 7035	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
92	75	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑇 ∈ ℝ)
93	91, 92	readdcld 11184	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
94	93, 64	fmptd 7062	. . . 4 ⊢ (𝜑 → 𝑆:(0...𝑀)⟶ℝ)
95	75	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ)
96	42, 45, 95, 29	ltadd1dd 11766	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
97	40, 93	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
98	64	fvmpt2 6959	. . . . . 6 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑄‘𝑖) + 𝑇) ∈ ℝ) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
99	41, 97, 98	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
100		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
101	100	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑗) + 𝑇))
102	101	cbvmptv 5218	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))
103	64, 102	eqtri 2764	. . . . . . 7 ⊢ 𝑆 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))
104	103	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇)))
105		fveq2 6842	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (𝑄‘𝑗) = (𝑄‘(𝑖 + 1)))
106	105	oveq1d 7372	. . . . . . 7 ⊢ (𝑗 = (𝑖 + 1) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
107	106	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
108	45, 95	readdcld 11184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
109	104, 107, 44, 108	fvmptd 6955	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) + 𝑇))
110	96, 99, 109	3brtr4d 5137	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
111	30	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝐹:ℝ⟶ℂ)
112	77, 76	eqeltrd 2838	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘0) ∈ ℝ)
113	112	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝑆‘0) ∈ ℝ)
114	86, 85	eqeltrd 2838	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝑀) ∈ ℝ)
115	114	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝑆‘𝑀) ∈ ℝ)
116	113, 115	iccssred 13351	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → ((𝑆‘0)[,](𝑆‘𝑀)) ⊆ ℝ)
117		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀)))
118	116, 117	sseldd 3945	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝑥 ∈ ℝ)
119	111, 118	ffvelcdmd 7036	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝐹‘𝑥) ∈ ℂ)
120	99, 97	eqeltrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) ∈ ℝ)
121	109, 108	eqeltrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
122		ioosscn 13326	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
123	122	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
124		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
125	124	rexbidv 3175	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)))
126		oveq1 7364	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
127	126	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
128	127	cbvrexvw 3226	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇))
129	125, 128	bitrdi 286	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)))
130	129	cbvrabv 3417	. . . . . . . 8 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}
131		fdm 6677	. . . . . . . . . . . 12 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
132	30, 131	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = ℝ)
133	132	feq2d 6654	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:ℝ⟶ℂ))
134	30, 133	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
135	134	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:dom 𝐹⟶ℂ)
136		elioore 13294	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑧 ∈ ℝ)
137	136	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ℝ)
138	75	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
139	137, 138	readdcld 11184	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) ∈ ℝ)
140	139	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) ∈ ℝ)
141	140	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ℝ)
142		simp3 1138	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 = (𝑧 + 𝑇))
143	132	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → dom 𝐹 = ℝ)
144	143	3adant1r 1177	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → dom 𝐹 = ℝ)
145	141, 142, 144	3eltr4d 2853	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 ∈ dom 𝐹)
146	145	3exp 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹)))
147	146	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ℂ) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹)))
148	147	rexlimdv 3150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ℂ) → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹))
149	148	ralrimiva 3143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑤 ∈ ℂ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹))
150		rabss 4029	. . . . . . . . 9 ⊢ ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ⊆ dom 𝐹 ↔ ∀𝑤 ∈ ℂ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹))
151	149, 150	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ⊆ dom 𝐹)
152		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝜑)
153	32	rexrd 11205	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
154	153	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐴 ∈ ℝ^*)
155	34	rexrd 11205	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
156	155	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐵 ∈ ℝ^*)
157	3, 2, 1	fourierdlem15 44353	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
158	157	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
159		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
160		ioossicc 13350	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
161	160	sseli 3940	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
162	161	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
163	154, 156, 158, 159, 162	fourierdlem1 44339	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ (𝐴[,]𝐵))
164		fourierdlem81.fper	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
165	152, 163, 164	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
166	123, 95, 130, 135, 151, 165, 53	cncfperiod 44110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) ∈ ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}–cn→ℂ))
167	125	elrab 3645	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↔ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)))
168	167	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} → ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
169		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
170		nfv 1917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑖 ∈ (0..^𝑀))
171		nfre1 3268	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)
172	170, 171	nfan 1902	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
173		nfv 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1)))
174		simp3 1138	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
175	139	3adant3 1132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ℝ)
176	174, 175	eqeltrd 2838	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ℝ)
177	176	3adant1r 1177	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ℝ)
178	42	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
179	136	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ℝ)
180	75	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
181		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
182	42	rexrd 11205	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
183	182	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
184	45	rexrd 11205	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
185	184	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
186		elioo2 13305	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^*) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (𝑧 ∈ ℝ ∧ (𝑄‘𝑖) < 𝑧 ∧ 𝑧 < (𝑄‘(𝑖 + 1)))))
187	183, 185, 186	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (𝑧 ∈ ℝ ∧ (𝑄‘𝑖) < 𝑧 ∧ 𝑧 < (𝑄‘(𝑖 + 1)))))
188	181, 187	mpbid 231	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 ∈ ℝ ∧ (𝑄‘𝑖) < 𝑧 ∧ 𝑧 < (𝑄‘(𝑖 + 1))))
189	188	simp2d 1143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑧)
190	178, 179, 180, 189	ltadd1dd 11766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) + 𝑇) < (𝑧 + 𝑇))
191	190	3adant3 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → ((𝑄‘𝑖) + 𝑇) < (𝑧 + 𝑇))
192	99	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
193		simp3 1138	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
194	191, 192, 193	3brtr4d 5137	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) < 𝑥)
195	45	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
196	188	simp3d 1144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 < (𝑄‘(𝑖 + 1)))
197	179, 195, 180, 196	ltadd1dd 11766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
198	197	3adant3 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
199	109	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑆‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) + 𝑇))
200	198, 193, 199	3brtr4d 5137	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 < (𝑆‘(𝑖 + 1)))
201	177, 194, 200	3jca 1128	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1))))
202	201	3exp 1119	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑥 = (𝑧 + 𝑇) → (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1))))))
203	202	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑥 = (𝑧 + 𝑇) → (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1))))))
204	172, 173, 203	rexlimd 3249	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇) → (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1)))))
205	169, 204	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1))))
206	120	rexrd 11205	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) ∈ ℝ^*)
207	206	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) ∈ ℝ^*)
208	121	rexrd 11205	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
209	208	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
210		elioo2 13305	. . . . . . . . . . . . . 14 ⊢ (((𝑆‘𝑖) ∈ ℝ^* ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ^*) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↔ (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1)))))
211	207, 209, 210	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↔ (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1)))))
212	205, 211	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
213	168, 212	sylan2 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
214		elioore 13294	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
215	214	recnd 11183	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℂ)
216	215	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
217	182	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
218	184	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
219	214	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
220	75	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
221	219, 220	resubcld 11583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
222	221	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
223	99	oveq1d 7372	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
224	42	recnd 11183	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
225	95	recnd 11183	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℂ)
226	224, 225	pncand 11513	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇) − 𝑇) = (𝑄‘𝑖))
227	223, 226	eqtr2d 2777	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
228	227	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
229	120	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ)
230	214	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
231	75	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
232		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
233	206	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ^*)
234	208	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
235	233, 234, 210	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↔ (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1)))))
236	232, 235	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1))))
237	236	simp2d 1143	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) < 𝑥)
238	229, 230, 231, 237	ltsub1dd 11767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑆‘𝑖) − 𝑇) < (𝑥 − 𝑇))
239	228, 238	eqbrtrd 5127	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑥 − 𝑇))
240	121	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
241	236	simp3d 1144	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 < (𝑆‘(𝑖 + 1)))
242	230, 240, 231, 241	ltsub1dd 11767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) < ((𝑆‘(𝑖 + 1)) − 𝑇))
243	109	oveq1d 7372	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
244	45	recnd 11183	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
245	244, 225	pncand 11513	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
246	243, 245	eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
247	246	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
248	242, 247	breqtrd 5131	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
249	217, 218, 222, 239, 248	eliood 43726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
250	219	recnd 11183	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
251	220	recnd 11183	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℂ)
252	250, 251	npcand 11516	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
253	252	eqcomd 2742	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
254	253	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
255		oveq1 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑧 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
256	255	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = ((𝑥 − 𝑇) + 𝑇)))
257	256	rspcev 3581	. . . . . . . . . . . . 13 ⊢ (((𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = ((𝑥 − 𝑇) + 𝑇)) → ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
258	249, 254, 257	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
259	216, 258, 167	sylanbrc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)})
260	213, 259	impbida 799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↔ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))))
261	260	eqrdv 2734	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
262	261	reseq2d 5937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) = (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))))
263	30	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
264		ioossre 13325	. . . . . . . . . 10 ⊢ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ℝ
265	264	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ℝ)
266	263, 265	feqresmpt 6911	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
267	262, 266	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) = (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
268	261	oveq1d 7372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}–cn→ℂ) = (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
269	166, 267, 268	3eltr3d 2852	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
270	49, 132	sseqtrrid 3997	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
271	270	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
272		eqid 2736	. . . . . . . 8 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}
273		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝜑)
274	153	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐴 ∈ ℝ^*)
275	155	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐵 ∈ ℝ^*)
276	157	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
277		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
278	160, 181	sselid 3942	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
279	274, 275, 276, 277, 278	fourierdlem1 44339	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ (𝐴[,]𝐵))
280		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑧 ∈ (𝐴[,]𝐵)))
281	280	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵))))
282		oveq1 7364	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 + 𝑇) = (𝑧 + 𝑇))
283	282	fveq2d 6846	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑧 + 𝑇)))
284		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
285	283, 284	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧)))
286	281, 285	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))))
287	286, 164	chvarvv 2002	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
288	273, 279, 287	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
289	135, 123, 271, 225, 272, 151, 288, 54	limcperiod 43859	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) lim_ℂ ((𝑄‘(𝑖 + 1)) + 𝑇)))
290	109	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) = (𝑆‘(𝑖 + 1)))
291	267, 290	oveq12d 7375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) lim_ℂ ((𝑄‘(𝑖 + 1)) + 𝑇)) = ((𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) lim_ℂ (𝑆‘(𝑖 + 1))))
292	289, 291	eleqtrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) lim_ℂ (𝑆‘(𝑖 + 1))))
293	135, 123, 271, 225, 272, 151, 288, 55	limcperiod 43859	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) lim_ℂ ((𝑄‘𝑖) + 𝑇)))
294	99	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) = (𝑆‘𝑖))
295	267, 294	oveq12d 7375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) lim_ℂ ((𝑄‘𝑖) + 𝑇)) = ((𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) lim_ℂ (𝑆‘𝑖)))
296	293, 295	eleqtrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) lim_ℂ (𝑆‘𝑖)))
297	120, 121, 269, 292, 296	iblcncfioo 44209	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
298	30	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℂ)
299	120	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ)
300	121	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
301		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
302		eliccre 43733	. . . . . . 7 ⊢ (((𝑆‘𝑖) ∈ ℝ ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
303	299, 300, 301, 302	syl3anc 1371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
304	298, 303	ffvelcdmd 7036	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘𝑥) ∈ ℂ)
305	120, 121, 297, 304	ibliooicc 44202	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
306	15, 20, 94, 110, 119, 305	itgspltprt 44210	. . 3 ⊢ (𝜑 → ∫((𝑆‘0)[,](𝑆‘𝑀))(𝐹‘𝑥) d𝑥 = Σ𝑖 ∈ (0..^𝑀)∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
307		iftrue 4492	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑆‘𝑖) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = 𝑅)
308	307	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = 𝑅)
309		fourierdlem81.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))))
310		iftrue 4492	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑄‘𝑖) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = 𝑅)
311		iftrue 4492	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑄‘𝑖) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))) = 𝑅)
312	310, 311	eqtr4d 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑄‘𝑖) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
313	312	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝑄‘𝑖)) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
314		iffalse 4495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑥 = (𝑄‘𝑖) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
315	314	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑥 = (𝑄‘𝑖) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
316		iftrue 4492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = 𝐿)
317	316	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑥 = (𝑄‘𝑖) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = 𝐿)
318		iffalse 4495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑥 = (𝑄‘𝑖) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))
319	318	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑥 = (𝑄‘𝑖) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))
320		iftrue 4492	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = 𝐿)
321	320	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑥 = (𝑄‘𝑖) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = 𝐿)
322	319, 321	eqtr2d 2777	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑥 = (𝑄‘𝑖) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝐿 = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
323	315, 317, 322	3eqtrd 2780	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑥 = (𝑄‘𝑖) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
324	323	adantll 712	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
325	314	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
326		iffalse 4495	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
327	326	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
328	318	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))
329		iffalse 4495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))
330	329	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))
331	182	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) ∈ ℝ^*)
332	184	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
333	60	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
334	42	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ∈ ℝ)
335	60	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ∈ ℝ)
336	182	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ∈ ℝ^*)
337	184	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
338		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
339		iccgelb 13320	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ≤ 𝑥)
340	336, 337, 338, 339	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ≤ 𝑥)
341		neqne 2951	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑥 = (𝑄‘𝑖) → 𝑥 ≠ (𝑄‘𝑖))
342	341	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ≠ (𝑄‘𝑖))
343	334, 335, 340, 342	leneltd 11309	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) < 𝑥)
344	343	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) < 𝑥)
345	45	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
346	182	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
347	184	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
348		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
349		iccleub 13319	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
350	346, 347, 348, 349	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
351	350	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
352		neqne 2951	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → 𝑥 ≠ (𝑄‘(𝑖 + 1)))
353	352	necomd 2999	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → (𝑄‘(𝑖 + 1)) ≠ 𝑥)
354	353	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ≠ 𝑥)
355	333, 345, 351, 354	leneltd 11309	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝑥 < (𝑄‘(𝑖 + 1)))
356	331, 332, 333, 344, 355	eliood 43726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
357		fvres 6861	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
358	356, 357	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
359	328, 330, 358	3eqtrrd 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝐹‘𝑥) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
360	325, 327, 359	3eqtrd 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
361	324, 360	pm2.61dan 811	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
362	313, 361	pm2.61dan 811	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
363	362	mpteq2dva 5205	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))) = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))))
364	309, 363	eqtrid 2788	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))))
365		eqeq1 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑄‘𝑖) ↔ 𝑤 = (𝑄‘𝑖)))
366		eqeq1 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑄‘(𝑖 + 1)) ↔ 𝑤 = (𝑄‘(𝑖 + 1))))
367		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))
368	366, 367	ifbieq2d 4512	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤)))
369	365, 368	ifbieq2d 4512	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))) = if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))))
370	369	cbvmptv 5218	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))) = (𝑤 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))))
371	364, 370	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 = (𝑤 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤)))))
372	371	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → 𝐺 = (𝑤 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤)))))
373		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → 𝑤 = (𝑥 − 𝑇))
374		oveq1 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑆‘𝑖) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
375	374	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
376	227	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) = (𝑄‘𝑖))
377	376	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → ((𝑆‘𝑖) − 𝑇) = (𝑄‘𝑖))
378	373, 375, 377	3eqtrd 2780	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → 𝑤 = (𝑄‘𝑖))
379	378	iftrued 4494	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))) = 𝑅)
380	374	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
381	42, 45, 29	ltled 11303	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1)))
382		lbicc2 13381	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
383	182, 184, 381, 382	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
384	376, 383	eqeltrd 2838	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
385	384	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → ((𝑆‘𝑖) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
386	380, 385	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
387		limccl 25239	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ⊆ ℂ
388	387, 55	sselid 3942	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ℂ)
389	388	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → 𝑅 ∈ ℂ)
390	372, 379, 386, 389	fvmptd 6955	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝐺‘(𝑥 − 𝑇)) = 𝑅)
391	308, 390	eqtr4d 2779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = (𝐺‘(𝑥 − 𝑇)))
392	391	adantlr 713	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = (𝐺‘(𝑥 − 𝑇)))
393		iffalse 4495	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = (𝑆‘𝑖) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)))
394	393	adantl 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)))
395	371	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝐺 = (𝑤 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤)))))
396		eqeq1 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → (𝑤 = (𝑄‘𝑖) ↔ ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖)))
397		eqeq1 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → (𝑤 = (𝑄‘(𝑖 + 1)) ↔ ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1))))
398		fveq2 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑆‘(𝑖 + 1)) − 𝑇)))
399	397, 398	ifbieq2d 4512	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤)) = if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑆‘(𝑖 + 1)) − 𝑇))))
400	396, 399	ifbieq2d 4512	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))) = if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖), 𝑅, if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑆‘(𝑖 + 1)) − 𝑇)))))
401	400	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇)) → if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))) = if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖), 𝑅, if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑆‘(𝑖 + 1)) − 𝑇)))))
402		eqeq1 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)) → (((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖) ↔ (𝑄‘(𝑖 + 1)) = (𝑄‘𝑖)))
403		iftrue 4492	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)) → if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑆‘(𝑖 + 1)) − 𝑇))) = 𝐿)
404	402, 403	ifbieq2d 4512	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)) → if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖), 𝑅, if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑆‘(𝑖 + 1)) − 𝑇)))) = if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿))
405	246, 404	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖), 𝑅, if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑆‘(𝑖 + 1)) − 𝑇)))) = if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿))
406	405	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇)) → if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖), 𝑅, if(((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑆‘(𝑖 + 1)) − 𝑇)))) = if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿))
407	42, 29	gtned 11290	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ≠ (𝑄‘𝑖))
408	407	neneqd 2948	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ (𝑄‘(𝑖 + 1)) = (𝑄‘𝑖))
409	408	iffalsed 4497	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿) = 𝐿)
410	409	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇)) → if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿) = 𝐿)
411	401, 406, 410	3eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇)) → if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))) = 𝐿)
412	411	adantlr 713	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) ∧ 𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇)) → if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))) = 𝐿)
413		ubicc2 13382	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
414	182, 184, 381, 413	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
415	246, 414	eqeltrd 2838	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
416	415	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → ((𝑆‘(𝑖 + 1)) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
417		limccl 25239	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ℂ
418	417, 54	sselid 3942	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ℂ)
419	418	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝐿 ∈ ℂ)
420	395, 412, 416, 419	fvmptd 6955	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)) = 𝐿)
421		oveq1 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → (𝑥 − 𝑇) = ((𝑆‘(𝑖 + 1)) − 𝑇))
422	421	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → (𝐺‘(𝑥 − 𝑇)) = (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)))
423	422	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘(𝑥 − 𝑇)) = (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)))
424		iftrue 4492	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = 𝐿)
425	424	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = 𝐿)
426	420, 423, 425	3eqtr4rd 2787	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = (𝐺‘(𝑥 − 𝑇)))
427	426	ad4ant14 750	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = (𝐺‘(𝑥 − 𝑇)))
428		iffalse 4495	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = (𝑆‘(𝑖 + 1)) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))
429	428	adantl 482	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))
430	371	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝐺 = (𝑤 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤)))))
431	430	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝐺 = (𝑤 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤)))))
432		eqeq1 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑥 − 𝑇) → (𝑤 = (𝑄‘𝑖) ↔ (𝑥 − 𝑇) = (𝑄‘𝑖)))
433		eqeq1 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑥 − 𝑇) → (𝑤 = (𝑄‘(𝑖 + 1)) ↔ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))))
434		fveq2 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑥 − 𝑇) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
435	433, 434	ifbieq2d 4512	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑥 − 𝑇) → if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤)) = if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))))
436	432, 435	ifbieq2d 4512	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑥 − 𝑇) → if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))) = if((𝑥 − 𝑇) = (𝑄‘𝑖), 𝑅, if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))))
437	436	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) ∧ 𝑤 = (𝑥 − 𝑇)) → if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))) = if((𝑥 − 𝑇) = (𝑄‘𝑖), 𝑅, if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))))
438	303	recnd 11183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
439	225	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℂ)
440	438, 439	npcand 11516	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
441	440	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
442	441	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
443		oveq1 7364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 − 𝑇) = (𝑄‘𝑖) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
444	443	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
445	294	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → ((𝑄‘𝑖) + 𝑇) = (𝑆‘𝑖))
446	442, 444, 445	3eqtrd 2780	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → 𝑥 = (𝑆‘𝑖))
447	446	stoic1a 1774	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → ¬ (𝑥 − 𝑇) = (𝑄‘𝑖))
448	447	iffalsed 4497	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → if((𝑥 − 𝑇) = (𝑄‘𝑖), 𝑅, if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))) = if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))))
449	448	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) ∧ 𝑤 = (𝑥 − 𝑇)) → if((𝑥 − 𝑇) = (𝑄‘𝑖), 𝑅, if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))) = if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))))
450	441	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
451		oveq1 7364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
452	451	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
453	290	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))) → ((𝑄‘(𝑖 + 1)) + 𝑇) = (𝑆‘(𝑖 + 1)))
454	450, 452, 453	3eqtrd 2780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))) → 𝑥 = (𝑆‘(𝑖 + 1)))
455	454	stoic1a 1774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ¬ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)))
456	455	iffalsed 4497	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
457	456	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
458	457	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) ∧ 𝑤 = (𝑥 − 𝑇)) → if((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
459	437, 449, 458	3eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) ∧ 𝑤 = (𝑥 − 𝑇)) → if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
460	42	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
461	45	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
462	75	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
463	303, 462	resubcld 11583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
464	227	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
465	206	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ^*)
466	208	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
467		iccgelb 13320	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑖) ∈ ℝ^* ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ≤ 𝑥)
468	465, 466, 301, 467	syl3anc 1371	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ≤ 𝑥)
469	299, 303, 462, 468	lesub1dd 11771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑆‘𝑖) − 𝑇) ≤ (𝑥 − 𝑇))
470	464, 469	eqbrtrd 5127	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
471		iccleub 13319	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑖) ∈ ℝ^* ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
472	465, 466, 301, 471	syl3anc 1371	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
473	303, 300, 462, 472	lesub1dd 11771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ≤ ((𝑆‘(𝑖 + 1)) − 𝑇))
474	246	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
475	473, 474	breqtrd 5131	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ≤ (𝑄‘(𝑖 + 1)))
476	460, 461, 463, 470, 475	eliccd 43732	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
477	476	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
478	135, 271	fssresd 6709	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
479	478	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
480	182	ad3antrrr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘𝑖) ∈ ℝ^*)
481	184	ad3antrrr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
482	303	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
483	95	ad3antrrr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑇 ∈ ℝ)
484	482, 483	resubcld 11583	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ℝ)
485	42	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) ∈ ℝ)
486	463	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ∈ ℝ)
487	470	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
488	447	neqned 2950	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ≠ (𝑄‘𝑖))
489	485, 486, 487, 488	leneltd 11309	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) < (𝑥 − 𝑇))
490	489	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘𝑖) < (𝑥 − 𝑇))
491	463	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ℝ)
492	45	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
493	475	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ≤ (𝑄‘(𝑖 + 1)))
494		eqcom 2743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇))
495	454	ex 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) → 𝑥 = (𝑆‘(𝑖 + 1))))
496	494, 495	biimtrrid 242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇) → 𝑥 = (𝑆‘(𝑖 + 1))))
497	496	con3dimp 409	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ¬ (𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇))
498	497	neqned 2950	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ≠ (𝑥 − 𝑇))
499	491, 492, 493, 498	leneltd 11309	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
500	499	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
501	480, 481, 484, 490, 500	eliood 43726	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
502	479, 501	ffvelcdmd 7036	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)) ∈ ℂ)
503	431, 459, 477, 502	fvmptd 6955	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘(𝑥 − 𝑇)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
504		fvres 6861	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
505	501, 504	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
506	503, 505	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘(𝑥 − 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
507	465	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘𝑖) ∈ ℝ^*)
508	466	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
509	120	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) ∈ ℝ)
510	303	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → 𝑥 ∈ ℝ)
511	468	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) ≤ 𝑥)
512		neqne 2951	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = (𝑆‘𝑖) → 𝑥 ≠ (𝑆‘𝑖))
513	512	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → 𝑥 ≠ (𝑆‘𝑖))
514	509, 510, 511, 513	leneltd 11309	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) < 𝑥)
515	514	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘𝑖) < 𝑥)
516	300	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
517	472	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
518		neqne 2951	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = (𝑆‘(𝑖 + 1)) → 𝑥 ≠ (𝑆‘(𝑖 + 1)))
519	518	necomd 2999	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = (𝑆‘(𝑖 + 1)) → (𝑆‘(𝑖 + 1)) ≠ 𝑥)
520	519	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘(𝑖 + 1)) ≠ 𝑥)
521	482, 516, 517, 520	leneltd 11309	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑥 < (𝑆‘(𝑖 + 1)))
522	507, 508, 482, 515, 521	eliood 43726	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
523		fvres 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
524	522, 523	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
525	440	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘𝑥))
526	525	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘𝑥) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
527	526	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐹‘𝑥) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
528	438, 439	subcld 11512	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℂ)
529		elex 3463	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 − 𝑇) ∈ ℂ → (𝑥 − 𝑇) ∈ V)
530	528, 529	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ V)
531	530	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ V)
532		simp-4l 781	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝜑)
533	153	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
534	155	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
535	157	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
536		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
537	533, 534, 535, 536	fourierdlem8 44346	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
538	537	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
539	538, 476	sseldd 3945	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
540	539	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
541	532, 540	jca 512	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
542		eleq1 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
543	542	anbi2d 629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))))
544		oveq1 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
545	544	fveq2d 6846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
546		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 − 𝑇)))
547	545, 546	eqeq12d 2752	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦) ↔ (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
548	543, 547	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 − 𝑇) → (((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)) ↔ ((𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇)))))
549		eleq1 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
550	549	anbi2d 629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵))))
551		oveq1 7364	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
552	551	fveq2d 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
553		fveq2 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
554	552, 553	eqeq12d 2752	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)))
555	550, 554	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))))
556	555, 164	chvarvv 2002	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
557	548, 556	vtoclg 3525	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 − 𝑇) ∈ V → ((𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
558	531, 541, 557	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
559	524, 527, 558	3eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥) = (𝐹‘(𝑥 − 𝑇)))
560	506, 559	eqtr4d 2779	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘(𝑥 − 𝑇)) = ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))
561	429, 560	eqtr4d 2779	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = (𝐺‘(𝑥 − 𝑇)))
562	427, 561	pm2.61dan 811	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = (𝐺‘(𝑥 − 𝑇)))
563	394, 562	eqtrd 2776	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = (𝐺‘(𝑥 − 𝑇)))
564	392, 563	pm2.61dan 811	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = (𝐺‘(𝑥 − 𝑇)))
565	308, 389	eqeltrd 2838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) ∈ ℂ)
566	565	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) ∈ ℂ)
567	425, 419	eqeltrd 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) ∈ ℂ)
568	567	ad4ant14 750	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) ∈ ℂ)
569	263, 265	fssresd 6709	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))):((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))⟶ℂ)
570	569	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))):((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))⟶ℂ)
571	570, 522	ffvelcdmd 7036	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥) ∈ ℂ)
572	429, 571	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) ∈ ℂ)
573	568, 572	pm2.61dan 811	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) ∈ ℂ)
574	394, 573	eqeltrd 2838	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) ∈ ℂ)
575	566, 574	pm2.61dan 811	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) ∈ ℂ)
576		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)))) = (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))
577	576	fvmpt2 6959	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ∧ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) ∈ ℂ) → ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) = if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))
578	301, 575, 577	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) = if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))
579		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ (0..^𝑀))
580		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))) = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
581	579, 580, 42, 45, 53, 54, 55	cncfiooicc 44125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))) ∈ (((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))–cn→ℂ))
582	364, 581	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 ∈ (((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))–cn→ℂ))
583		cncff 24256	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))–cn→ℂ) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
584	582, 583	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
585	584	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
586	585, 476	ffvelcdmd 7036	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐺‘(𝑥 − 𝑇)) ∈ ℂ)
587		fourierdlem81.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ (𝐺‘(𝑥 − 𝑇)))
588	587	fvmpt2 6959	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ∧ (𝐺‘(𝑥 − 𝑇)) ∈ ℂ) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
589	301, 586, 588	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
590	564, 578, 589	3eqtr4rd 2787	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐻‘𝑥) = ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥))
591	590	itgeq2dv 25146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐻‘𝑥) d𝑥 = ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) d𝑥)
592		ioossicc 13350	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))
593	592	sseli 3940	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
594	593	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
595	593, 575	sylan2 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) ∈ ℂ)
596	594, 595, 577	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) = if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))
597	229, 237	gtned 11290	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ≠ (𝑆‘𝑖))
598	597	neneqd 2948	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ¬ 𝑥 = (𝑆‘𝑖))
599	598	iffalsed 4497	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))) = if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)))
600	230, 241	ltned 11291	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ≠ (𝑆‘(𝑖 + 1)))
601	600	neneqd 2948	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ¬ 𝑥 = (𝑆‘(𝑖 + 1)))
602	601	iffalsed 4497	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))
603	523	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
604	602, 603	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = (𝐹‘𝑥))
605	596, 599, 604	3eqtrd 2780	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) = (𝐹‘𝑥))
606	605	itgeq2dv 25146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) d𝑥 = ∫((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
607	578, 575	eqeltrd 2838	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) ∈ ℂ)
608	120, 121, 607	itgioo 25180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) d𝑥 = ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) d𝑥)
609	120, 121, 304	itgioo 25180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥 = ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
610	606, 608, 609	3eqtr3d 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥))))‘𝑥) d𝑥 = ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
611	591, 610	eqtr2d 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥 = ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐻‘𝑥) d𝑥)
612	99, 109	oveq12d 7375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) = (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)))
613	612	itgeq1d 44188	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐻‘𝑥) d𝑥 = ∫(((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))(𝐻‘𝑥) d𝑥)
614		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)))
615	612	eqcomd 2742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) = ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
616	615	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) = ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
617	614, 616	eleqtrd 2840	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
618	584	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
619	42	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ∈ ℝ)
620	45	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
621	97	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
622	108	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
623		eliccre 43733	. . . . . . . . . . . . 13 ⊢ ((((𝑄‘𝑖) + 𝑇) ∈ ℝ ∧ ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
624	621, 622, 614, 623	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
625	75	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑇 ∈ ℝ)
626	624, 625	resubcld 11583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
627	226	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
628	627	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
629	621	rexrd 11205	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
630	622	rexrd 11205	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
631		iccgelb 13320	. . . . . . . . . . . . . 14 ⊢ ((((𝑄‘𝑖) + 𝑇) ∈ ℝ^* ∧ ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ≤ 𝑥)
632	629, 630, 614, 631	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ≤ 𝑥)
633	621, 624, 625, 632	lesub1dd 11771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘𝑖) + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
634	628, 633	eqbrtrd 5127	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
635		iccleub 13319	. . . . . . . . . . . . . 14 ⊢ ((((𝑄‘𝑖) + 𝑇) ∈ ℝ^* ∧ ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ≤ ((𝑄‘(𝑖 + 1)) + 𝑇))
636	629, 630, 614, 635	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ≤ ((𝑄‘(𝑖 + 1)) + 𝑇))
637	624, 622, 625, 636	lesub1dd 11771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ≤ (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
638	245	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
639	637, 638	breqtrd 5131	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ≤ (𝑄‘(𝑖 + 1)))
640	619, 620, 626, 634, 639	eliccd 43732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
641	618, 640	ffvelcdmd 7036	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐺‘(𝑥 − 𝑇)) ∈ ℂ)
642	617, 641, 588	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
643		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑦 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) ↦ (𝐺‘(𝑦 − 𝑇))) = (𝑦 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) ↦ (𝐺‘(𝑦 − 𝑇))))
644		oveq1 7364	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝑇) = (𝑥 − 𝑇))
645	644	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐺‘(𝑦 − 𝑇)) = (𝐺‘(𝑥 − 𝑇)))
646	645	adantl 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) ∧ 𝑦 = 𝑥) → (𝐺‘(𝑦 − 𝑇)) = (𝐺‘(𝑥 − 𝑇)))
647	643, 646, 614, 641	fvmptd 6955	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑦 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) ↦ (𝐺‘(𝑦 − 𝑇)))‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
648	642, 647	eqtr4d 2779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐻‘𝑥) = ((𝑦 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) ↦ (𝐺‘(𝑦 − 𝑇)))‘𝑥))
649	648	itgeq2dv 25146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫(((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))(𝐻‘𝑥) d𝑥 = ∫(((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))((𝑦 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) ↦ (𝐺‘(𝑦 − 𝑇)))‘𝑥) d𝑥)
650	74	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ⁺)
651	645	cbvmptv 5218	. . . . . . 7 ⊢ (𝑦 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) ↦ (𝐺‘(𝑦 − 𝑇))) = (𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) ↦ (𝐺‘(𝑥 − 𝑇)))
652	42, 45, 381, 582, 650, 651	itgiccshift 44211	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫(((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))((𝑦 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) ↦ (𝐺‘(𝑦 − 𝑇)))‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐺‘𝑥) d𝑥)
653	613, 649, 652	3eqtrd 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐻‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐺‘𝑥) d𝑥)
654	132	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = ℝ)
655	59, 654	sseqtrrd 3985	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
656	42, 45, 135, 53, 655, 55, 54, 309	itgioocnicc 44208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ∈ 𝐿¹ ∧ ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐺‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥))
657	656	simprd 496	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐺‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
658	611, 653, 657	3eqtrd 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
659	658	sumeq2dv 15588	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (0..^𝑀)∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥 = Σ𝑖 ∈ (0..^𝑀)∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
660	90, 306, 659	3eqtrrd 2781	. 2 ⊢ (𝜑 → Σ𝑖 ∈ (0..^𝑀)∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥)
661	14, 63, 660	3eqtrrd 2781	1 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 Vcvv 3445 ⊆ wss 3910 ifcif 4486 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ↾ cres 5635 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 ℕcn 12153 ℕ₀cn0 12413 ℤ_≥cuz 12763 ℝ⁺crp 12915 (,)cioo 13264 [,]cicc 13267 ...cfz 13424 ..^cfzo 13567 Σcsu 15570 –cn→ccncf 24239 𝐿¹cibl 24981 ∫citg 24982 lim_ℂ climc 25226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-symdif 4202 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-ovol 24828 df-vol 24829 df-mbf 24983 df-itg1 24984 df-itg2 24985 df-ibl 24986 df-itg 24987 df-0p 25034 df-ditg 25211 df-limc 25230 df-dv 25231

This theorem is referenced by: fourierdlem92 44429

W3C validator