fourierdlem81 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem81 41313

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 41235	. . . . . . . . . 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 224	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simprd 491	. . . . . . 7 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
8	7	simpld 490	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
9	8	simpld 490	. . . . 5 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
10	9	eqcomd 2783	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
11	8	simprd 491	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
12	11	eqcomd 2783	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
13	10, 12	oveq12d 6940	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
14	13	itgeq1d 41082	. 2 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝑄‘0)[,](𝑄‘𝑀))(𝐹‘𝑥) d𝑥)
15		0zd 11740	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
16		nnuz 12029	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
17		0p1e1 11504	. . . . . 6 ⊢ (0 + 1) = 1
18	17	fveq2i 6449	. . . . 5 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
19	16, 18	eqtr4i 2804	. . . 4 ⊢ ℕ = (ℤ_≥‘(0 + 1))
20	2, 19	syl6eleq 2868	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(0 + 1)))
21	6	simpld 490	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)))
22		reex 10363	. . . . . 6 ⊢ ℝ ∈ V
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
24		ovex 6954	. . . . . 6 ⊢ (0...𝑀) ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
26	23, 25	elmapd 8154	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
27	21, 26	mpbid 224	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
28	7	simprd 491	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
29	28	r19.21bi 3113	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
30		fourierdlem81.f	. . . . 5 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
31	30	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → 𝐹:ℝ⟶ℂ)
32		fourierdlem81.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
33	9, 32	eqeltrd 2858	. . . . . 6 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
34		fourierdlem81.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
35	11, 34	eqeltrd 2858	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
36	33, 35	iccssred 40621	. . . . 5 ⊢ (𝜑 → ((𝑄‘0)[,](𝑄‘𝑀)) ⊆ ℝ)
37	36	sselda 3820	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → 𝑥 ∈ ℝ)
38	31, 37	ffvelrnd 6624	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → (𝐹‘𝑥) ∈ ℂ)
39	27	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
40		elfzofz 12804	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
41	40	adantl 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
42	39, 41	ffvelrnd 6624	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
43		fzofzp1 12884	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
44	43	adantl 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
45	39, 44	ffvelrnd 6624	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
46	30	feqmptd 6509	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
47	46	reseq1d 5641	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
48	47	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
49		ioossre 12547	. . . . . . . 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 5702	. . . . . 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 2814	. . . . 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 41103	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ 𝐿¹)
57	52, 56	eqeltrd 2858	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
58	30	ad2antrr 716	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℂ)
59	42, 45	iccssred 40621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ ℝ)
60	59	sselda 3820	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
61	58, 60	ffvelrnd 6624	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹‘𝑥) ∈ ℂ)
62	42, 45, 57, 61	ibliooicc 41096	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
63	15, 20, 27, 29, 38, 62	itgspltprt 41104	. 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 6446	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
67	66	oveq1d 6937	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
68	67	adantl 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
69	2	nnnn0d 11702	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70		nn0uz 12028	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
71	69, 70	syl6eleq 2868	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
72		eluzfz1 12665	. . . . . . . 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 12181	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℝ)
76	33, 75	readdcld 10406	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) ∈ ℝ)
77	65, 68, 73, 76	fvmptd 6548	. . . . . 6 ⊢ (𝜑 → (𝑆‘0) = ((𝑄‘0) + 𝑇))
78	9	oveq1d 6937	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) = (𝐴 + 𝑇))
79	77, 78	eqtr2d 2814	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑇) = (𝑆‘0))
80		fveq2 6446	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
81	80	oveq1d 6937	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
82	81	adantl 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
83		eluzfz2 12666	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
84	71, 83	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
85	35, 75	readdcld 10406	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) ∈ ℝ)
86	65, 82, 84, 85	fvmptd 6548	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑀) = ((𝑄‘𝑀) + 𝑇))
87	11	oveq1d 6937	. . . . . 6 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) = (𝐵 + 𝑇))
88	86, 87	eqtr2d 2814	. . . . 5 ⊢ (𝜑 → (𝐵 + 𝑇) = (𝑆‘𝑀))
89	79, 88	oveq12d 6940	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = ((𝑆‘0)[,](𝑆‘𝑀)))
90	89	itgeq1d 41082	. . 3 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝑆‘0)[,](𝑆‘𝑀))(𝐹‘𝑥) d𝑥)
91	27	ffvelrnda 6623	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
92	75	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑇 ∈ ℝ)
93	91, 92	readdcld 10406	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
94	93, 64	fmptd 6648	. . . 4 ⊢ (𝜑 → 𝑆:(0...𝑀)⟶ℝ)
95	75	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ)
96	42, 45, 95, 29	ltadd1dd 10986	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
97	40, 93	sylan2 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
98	64	fvmpt2 6552	. . . . . 6 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑄‘𝑖) + 𝑇) ∈ ℝ) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
99	41, 97, 98	syl2anc 579	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
100		fveq2 6446	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
101	100	oveq1d 6937	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑗) + 𝑇))
102	101	cbvmptv 4985	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))
103	64, 102	eqtri 2801	. . . . . . 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 6446	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (𝑄‘𝑗) = (𝑄‘(𝑖 + 1)))
106	105	oveq1d 6937	. . . . . . 7 ⊢ (𝑗 = (𝑖 + 1) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
107	106	adantl 475	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
108	45, 95	readdcld 10406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
109	104, 107, 44, 108	fvmptd 6548	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) + 𝑇))
110	96, 99, 109	3brtr4d 4918	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
111	30	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝐹:ℝ⟶ℂ)
112	77, 76	eqeltrd 2858	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘0) ∈ ℝ)
113	112	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝑆‘0) ∈ ℝ)
114	86, 85	eqeltrd 2858	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝑀) ∈ ℝ)
115	114	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝑆‘𝑀) ∈ ℝ)
116	113, 115	iccssred 40621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → ((𝑆‘0)[,](𝑆‘𝑀)) ⊆ ℝ)
117		simpr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀)))
118	116, 117	sseldd 3821	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝑥 ∈ ℝ)
119	111, 118	ffvelrnd 6624	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝐹‘𝑥) ∈ ℂ)
120	99, 97	eqeltrd 2858	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) ∈ ℝ)
121	109, 108	eqeltrd 2858	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
122		ioosscn 40610	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
123	122	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
124		eqeq1 2781	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
125	124	rexbidv 3236	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)))
126		oveq1 6929	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
127	126	eqeq2d 2787	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
128	127	cbvrexv 3367	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇))
129	125, 128	syl6bb 279	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)))
130	129	cbvrabv 3395	. . . . . . . 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 6299	. . . . . . . . . . . 12 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
132	30, 131	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = ℝ)
133	132	feq2d 6277	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:ℝ⟶ℂ))
134	30, 133	mpbird 249	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
135	134	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:dom 𝐹⟶ℂ)
136		elioore 12517	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑧 ∈ ℝ)
137	136	adantl 475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ℝ)
138	75	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
139	137, 138	readdcld 10406	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) ∈ ℝ)
140	139	adantlr 705	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) ∈ ℝ)
141	140	3adant3 1123	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ℝ)
142		simp3 1129	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 = (𝑧 + 𝑇))
143	132	3ad2ant1 1124	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → dom 𝐹 = ℝ)
144	143	3adant1r 1180	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → dom 𝐹 = ℝ)
145	141, 142, 144	3eltr4d 2873	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 ∈ dom 𝐹)
146	145	3exp 1109	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹)))
147	146	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ℂ) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹)))
148	147	rexlimdv 3211	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ℂ) → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹))
149	148	ralrimiva 3147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑤 ∈ ℂ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹))
150		rabss 3899	. . . . . . . . 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 226	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ⊆ dom 𝐹)
152		simpll 757	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝜑)
153	32	rexrd 10426	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
154	153	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐴 ∈ ℝ^*)
155	34	rexrd 10426	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
156	155	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐵 ∈ ℝ^*)
157	3, 2, 1	fourierdlem15 41248	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
158	157	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
159		simplr 759	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
160		ioossicc 12571	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
161	160	sseli 3816	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
162	161	adantl 475	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
163	154, 156, 158, 159, 162	fourierdlem1 41234	. . . . . . . . 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 579	. . . . . . . 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 41002	. . . . . . 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 3571	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} → ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
169		simpr 479	. . . . . . . . . . . . . 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 1957	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑖 ∈ (0..^𝑀))
171		nfre1 3185	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)
172	170, 171	nfan 1946	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
173		nfv 1957	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1)))
174		simp3 1129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
175	139	3adant3 1123	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ℝ)
176	174, 175	eqeltrd 2858	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ℝ)
177	176	3adant1r 1180	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ℝ)
178	42	adantr 474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
179	136	adantl 475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ℝ)
180	75	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
181		simpr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
182	42	rexrd 10426	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
183	182	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
184	45	rexrd 10426	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
185	184	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
186		elioo2 12528	. . . . . . . . . . . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 ∈ ℝ ∧ (𝑄‘𝑖) < 𝑧 ∧ 𝑧 < (𝑄‘(𝑖 + 1))))
189	188	simp2d 1134	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑧)
190	178, 179, 180, 189	ltadd1dd 10986	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) + 𝑇) < (𝑧 + 𝑇))
191	190	3adant3 1123	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → ((𝑄‘𝑖) + 𝑇) < (𝑧 + 𝑇))
192	99	3ad2ant1 1124	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
193		simp3 1129	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
194	191, 192, 193	3brtr4d 4918	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) < 𝑥)
195	45	adantr 474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
196	188	simp3d 1135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 < (𝑄‘(𝑖 + 1)))
197	179, 195, 180, 196	ltadd1dd 10986	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
198	197	3adant3 1123	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
199	109	3ad2ant1 1124	. . . . . . . . . . . . . . . . . . 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 4918	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 < (𝑆‘(𝑖 + 1)))
201	177, 194, 200	3jca 1119	. . . . . . . . . . . . . . . . 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 1109	. . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . 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 3207	. . . . . . . . . . . . . 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 10426	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) ∈ ℝ^*)
207	206	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) ∈ ℝ^*)
208	121	rexrd 10426	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
209	208	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
210		elioo2 12528	. . . . . . . . . . . . . 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 579	. . . . . . . . . . . . 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 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
213	168, 212	sylan2 586	. . . . . . . . . . 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 12517	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
215	214	recnd 10405	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℂ)
216	215	adantl 475	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
217	182	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
218	184	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
219	214	adantl 475	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
220	75	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
221	219, 220	resubcld 10803	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
222	221	adantlr 705	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
223	99	oveq1d 6937	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
224	42	recnd 10405	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
225	95	recnd 10405	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℂ)
226	224, 225	pncand 10735	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇) − 𝑇) = (𝑄‘𝑖))
227	223, 226	eqtr2d 2814	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
228	227	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
229	120	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ)
230	214	adantl 475	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
231	75	ad2antrr 716	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
232		simpr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
233	206	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ^*)
234	208	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
235	233, 234, 210	syl2anc 579	. . . . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1))))
237	236	simp2d 1134	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) < 𝑥)
238	229, 230, 231, 237	ltsub1dd 10987	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑆‘𝑖) − 𝑇) < (𝑥 − 𝑇))
239	228, 238	eqbrtrd 4908	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑥 − 𝑇))
240	121	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
241	236	simp3d 1135	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 < (𝑆‘(𝑖 + 1)))
242	230, 240, 231, 241	ltsub1dd 10987	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) < ((𝑆‘(𝑖 + 1)) − 𝑇))
243	109	oveq1d 6937	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
244	45	recnd 10405	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
245	244, 225	pncand 10735	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
246	243, 245	eqtrd 2813	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
247	246	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
248	242, 247	breqtrd 4912	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
249	217, 218, 222, 239, 248	eliood 40614	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
250	219	recnd 10405	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
251	220	recnd 10405	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℂ)
252	250, 251	npcand 10738	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
253	252	eqcomd 2783	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
254	253	adantlr 705	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
255		oveq1 6929	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑧 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
256	255	eqeq2d 2787	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = ((𝑥 − 𝑇) + 𝑇)))
257	256	rspcev 3510	. . . . . . . . . . . . 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 579	. . . . . . . . . . . 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 578	. . . . . . . . . . 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 791	. . . . . . . . . 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 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
262	261	reseq2d 5642	. . . . . . . 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 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
264		ioossre 12547	. . . . . . . . . 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 6510	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
267	262, 266	eqtrd 2813	. . . . . . 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 6937	. . . . . . 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 2872	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
270	49, 132	syl5sseqr 3872	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
271	270	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
272		eqid 2777	. . . . . . . 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 757	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝜑)
274	153	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐴 ∈ ℝ^*)
275	155	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐵 ∈ ℝ^*)
276	157	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
277		simplr 759	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
278	160, 181	sseldi 3818	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
279	274, 275, 276, 277, 278	fourierdlem1 41234	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ (𝐴[,]𝐵))
280		eleq1 2846	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑧 ∈ (𝐴[,]𝐵)))
281	280	anbi2d 622	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵))))
282		oveq1 6929	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 + 𝑇) = (𝑧 + 𝑇))
283	282	fveq2d 6450	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑧 + 𝑇)))
284		fveq2 6446	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
285	283, 284	eqeq12d 2792	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧)))
286	281, 285	imbi12d 336	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))))
287	286, 164	chvarv 2360	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
288	273, 279, 287	syl2anc 579	. . . . . . . 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 40750	. . . . . . 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 2783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) = (𝑆‘(𝑖 + 1)))
291	267, 290	oveq12d 6940	. . . . . . 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 2860	. . . . . 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 40750	. . . . . . 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 2783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) = (𝑆‘𝑖))
295	267, 294	oveq12d 6940	. . . . . . 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 2860	. . . . . 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 41103	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
298	30	ad2antrr 716	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℂ)
299	120	adantr 474	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ)
300	121	adantr 474	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
301		simpr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
302		eliccre 40622	. . . . . . 7 ⊢ (((𝑆‘𝑖) ∈ ℝ ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
303	299, 300, 301, 302	syl3anc 1439	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
304	298, 303	ffvelrnd 6624	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘𝑥) ∈ ℂ)
305	120, 121, 297, 304	ibliooicc 41096	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
306	15, 20, 94, 110, 119, 305	itgspltprt 41104	. . 3 ⊢ (𝜑 → ∫((𝑆‘0)[,](𝑆‘𝑀))(𝐹‘𝑥) d𝑥 = Σ𝑖 ∈ (0..^𝑀)∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
307		iftrue 4312	. . . . . . . . . . . 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 475	. . . . . . . . . . 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 4312	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑄‘𝑖) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = 𝑅)
311		iftrue 4312	. . . . . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . . 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 4315	. . . . . . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . . . . 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 4312	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = 𝐿)
317	316	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑥 = (𝑄‘𝑖) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = 𝐿)
318		iffalse 4315	. . . . . . . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . . . . . 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 4312	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = 𝐿)
321	320	adantl 475	. . . . . . . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . 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 704	. . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . 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 4315	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
327	326	adantl 475	. . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . . 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 4315	. . . . . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . . . . . 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 720	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) ∈ ℝ^*)
332	184	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
334	42	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ∈ ℝ)
335	60	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ∈ ℝ)
336	182	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ∈ ℝ^*)
337	184	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
338		simplr 759	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
339		iccgelb 12542	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ≤ 𝑥)
340	336, 337, 338, 339	syl3anc 1439	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ≤ 𝑥)
341		neqne 2976	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑥 = (𝑄‘𝑖) → 𝑥 ≠ (𝑄‘𝑖))
342	341	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ≠ (𝑄‘𝑖))
343	334, 335, 340, 342	leneltd 10530	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) < 𝑥)
344	343	adantr 474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) < 𝑥)
345	45	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
347	184	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
348		simpr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
349		iccleub 12541	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
350	346, 347, 348, 349	syl3anc 1439	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
351	350	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
352		neqne 2976	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → 𝑥 ≠ (𝑄‘(𝑖 + 1)))
353	352	necomd 3023	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → (𝑄‘(𝑖 + 1)) ≠ 𝑥)
354	353	adantl 475	. . . . . . . . . . . . . . . . . . . . . . 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 10530	. . . . . . . . . . . . . . . . . . . . . 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 40614	. . . . . . . . . . . . . . . . . . . . 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 6465	. . . . . . . . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . 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 803	. . . . . . . . . . . . . . . . 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 803	. . . . . . . . . . . . . . . 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 4979	. . . . . . . . . . . . . . 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	syl5eq 2825	. . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑄‘𝑖) ↔ 𝑤 = (𝑄‘𝑖)))
366		eqeq1 2781	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑄‘(𝑖 + 1)) ↔ 𝑤 = (𝑄‘(𝑖 + 1))))
367		fveq2 6446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))
368	366, 367	ifbieq2d 4331	. . . . . . . . . . . . . . . 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 4331	. . . . . . . . . . . . . . 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 4985	. . . . . . . . . . . . . 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	syl6eq 2829	. . . . . . . . . . . . 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 474	. . . . . . . . . . . 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 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → 𝑤 = (𝑥 − 𝑇))
374		oveq1 6929	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑆‘𝑖) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
375	374	ad2antlr 717	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
376	227	eqcomd 2783	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) = (𝑄‘𝑖))
377	376	ad2antrr 716	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → ((𝑆‘𝑖) − 𝑇) = (𝑄‘𝑖))
378	373, 375, 377	3eqtrd 2817	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → 𝑤 = (𝑄‘𝑖))
379	378	iftrued 4314	. . . . . . . . . . . 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 475	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
381	42, 45, 29	ltled 10524	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1)))
382		lbicc2 12602	. . . . . . . . . . . . . . . 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 1439	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
384	376, 383	eqeltrd 2858	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
385	384	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → ((𝑆‘𝑖) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
386	380, 385	eqeltrd 2858	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
387		limccl 24076	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ⊆ ℂ
388	387, 55	sseldi 3818	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ℂ)
389	388	adantr 474	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → 𝑅 ∈ ℂ)
390	372, 379, 386, 389	fvmptd 6548	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝐺‘(𝑥 − 𝑇)) = 𝑅)
391	308, 390	eqtr4d 2816	. . . . . . . . . 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 705	. . . . . . . . 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 4315	. . . . . . . . . . 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 475	. . . . . . . . . 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 474	. . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → (𝑤 = (𝑄‘𝑖) ↔ ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖)))
397		eqeq1 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → (𝑤 = (𝑄‘(𝑖 + 1)) ↔ ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1))))
398		fveq2 6446	. . . . . . . . . . . . . . . . . . 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 4331	. . . . . . . . . . . . . . . . . 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 4331	. . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)) → (((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖) ↔ (𝑄‘(𝑖 + 1)) = (𝑄‘𝑖)))
403		iftrue 4312	. . . . . . . . . . . . . . . . . . 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 4331	. . . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . 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 10511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ≠ (𝑄‘𝑖))
408	407	neneqd 2973	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ (𝑄‘(𝑖 + 1)) = (𝑄‘𝑖))
409	408	iffalsed 4317	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿) = 𝐿)
410	409	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇)) → if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿) = 𝐿)
411	401, 406, 410	3eqtrd 2817	. . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . 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 12603	. . . . . . . . . . . . . . . . 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 1439	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
415	246, 414	eqeltrd 2858	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
416	415	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → ((𝑆‘(𝑖 + 1)) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
417		limccl 24076	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ℂ
418	417, 54	sseldi 3818	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ℂ)
419	418	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝐿 ∈ ℂ)
420	395, 412, 416, 419	fvmptd 6548	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)) = 𝐿)
421		oveq1 6929	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → (𝑥 − 𝑇) = ((𝑆‘(𝑖 + 1)) − 𝑇))
422	421	fveq2d 6450	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → (𝐺‘(𝑥 − 𝑇)) = (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)))
423	422	adantl 475	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘(𝑥 − 𝑇)) = (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)))
424		iftrue 4312	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = 𝐿)
425	424	adantl 475	. . . . . . . . . . . . 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 2824	. . . . . . . . . . . 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 742	. . . . . . . . . . 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 4315	. . . . . . . . . . . . 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 475	. . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑥 − 𝑇) → (𝑤 = (𝑄‘𝑖) ↔ (𝑥 − 𝑇) = (𝑄‘𝑖)))
433		eqeq1 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑥 − 𝑇) → (𝑤 = (𝑄‘(𝑖 + 1)) ↔ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))))
434		fveq2 6446	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑥 − 𝑇) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
435	433, 434	ifbieq2d 4331	. . . . . . . . . . . . . . . . . 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 4331	. . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . 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 10405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
439	225	adantr 474	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℂ)
440	438, 439	npcand 10738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
441	440	eqcomd 2783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
442	441	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
443		oveq1 6929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 − 𝑇) = (𝑄‘𝑖) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
444	443	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
445	294	ad2antrr 716	. . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → 𝑥 = (𝑆‘𝑖))
447	446	stoic1a 1816	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → ¬ (𝑥 − 𝑇) = (𝑄‘𝑖))
448	447	iffalsed 4317	. . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
451		oveq1 6929	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
452	451	adantl 475	. . . . . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))) → 𝑥 = (𝑆‘(𝑖 + 1)))
455	454	stoic1a 1816	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ¬ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)))
456	455	iffalsed 4317	. . . . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
461	45	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
462	75	ad2antrr 716	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
463	303, 462	resubcld 10803	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
464	227	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
465	206	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ^*)
466	208	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
467		iccgelb 12542	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑖) ∈ ℝ^* ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ≤ 𝑥)
468	465, 466, 301, 467	syl3anc 1439	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ≤ 𝑥)
469	299, 303, 462, 468	lesub1dd 10991	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑆‘𝑖) − 𝑇) ≤ (𝑥 − 𝑇))
470	464, 469	eqbrtrd 4908	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
471		iccleub 12541	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑖) ∈ ℝ^* ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
472	465, 466, 301, 471	syl3anc 1439	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
473	303, 300, 462, 472	lesub1dd 10991	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ≤ ((𝑆‘(𝑖 + 1)) − 𝑇))
474	246	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
475	473, 474	breqtrd 4912	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ≤ (𝑄‘(𝑖 + 1)))
476	460, 461, 463, 470, 475	eliccd 40620	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
477	476	ad2antrr 716	. . . . . . . . . . . . . . 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 6321	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
479	478	ad3antrrr 720	. . . . . . . . . . . . . . . 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 720	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘𝑖) ∈ ℝ^*)
481	184	ad3antrrr 720	. . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
483	95	ad3antrrr 720	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑇 ∈ ℝ)
484	482, 483	resubcld 10803	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ℝ)
485	42	ad2antrr 716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) ∈ ℝ)
486	463	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ∈ ℝ)
487	470	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
488	447	neqned 2975	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ≠ (𝑄‘𝑖))
489	485, 486, 487, 488	leneltd 10530	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) < (𝑥 − 𝑇))
490	489	adantr 474	. . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ℝ)
492	45	ad2antrr 716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
493	475	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ≤ (𝑄‘(𝑖 + 1)))
494		eqcom 2784	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇))
495	454	ex 403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) → 𝑥 = (𝑆‘(𝑖 + 1))))
496	494, 495	syl5bir 235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇) → 𝑥 = (𝑆‘(𝑖 + 1))))
497	496	con3dimp 399	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ¬ (𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇))
498	497	neqned 2975	. . . . . . . . . . . . . . . . . . 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 10530	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
500	499	adantlr 705	. . . . . . . . . . . . . . . . 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 40614	. . . . . . . . . . . . . . . 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	ffvelrnd 6624	. . . . . . . . . . . . . . 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 6548	. . . . . . . . . . . . . 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 6465	. . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘𝑖) ∈ ℝ^*)
508	466	ad2antrr 716	. . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) ∈ ℝ)
510	303	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → 𝑥 ∈ ℝ)
511	468	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) ≤ 𝑥)
512		neqne 2976	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = (𝑆‘𝑖) → 𝑥 ≠ (𝑆‘𝑖))
513	512	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → 𝑥 ≠ (𝑆‘𝑖))
514	509, 510, 511, 513	leneltd 10530	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) < 𝑥)
515	514	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘𝑖) < 𝑥)
516	300	ad2antrr 716	. . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
518		neqne 2976	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = (𝑆‘(𝑖 + 1)) → 𝑥 ≠ (𝑆‘(𝑖 + 1)))
519	518	necomd 3023	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = (𝑆‘(𝑖 + 1)) → (𝑆‘(𝑖 + 1)) ≠ 𝑥)
520	519	adantl 475	. . . . . . . . . . . . . . . . 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 10530	. . . . . . . . . . . . . . . 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 40614	. . . . . . . . . . . . . . 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 6465	. . . . . . . . . . . . . . 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 6450	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘𝑥))
526	525	eqcomd 2783	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘𝑥) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
527	526	ad2antrr 716	. . . . . . . . . . . . . 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 10734	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℂ)
529		elex 3413	. . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ V)
532		simp-4l 773	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝜑)
533	153	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
534	155	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
535	157	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
536		simpr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
537	533, 534, 535, 536	fourierdlem8 41241	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
538	537	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
539	538, 476	sseldd 3821	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
540	539	ad2antrr 716	. . . . . . . . . . . . . . . 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 507	. . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
543	542	anbi2d 622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))))
544		oveq1 6929	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
545	544	fveq2d 6450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
546		fveq2 6446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 − 𝑇)))
547	545, 546	eqeq12d 2792	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦) ↔ (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
548	543, 547	imbi12d 336	. . . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
550	549	anbi2d 622	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵))))
551		oveq1 6929	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
552	551	fveq2d 6450	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
553		fveq2 6446	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
554	552, 553	eqeq12d 2792	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)))
555	550, 554	imbi12d 336	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))))
556	555, 164	chvarv 2360	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
557	548, 556	vtoclg 3466	. . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . 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 2816	. . . . . . . . . . . 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 2816	. . . . . . . . . . 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 803	. . . . . . . . . 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 2813	. . . . . . . . 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 803	. . . . . . . 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 2858	. . . . . . . . . . 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 705	. . . . . . . . . 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 2858	. . . . . . . . . . . . 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 742	. . . . . . . . . . . 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 6321	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))):((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))⟶ℂ)
570	569	ad3antrrr 720	. . . . . . . . . . . . . 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	ffvelrnd 6624	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . 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 803	. . . . . . . . . . 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 2858	. . . . . . . . . 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 803	. . . . . . . . 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 2777	. . . . . . . . . 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 6552	. . . . . . . . 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 579	. . . . . . . 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 1957	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ (0..^𝑀))
580		eqid 2777	. . . . . . . . . . . . . 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 41017	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 ∈ (((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))–cn→ℂ))
583		cncff 23104	. . . . . . . . . . . 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 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
586	585, 476	ffvelrnd 6624	. . . . . . . . 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 6552	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ∧ (𝐺‘(𝑥 − 𝑇)) ∈ ℂ) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
589	301, 586, 588	syl2anc 579	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
590	564, 578, 589	3eqtr4rd 2824	. . . . . . 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 23985	. . . . . 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 12571	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))
593	592	sseli 3816	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
594	593	adantl 475	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
595	593, 575	sylan2 586	. . . . . . . . . 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 579	. . . . . . . . 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 10511	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ≠ (𝑆‘𝑖))
598	597	neneqd 2973	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ¬ 𝑥 = (𝑆‘𝑖))
599	598	iffalsed 4317	. . . . . . . . 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 10512	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ≠ (𝑆‘(𝑖 + 1)))
601	600	neneqd 2973	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ¬ 𝑥 = (𝑆‘(𝑖 + 1)))
602	601	iffalsed 4317	. . . . . . . . . 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 475	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
604	602, 603	eqtrd 2813	. . . . . . . . 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 2817	. . . . . . . 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 23985	. . . . . . 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 2858	. . . . . . . 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 24019	. . . . . . 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 24019	. . . . . . 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 2821	. . . . . 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 2814	. . . . 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 6940	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) = (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)))
613	612	itgeq1d 41082	. . . . . 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 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)))
615	612	eqcomd 2783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) = ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
616	615	adantr 474	. . . . . . . . . 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 2860	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
618	584	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
619	42	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ∈ ℝ)
620	45	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
621	97	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
622	108	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
623		eliccre 40622	. . . . . . . . . . . . 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 1439	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
625	75	ad2antrr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑇 ∈ ℝ)
626	624, 625	resubcld 10803	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
627	226	eqcomd 2783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
628	627	adantr 474	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
629	621	rexrd 10426	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
630	622	rexrd 10426	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
631		iccgelb 12542	. . . . . . . . . . . . . 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 1439	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ≤ 𝑥)
633	621, 624, 625, 632	lesub1dd 10991	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘𝑖) + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
634	628, 633	eqbrtrd 4908	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
635		iccleub 12541	. . . . . . . . . . . . . 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 1439	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ≤ ((𝑄‘(𝑖 + 1)) + 𝑇))
637	624, 622, 625, 636	lesub1dd 10991	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ≤ (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
638	245	adantr 474	. . . . . . . . . . . 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 4912	. . . . . . . . . . 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 40620	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
641	618, 640	ffvelrnd 6624	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐺‘(𝑥 − 𝑇)) ∈ ℂ)
642	617, 641, 588	syl2anc 579	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
643		eqidd 2778	. . . . . . . . 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 6929	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝑇) = (𝑥 − 𝑇))
645	644	fveq2d 6450	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐺‘(𝑦 − 𝑇)) = (𝐺‘(𝑥 − 𝑇)))
646	645	adantl 475	. . . . . . . . 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 6548	. . . . . . . 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 2816	. . . . . . 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 23985	. . . . . 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 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ⁺)
651	645	cbvmptv 4985	. . . . . . 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 41105	. . . . . 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 2817	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐻‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐺‘𝑥) d𝑥)
654	132	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = ℝ)
655	59, 654	sseqtr4d 3860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
656	42, 45, 135, 53, 655, 55, 54, 309	itgioocnicc 41102	. . . . . 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 491	. . . . 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 2817	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
659	658	sumeq2dv 14841	. . 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 2818	. 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 2818	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 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 ∃wrex 3090 {crab 3093 Vcvv 3397 ⊆ wss 3791 ifcif 4306 class class class wbr 4886 ↦ cmpt 4965 dom cdm 5355 ↾ cres 5357 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↑_𝑚 cmap 8140 ℂcc 10270 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 ℝ^*cxr 10410 < clt 10411 ≤ cle 10412 − cmin 10606 ℕcn 11374 ℕ₀cn0 11642 ℤ_≥cuz 11992 ℝ⁺crp 12137 (,)cioo 12487 [,]cicc 12490 ...cfz 12643 ..^cfzo 12784 Σcsu 14824 –cn→ccncf 23087 𝐿¹cibl 23821 ∫citg 23822 lim_ℂ climc 24063

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-symdif 4066 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-omul 7848 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-cmp 21599 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-ovol 23668 df-vol 23669 df-mbf 23823 df-itg1 23824 df-itg2 23825 df-ibl 23826 df-itg 23827 df-0p 23874 df-ditg 24048 df-limc 24067 df-dv 24068

This theorem is referenced by: fourierdlem92 41324

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