fourierdlem81 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem81 46959

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 46881	. . . . . . . . . 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 18	. . . . . . . . 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 235	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simprd 501	. . . . . . 7 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
8	7	simpld 500	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
9	8	simpld 500	. . . . 5 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
10	9	eqcomd 2771	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
11	8	simprd 501	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
12	11	eqcomd 2771	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
13	10, 12	oveq12d 7437	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
14	13	itgeq1d 46729	. 2 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝑄‘0)[,](𝑄‘𝑀))(𝐹‘𝑥) d𝑥)
15		0zd 12618	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
16		nnuz 12917	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
17		0p1e1 12376	. . . . . 6 ⊢ (0 + 1) = 1
18	17	fveq2i 6888	. . . . 5 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
19	16, 18	eqtr4i 2791	. . . 4 ⊢ ℕ = (ℤ_≥‘(0 + 1))
20	2, 19	eleqtrdi 2875	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(0 + 1)))
21	6	simpld 500	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
22		reex 11206	. . . . . 6 ⊢ ℝ ∈ V
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
24		ovex 7452	. . . . . 6 ⊢ (0...𝑀) ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
26	23, 25	elmapd 8843	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
27	21, 26	mpbid 235	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
28	7	simprd 501	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
29	28	r19.21bi 3259	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
30		fourierdlem81.f	. . . . 5 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
31	30	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → 𝐹:ℝ⟶ℂ)
32		fourierdlem81.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
33	9, 32	eqeltrd 2865	. . . . . 6 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
34		fourierdlem81.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
35	11, 34	eqeltrd 2865	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
36	33, 35	iccssred 13477	. . . . 5 ⊢ (𝜑 → ((𝑄‘0)[,](𝑄‘𝑀)) ⊆ ℝ)
37	36	sselda 3938	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → 𝑥 ∈ ℝ)
38	31, 37	ffvelcdmd 7084	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → (𝐹‘𝑥) ∈ ℂ)
39	27	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
40		elfzofz 13721	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
41	40	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
42	39, 41	ffvelcdmd 7084	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
43		fzofzp1 13810	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
44	43	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
45	39, 44	ffvelcdmd 7084	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
46	30	feqmptd 6953	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
47	46	reseq1d 5979	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
48	47	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
49		ioossre 13450	. . . . . . . 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 6044	. . . . . 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 2801	. . . . 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 46750	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ 𝐿¹)
57	52, 56	eqeltrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
58	30	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℂ)
59	42, 45	iccssred 13477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ ℝ)
60	59	sselda 3938	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
61	58, 60	ffvelcdmd 7084	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹‘𝑥) ∈ ℂ)
62	42, 45, 57, 61	ibliooicc 46743	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
63	15, 20, 27, 29, 38, 62	itgspltprt 46751	. 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 6885	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
67	66	oveq1d 7434	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
68	67	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
69	2	nnnn0d 12580	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70		nn0uz 12916	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
71	69, 70	eleqtrdi 2875	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
72		eluzfz1 13575	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
73	71, 72	syl 18	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0...𝑀))
74		fourierdlem81.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
75	74	rpred 13076	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℝ)
76	33, 75	readdcld 11253	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) ∈ ℝ)
77	65, 68, 73, 76	fvmptd 7001	. . . . . 6 ⊢ (𝜑 → (𝑆‘0) = ((𝑄‘0) + 𝑇))
78	9	oveq1d 7434	. . . . . 6 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) = (𝐴 + 𝑇))
79	77, 78	eqtr2d 2801	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑇) = (𝑆‘0))
80		fveq2 6885	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
81	80	oveq1d 7434	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
82	81	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
83		eluzfz2 13576	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
84	71, 83	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
85	35, 75	readdcld 11253	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) ∈ ℝ)
86	65, 82, 84, 85	fvmptd 7001	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑀) = ((𝑄‘𝑀) + 𝑇))
87	11	oveq1d 7434	. . . . . 6 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) = (𝐵 + 𝑇))
88	86, 87	eqtr2d 2801	. . . . 5 ⊢ (𝜑 → (𝐵 + 𝑇) = (𝑆‘𝑀))
89	79, 88	oveq12d 7437	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = ((𝑆‘0)[,](𝑆‘𝑀)))
90	89	itgeq1d 46729	. . 3 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝑆‘0)[,](𝑆‘𝑀))(𝐹‘𝑥) d𝑥)
91	27	ffvelcdmda 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
92	75	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑇 ∈ ℝ)
93	91, 92	readdcld 11253	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
94	93, 64	fmptd 7113	. . . 4 ⊢ (𝜑 → 𝑆:(0...𝑀)⟶ℝ)
95	75	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ)
96	42, 45, 95, 29	ltadd1dd 11840	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
97	40, 93	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
98	64	fvmpt2 7005	. . . . . 6 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑄‘𝑖) + 𝑇) ∈ ℝ) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
99	41, 97, 98	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
100		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
101	100	oveq1d 7434	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑗) + 𝑇))
102	101	cbvmptv 5217	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))
103	64, 102	eqtri 2788	. . . . . . 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 6885	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (𝑄‘𝑗) = (𝑄‘(𝑖 + 1)))
106	105	oveq1d 7434	. . . . . . 7 ⊢ (𝑗 = (𝑖 + 1) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
107	106	adantl 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
108	45, 95	readdcld 11253	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
109	104, 107, 44, 108	fvmptd 7001	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) + 𝑇))
110	96, 99, 109	3brtr4d 5145	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
111	30	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝐹:ℝ⟶ℂ)
112	77, 76	eqeltrd 2865	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘0) ∈ ℝ)
113	112	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝑆‘0) ∈ ℝ)
114	86, 85	eqeltrd 2865	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝑀) ∈ ℝ)
115	114	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝑆‘𝑀) ∈ ℝ)
116	113, 115	iccssred 13477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → ((𝑆‘0)[,](𝑆‘𝑀)) ⊆ ℝ)
117		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀)))
118	116, 117	sseldd 3939	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → 𝑥 ∈ ℝ)
119	111, 118	ffvelcdmd 7084	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘0)[,](𝑆‘𝑀))) → (𝐹‘𝑥) ∈ ℂ)
120	99, 97	eqeltrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) ∈ ℝ)
121	109, 108	eqeltrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
122		ioosscn 13451	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
123	122	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
124		eqeq1 2769	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
125	124	rexbidv 3191	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)))
126		oveq1 7426	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
127	126	eqeq2d 2776	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
128	127	cbvrexvw 3246	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇))
129	125, 128	bitrdi 290	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)))
130	129	cbvrabv 3428	. . . . . . . 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 6719	. . . . . . . . . . . 12 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
132	30, 131	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = ℝ)
133	132	feq2d 6693	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:ℝ⟶ℂ))
134	30, 133	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
135	134	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:dom 𝐹⟶ℂ)
136		elioore 13418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑧 ∈ ℝ)
137	136	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ℝ)
138	75	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
139	137, 138	readdcld 11253	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) ∈ ℝ)
140	139	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) ∈ ℝ)
141	140	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ℝ)
142		simp3 1156	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 = (𝑧 + 𝑇))
143	132	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → dom 𝐹 = ℝ)
144	143	3adant1r 1196	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → dom 𝐹 = ℝ)
145	141, 142, 144	3eltr4d 2880	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 ∈ dom 𝐹)
146	145	3exp 1137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹)))
147	146	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ℂ) → (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹)))
148	147	rexlimdv 3166	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ℂ) → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹))
149	148	ralrimiva 3159	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑤 ∈ ℂ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ dom 𝐹))
150		rabss 4025	. . . . . . . . 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 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ⊆ dom 𝐹)
152		simpll 779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝜑)
153	32	rexrd 11274	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
154	153	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐴 ∈ ℝ^*)
155	34	rexrd 11274	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
156	155	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐵 ∈ ℝ^*)
157	3, 2, 1	fourierdlem15 46894	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
158	157	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
159		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
160		ioossicc 13476	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
161	160	sseli 3934	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
162	161	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
163	154, 156, 158, 159, 162	fourierdlem1 46880	. . . . . . . . 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 596	. . . . . . . 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 46651	. . . . . . 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 3652	. . . . . . . . . . . . 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 503	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} → ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
169		simpr 490	. . . . . . . . . . . . . 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 1947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑖 ∈ (0..^𝑀))
171		nfre1 3292	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)
172	170, 171	nfan 1932	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇))
173		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1)))
174		simp3 1156	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
175	139	3adant3 1150	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ℝ)
176	174, 175	eqeltrd 2865	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ℝ)
177	176	3adant1r 1196	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ℝ)
178	42	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
179	136	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ℝ)
180	75	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
181		simpr 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
182	42	rexrd 11274	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
183	182	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
184	45	rexrd 11274	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
185	184	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
186		elioo2 13429	. . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 ∈ ℝ ∧ (𝑄‘𝑖) < 𝑧 ∧ 𝑧 < (𝑄‘(𝑖 + 1))))
189	188	simp2d 1161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑧)
190	178, 179, 180, 189	ltadd1dd 11840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) + 𝑇) < (𝑧 + 𝑇))
191	190	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → ((𝑄‘𝑖) + 𝑇) < (𝑧 + 𝑇))
192	99	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
193		simp3 1156	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
194	191, 192, 193	3brtr4d 5145	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) < 𝑥)
195	45	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
196	188	simp3d 1162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 < (𝑄‘(𝑖 + 1)))
197	179, 195, 180, 196	ltadd1dd 11840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑧 + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
198	197	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
199	109	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 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 5145	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 < (𝑆‘(𝑖 + 1)))
201	177, 194, 200	3jca 1146	. . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 3274	. . . . . . . . . . . . . 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 16	. . . . . . . . . . . . 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 11274	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) ∈ ℝ^*)
207	206	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑆‘𝑖) ∈ ℝ^*)
208	121	rexrd 11274	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
209	208	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
210		elioo2 13429	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
213	168, 212	sylan2 605	. . . . . . . . . . 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 13418	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
215	214	recnd 11252	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℂ)
216	215	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
217	182	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
218	184	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
219	214	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
220	75	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
221	219, 220	resubcld 11657	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
222	221	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
223	99	oveq1d 7434	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
224	42	recnd 11252	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
225	95	recnd 11252	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℂ)
226	224, 225	pncand 11585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇) − 𝑇) = (𝑄‘𝑖))
227	223, 226	eqtr2d 2801	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
228	227	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
229	120	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ)
230	214	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
231	75	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
232		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
233	206	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ^*)
234	208	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
235	233, 234, 210	syl2anc 596	. . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 ∈ ℝ ∧ (𝑆‘𝑖) < 𝑥 ∧ 𝑥 < (𝑆‘(𝑖 + 1))))
237	236	simp2d 1161	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) < 𝑥)
238	229, 230, 231, 237	ltsub1dd 11841	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑆‘𝑖) − 𝑇) < (𝑥 − 𝑇))
239	228, 238	eqbrtrd 5135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑥 − 𝑇))
240	121	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
241	236	simp3d 1162	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 < (𝑆‘(𝑖 + 1)))
242	230, 240, 231, 241	ltsub1dd 11841	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) < ((𝑆‘(𝑖 + 1)) − 𝑇))
243	109	oveq1d 7434	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
244	45	recnd 11252	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
245	244, 225	pncand 11585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
246	243, 245	eqtrd 2800	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
247	246	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
248	242, 247	breqtrd 5139	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
249	217, 218, 222, 239, 248	eliood 46272	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
250	219	recnd 11252	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
251	220	recnd 11252	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℂ)
252	250, 251	npcand 11588	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
253	252	eqcomd 2771	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
254	253	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
255		oveq1 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑧 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
256	255	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = ((𝑥 − 𝑇) + 𝑇)))
257	256	rspcev 3583	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 813	. . . . . . . . . 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 2763	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
262	261	reseq2d 5980	. . . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
264		ioossre 13450	. . . . . . . . . 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 6954	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
267	262, 266	eqtrd 2800	. . . . . . 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 7434	. . . . . . 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 2879	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
270	49, 132	sseqtrrid 3981	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
271	270	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
272		eqid 2765	. . . . . . . 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 779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝜑)
274	153	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐴 ∈ ℝ^*)
275	155	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐵 ∈ ℝ^*)
276	157	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
277		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
278	160, 181	sselid 3936	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
279	274, 275, 276, 277, 278	fourierdlem1 46880	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ (𝐴[,]𝐵))
280		eleq1 2853	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑧 ∈ (𝐴[,]𝐵)))
281	280	anbi2d 642	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵))))
282		oveq1 7426	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 + 𝑇) = (𝑧 + 𝑇))
283	282	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑧 + 𝑇)))
284		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
285	283, 284	eqeq12d 2781	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧)))
286	281, 285	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))))
287	286, 164	chvarvv 2022	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
288	273, 279, 287	syl2anc 596	. . . . . . . 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 46402	. . . . . . 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 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) = (𝑆‘(𝑖 + 1)))
291	267, 290	oveq12d 7437	. . . . . . 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 2867	. . . . . 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 46402	. . . . . . 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 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) = (𝑆‘𝑖))
295	267, 294	oveq12d 7437	. . . . . . 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 2867	. . . . . 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 46750	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
298	30	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℂ)
299	120	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ)
300	121	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ)
301		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
302		eliccre 46279	. . . . . . 7 ⊢ (((𝑆‘𝑖) ∈ ℝ ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
303	299, 300, 301, 302	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
304	298, 303	ffvelcdmd 7084	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘𝑥) ∈ ℂ)
305	120, 121, 297, 304	ibliooicc 46743	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
306	15, 20, 94, 110, 119, 305	itgspltprt 46751	. . 3 ⊢ (𝜑 → ∫((𝑆‘0)[,](𝑆‘𝑀))(𝐹‘𝑥) d𝑥 = Σ𝑖 ∈ (0..^𝑀)∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
307		iftrue 4495	. . . . . . . . . . . 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 487	. . . . . . . . . . 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 4495	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑄‘𝑖) → if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))) = 𝑅)
311		iftrue 4495	. . . . . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . 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 4498	. . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 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 4495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = 𝐿)
317	316	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑥 = (𝑄‘𝑖) ∧ 𝑥 = (𝑄‘(𝑖 + 1))) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = 𝐿)
318		iffalse 4498	. . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . 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 4495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = 𝐿)
321	320	adantl 487	. . . . . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . . . 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 4498	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
327	326	adantl 487	. . . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . . . . 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 4498	. . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) ∈ ℝ^*)
332	184	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
334	42	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ∈ ℝ)
335	60	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ∈ ℝ)
336	182	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ∈ ℝ^*)
337	184	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
338		simplr 781	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
339		iccgelb 13445	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ≤ 𝑥)
340	336, 337, 338, 339	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) ≤ 𝑥)
341		neqne 2968	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑥 = (𝑄‘𝑖) → 𝑥 ≠ (𝑄‘𝑖))
342	341	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → 𝑥 ≠ (𝑄‘𝑖))
343	334, 335, 340, 342	leneltd 11379	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) → (𝑄‘𝑖) < 𝑥)
344	343	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) < 𝑥)
345	45	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
347	184	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
348		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
349		iccleub 13444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
350	346, 347, 348, 349	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
351	350	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝑖)) ∧ ¬ 𝑥 = (𝑄‘(𝑖 + 1))) → 𝑥 ≤ (𝑄‘(𝑖 + 1)))
352		neqne 2968	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → 𝑥 ≠ (𝑄‘(𝑖 + 1)))
353	352	necomd 3015	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑥 = (𝑄‘(𝑖 + 1)) → (𝑄‘(𝑖 + 1)) ≠ 𝑥)
354	353	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 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 11379	. . . . . . . . . . . . . . . . . . . . . 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 46272	. . . . . . . . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
358	356, 357	syl 18	. . . . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . 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 5206	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))) = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))))
364	309, 363	eqtrid 2812	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑄‘𝑖) ↔ 𝑤 = (𝑄‘𝑖)))
366		eqeq1 2769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑄‘(𝑖 + 1)) ↔ 𝑤 = (𝑄‘(𝑖 + 1))))
367		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))
368	366, 367	ifbieq2d 4516	. . . . . . . . . . . . . . . 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 4516	. . . . . . . . . . . . . . 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 5217	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))) = (𝑤 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑤 = (𝑄‘𝑖), 𝑅, if(𝑤 = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤))))
371	364, 370	eqtrdi 2816	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 490	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → 𝑤 = (𝑥 − 𝑇))
374		oveq1 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑆‘𝑖) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
375	374	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
376	227	eqcomd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) = (𝑄‘𝑖))
377	376	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → ((𝑆‘𝑖) − 𝑇) = (𝑄‘𝑖))
378	373, 375, 377	3eqtrd 2804	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) ∧ 𝑤 = (𝑥 − 𝑇)) → 𝑤 = (𝑄‘𝑖))
379	378	iftrued 4497	. . . . . . . . . . . 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 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) = ((𝑆‘𝑖) − 𝑇))
381	42, 45, 29	ltled 11373	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1)))
382		lbicc2 13507	. . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
384	376, 383	eqeltrd 2865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
385	384	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → ((𝑆‘𝑖) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
386	380, 385	eqeltrd 2865	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
387		limccl 26085	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ⊆ ℂ
388	387, 55	sselid 3936	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ℂ)
389	388	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → 𝑅 ∈ ℂ)
390	372, 379, 386, 389	fvmptd 7001	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘𝑖)) → (𝐺‘(𝑥 − 𝑇)) = 𝑅)
391	308, 390	eqtr4d 2803	. . . . . . . . . 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 728	. . . . . . . . 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 4498	. . . . . . . . . . 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 487	. . . . . . . . . 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 486	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → (𝑤 = (𝑄‘𝑖) ↔ ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖)))
397		eqeq1 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇) → (𝑤 = (𝑄‘(𝑖 + 1)) ↔ ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1))))
398		fveq2 6885	. . . . . . . . . . . . . . . . . . 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 4516	. . . . . . . . . . . . . . . . . 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 4516	. . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)) → (((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘𝑖) ↔ (𝑄‘(𝑖 + 1)) = (𝑄‘𝑖)))
403		iftrue 4495	. . . . . . . . . . . . . . . . . . 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 4516	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 11360	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ≠ (𝑄‘𝑖))
408	407	neneqd 2965	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ (𝑄‘(𝑖 + 1)) = (𝑄‘𝑖))
409	408	iffalsed 4500	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿) = 𝐿)
410	409	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 = ((𝑆‘(𝑖 + 1)) − 𝑇)) → if((𝑄‘(𝑖 + 1)) = (𝑄‘𝑖), 𝑅, 𝐿) = 𝐿)
411	401, 406, 410	3eqtrd 2804	. . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 13508	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
415	246, 414	eqeltrd 2865	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘(𝑖 + 1)) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
416	415	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → ((𝑆‘(𝑖 + 1)) − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
417		limccl 26085	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ℂ
418	417, 54	sselid 3936	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ℂ)
419	418	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝐿 ∈ ℂ)
420	395, 412, 416, 419	fvmptd 7001	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)) = 𝐿)
421		oveq1 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → (𝑥 − 𝑇) = ((𝑆‘(𝑖 + 1)) − 𝑇))
422	421	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → (𝐺‘(𝑥 − 𝑇)) = (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)))
423	422	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐺‘(𝑥 − 𝑇)) = (𝐺‘((𝑆‘(𝑖 + 1)) − 𝑇)))
424		iftrue 4495	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑆‘(𝑖 + 1)) → if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥)) = 𝐿)
425	424	adantl 487	. . . . . . . . . . . . 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 2811	. . . . . . . . . . . 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 765	. . . . . . . . . . 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 4498	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑥 − 𝑇) → (𝑤 = (𝑄‘𝑖) ↔ (𝑥 − 𝑇) = (𝑄‘𝑖)))
433		eqeq1 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑥 − 𝑇) → (𝑤 = (𝑄‘(𝑖 + 1)) ↔ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))))
434		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑥 − 𝑇) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑤) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
435	433, 434	ifbieq2d 4516	. . . . . . . . . . . . . . . . . 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 4516	. . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . 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 11252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
439	225	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℂ)
440	438, 439	npcand 11588	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
441	440	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
442	441	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
443		oveq1 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 − 𝑇) = (𝑄‘𝑖) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
444	443	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
445	294	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘𝑖)) → 𝑥 = (𝑆‘𝑖))
447	446	stoic1a 1805	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → ¬ (𝑥 − 𝑇) = (𝑄‘𝑖))
448	447	iffalsed 4500	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
451		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) → ((𝑥 − 𝑇) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
452	451	adantl 487	. . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1))) → 𝑥 = (𝑆‘(𝑖 + 1)))
455	454	stoic1a 1805	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ¬ (𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)))
456	455	iffalsed 4500	. . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
461	45	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
462	75	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑇 ∈ ℝ)
463	303, 462	resubcld 11657	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℝ)
464	227	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) = ((𝑆‘𝑖) − 𝑇))
465	206	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ∈ ℝ^*)
466	208	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘(𝑖 + 1)) ∈ ℝ^*)
467		iccgelb 13445	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑖) ∈ ℝ^* ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ≤ 𝑥)
468	465, 466, 301, 467	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑆‘𝑖) ≤ 𝑥)
469	299, 303, 462, 468	lesub1dd 11845	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑆‘𝑖) − 𝑇) ≤ (𝑥 − 𝑇))
470	464, 469	eqbrtrd 5135	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
471		iccleub 13444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆‘𝑖) ∈ ℝ^* ∧ (𝑆‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
472	465, 466, 301, 471	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
473	303, 300, 462, 472	lesub1dd 11845	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ≤ ((𝑆‘(𝑖 + 1)) − 𝑇))
474	246	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑆‘(𝑖 + 1)) − 𝑇) = (𝑄‘(𝑖 + 1)))
475	473, 474	breqtrd 5139	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ≤ (𝑄‘(𝑖 + 1)))
476	460, 461, 463, 470, 475	eliccd 46278	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
477	476	ad2antrr 739	. . . . . . . . . . . . . . 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 6749	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
479	478	ad3antrrr 743	. . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘𝑖) ∈ ℝ^*)
481	184	ad3antrrr 743	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
483	95	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑇 ∈ ℝ)
484	482, 483	resubcld 11657	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ℝ)
485	42	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) ∈ ℝ)
486	463	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ∈ ℝ)
487	470	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
488	447	neqned 2967	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑥 − 𝑇) ≠ (𝑄‘𝑖))
489	485, 486, 487, 488	leneltd 11379	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑄‘𝑖) < (𝑥 − 𝑇))
490	489	adantr 486	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ℝ)
492	45	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
493	475	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ≤ (𝑄‘(𝑖 + 1)))
494		eqcom 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇))
495	454	ex 418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑥 − 𝑇) = (𝑄‘(𝑖 + 1)) → 𝑥 = (𝑆‘(𝑖 + 1))))
496	494, 495	biimtrrid 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇) → 𝑥 = (𝑆‘(𝑖 + 1))))
497	496	con3dimp 414	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → ¬ (𝑄‘(𝑖 + 1)) = (𝑥 − 𝑇))
498	497	neqned 2967	. . . . . . . . . . . . . . . . . . 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 11379	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
500	499	adantlr 728	. . . . . . . . . . . . . . . . 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 46272	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
502	479, 501	ffvelcdmd 7084	. . . . . . . . . . . . . . 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 7001	. . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
505	501, 504	syl 18	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘𝑖) ∈ ℝ^*)
508	466	ad2antrr 739	. . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) ∈ ℝ)
510	303	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → 𝑥 ∈ ℝ)
511	468	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) ≤ 𝑥)
512		neqne 2968	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = (𝑆‘𝑖) → 𝑥 ≠ (𝑆‘𝑖))
513	512	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → 𝑥 ≠ (𝑆‘𝑖))
514	509, 510, 511, 513	leneltd 11379	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) → (𝑆‘𝑖) < 𝑥)
515	514	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑆‘𝑖) < 𝑥)
516	300	ad2antrr 739	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝑥 ≤ (𝑆‘(𝑖 + 1)))
518		neqne 2968	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = (𝑆‘(𝑖 + 1)) → 𝑥 ≠ (𝑆‘(𝑖 + 1)))
519	518	necomd 3015	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = (𝑆‘(𝑖 + 1)) → (𝑆‘(𝑖 + 1)) ≠ 𝑥)
520	519	adantl 487	. . . . . . . . . . . . . . . . 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 11379	. . . . . . . . . . . . . . . 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 46272	. . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
524	522, 523	syl 18	. . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘𝑥))
526	525	eqcomd 2771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐹‘𝑥) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
527	526	ad2antrr 739	. . . . . . . . . . . . . 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 11584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ ℂ)
529		elex 3478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 − 𝑇) ∈ ℂ → (𝑥 − 𝑇) ∈ V)
530	528, 529	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ V)
531	530	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝑥 − 𝑇) ∈ V)
532		simp-4l 795	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → 𝜑)
533	153	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
534	155	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
535	157	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
536		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
537	533, 534, 535, 536	fourierdlem8 46887	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
538	537	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
539	538, 476	sseldd 3939	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
540	539	ad2antrr 739	. . . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
543	542	anbi2d 642	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))))
544		oveq1 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
545	544	fveq2d 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
546		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 − 𝑇)))
547	545, 546	eqeq12d 2781	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦) ↔ (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
548	543, 547	imbi12d 347	. . . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
550	549	anbi2d 642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵))))
551		oveq1 7426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
552	551	fveq2d 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
553		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
554	552, 553	eqeq12d 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)))
555	550, 554	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))))
556	555, 164	chvarvv 2022	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
557	548, 556	vtoclg 3524	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 − 𝑇) ∈ V → ((𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
558	531, 541, 557	sylc 66	. . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . 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 2803	. . . . . . . . . . . 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 2803	. . . . . . . . . . 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 825	. . . . . . . . . 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 2800	. . . . . . . . 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 825	. . . . . . . 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 2865	. . . . . . . . . . 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 728	. . . . . . . . . 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 2865	. . . . . . . . . . . . 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 765	. . . . . . . . . . . 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 6749	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))):((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))⟶ℂ)
570	569	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) ∧ ¬ 𝑥 = (𝑆‘𝑖)) ∧ ¬ 𝑥 = (𝑆‘(𝑖 + 1))) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))):((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))⟶ℂ)
571	570, 522	ffvelcdmd 7084	. . . . . . . . . . . . 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 2865	. . . . . . . . . . . 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 825	. . . . . . . . . . 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 2865	. . . . . . . . . 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 825	. . . . . . . . 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 2765	. . . . . . . . . 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 7005	. . . . . . . . 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 596	. . . . . . . 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 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ (0..^𝑀))
580		eqid 2765	. . . . . . . . . . . . . 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 46666	. . . . . . . . . . . . 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 2865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 ∈ (((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))–cn→ℂ))
583		cncff 25103	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))–cn→ℂ) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
584	582, 583	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
585	584	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
586	585, 476	ffvelcdmd 7084	. . . . . . . . 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 7005	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ∧ (𝐺‘(𝑥 − 𝑇)) ∈ ℂ) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
589	301, 586, 588	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
590	564, 578, 589	3eqtr4rd 2811	. . . . . . 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 25992	. . . . . 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 13476	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))
593	592	sseli 3934	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
594	593	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
595	593, 575	sylan2 605	. . . . . . . . . 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 596	. . . . . . . . 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 11360	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ≠ (𝑆‘𝑖))
598	597	neneqd 2965	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ¬ 𝑥 = (𝑆‘𝑖))
599	598	iffalsed 4500	. . . . . . . . 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 11361	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → 𝑥 ≠ (𝑆‘(𝑖 + 1)))
601	600	neneqd 2965	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ¬ 𝑥 = (𝑆‘(𝑖 + 1)))
602	601	iffalsed 4500	. . . . . . . . . 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 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
604	602, 603	eqtrd 2800	. . . . . . . . 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 2804	. . . . . . . 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 25992	. . . . . . 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 2865	. . . . . . . 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 26026	. . . . . . 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 26026	. . . . . . 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 2808	. . . . . 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 2801	. . . . 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 7437	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) = (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)))
613	612	itgeq1d 46729	. . . . . 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 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)))
615	612	eqcomd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇)) = ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
616	615	adantr 486	. . . . . . . . . 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 2867	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))))
618	584	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐺:((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
619	42	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ∈ ℝ)
620	45	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
621	97	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
622	108	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
623		eliccre 46279	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
625	75	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑇 ∈ ℝ)
626	624, 625	resubcld 11657	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
627	226	eqcomd 2771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
628	627	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
629	621	rexrd 11274	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
630	622	rexrd 11274	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
631		iccgelb 13445	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ≤ 𝑥)
633	621, 624, 625, 632	lesub1dd 11845	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘𝑖) + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
634	628, 633	eqbrtrd 5135	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ≤ (𝑥 − 𝑇))
635		iccleub 13444	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ≤ ((𝑄‘(𝑖 + 1)) + 𝑇))
637	624, 622, 625, 636	lesub1dd 11845	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ≤ (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
638	245	adantr 486	. . . . . . . . . . . 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 5139	. . . . . . . . . . 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 46278	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
641	618, 640	ffvelcdmd 7084	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐺‘(𝑥 − 𝑇)) ∈ ℂ)
642	617, 641, 588	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)[,]((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐻‘𝑥) = (𝐺‘(𝑥 − 𝑇)))
643		eqidd 2766	. . . . . . . . 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 7426	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝑇) = (𝑥 − 𝑇))
645	644	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐺‘(𝑦 − 𝑇)) = (𝐺‘(𝑥 − 𝑇)))
646	645	adantl 487	. . . . . . . . 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 7001	. . . . . . . 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 2803	. . . . . . 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 25992	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ⁺)
651	645	cbvmptv 5217	. . . . . . 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 46752	. . . . . 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 2804	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐻‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐺‘𝑥) d𝑥)
654	132	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = ℝ)
655	59, 654	sseqtrrd 3975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
656	42, 45, 135, 53, 655, 55, 54, 309	itgioocnicc 46749	. . . . . 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 501	. . . . 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 2804	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∫((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥 = ∫((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹‘𝑥) d𝑥)
659	658	sumeq2dv 15777	. . 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 2805	. 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 2805	1 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 {crab 3418 Vcvv 3457 ⊆ wss 3906 ifcif 4489 class class class wbr 5111 ↦ cmpt 5194 dom cdm 5663 ↾ cres 5665 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ↑_m cmap 8830 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 ℝ^*cxr 11257 < clt 11258 ≤ cle 11259 − cmin 11456 ℕcn 12248 ℕ₀cn0 12519 ℤ_≥cuz 12878 ℝ⁺crp 13032 (,)cioo 13388 [,]cicc 13391 ...cfz 13551 ..^cfzo 13699 Σcsu 15761 –cn→ccncf 25086 𝐿¹cibl 25827 ∫citg 25828 lim_ℂ climc 26072

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cc 10434 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-symdif 4206 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-ofr 7685 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-omul 8464 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-acn 9944 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-mod 13921 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-limsup 15546 df-clim 15563 df-rlim 15564 df-sum 15762 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-mulg 19178 df-cntz 19431 df-cmn 19896 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cn 23434 df-cnp 23435 df-haus 23522 df-cmp 23594 df-tx 23770 df-hmeo 23963 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-ovol 25674 df-vol 25675 df-mbf 25829 df-itg1 25830 df-itg2 25831 df-ibl 25832 df-itg 25833 df-0p 25880 df-ditg 26057 df-limc 26076 df-dv 26077

This theorem is used by: fourierdlem92 46970

W3C validator