fourierdlem49 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem49 46729

Description: The given periodic function 𝐹 has a left limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem49.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem49.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem49.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem49.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem49.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem49.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem49.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem49.d	⊢ (𝜑 → 𝐷 ⊆ ℝ)
fourierdlem49.f	⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
fourierdlem49.dper	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)
fourierdlem49.per	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥))
fourierdlem49.cn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem49.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
fourierdlem49.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem49.z	⊢ 𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
fourierdlem49.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥)))

**Assertion**
Ref	Expression
fourierdlem49	⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅)

Distinct variable groups: 𝐴,𝑖,𝑚,𝑝 𝑥,𝐴,𝑖 𝐵,𝑖,𝑘 𝐵,𝑚,𝑝 𝑥,𝐵,𝑘 𝐷,𝑘,𝑥 𝑖,𝐸,𝑘,𝑥 𝑖,𝐹,𝑘,𝑥 𝑖,𝑀,𝑘 𝑚,𝑀,𝑝 𝑥,𝑀 𝑄,𝑖,𝑘 𝑄,𝑝 𝑥,𝑄 𝑇,𝑘,𝑥 𝑖,𝑋,𝑘,𝑥 𝑘,𝑍,𝑥 𝜑,𝑖,𝑘,𝑥 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝐴(𝑘) 𝐷(𝑖,𝑚,𝑝) 𝑃(𝑥,𝑖,𝑘,𝑚,𝑝) 𝑄(𝑚) 𝑇(𝑖,𝑚,𝑝) 𝐸(𝑚,𝑝) 𝐹(𝑚,𝑝) 𝐿(𝑥,𝑖,𝑘,𝑚,𝑝) 𝑋(𝑚,𝑝) 𝑍(𝑖,𝑚,𝑝)

Proof of Theorem fourierdlem49 Dummy variables 𝑗 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem49.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		fourierdlem49.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		fourierdlem49.altb	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵)
4		fourierdlem49.t	. . . . . 6 ⊢ 𝑇 = (𝐵 − 𝐴)
5		fourierdlem49.e	. . . . . . 7 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥)))
6		ovex 7429	. . . . . . . . . 10 ⊢ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) ∈ V
7		fourierdlem49.z	. . . . . . . . . . 11 ⊢ 𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
8	7	fvmpt2 6987	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) ∈ V) → (𝑍‘𝑥) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
9	6, 8	mpan2 701	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (𝑍‘𝑥) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
10	9	oveq2d 7412	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (𝑥 + (𝑍‘𝑥)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
11	10	mpteq2ia 5195	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
12	5, 11	eqtri 2785	. . . . . 6 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
13	1, 2, 3, 4, 12	fourierdlem4 46685	. . . . 5 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
14		fourierdlem49.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ)
15	13, 14	ffvelcdmd 7066	. . . 4 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (𝐴(,]𝐵))
16		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (𝐸‘𝑋) ∈ ran 𝑄)
17		fourierdlem49.q	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
18		fourierdlem49.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ)
19		fourierdlem49.p	. . . . . . . . . . . . . . 15 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
20	19	fourierdlem2 46683	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
21	18, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
22	17, 21	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
23	22	simpld 498	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
24		elmapi 8830	. . . . . . . . . . 11 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
26	25	ffnd 6692	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
27	26	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑄 Fn (0...𝑀))
28		fvelrnb 6927	. . . . . . . 8 ⊢ (𝑄 Fn (0...𝑀) → ((𝐸‘𝑋) ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋)))
29	27, 28	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ((𝐸‘𝑋) ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋)))
30	16, 29	mpbid 234	. . . . . 6 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋))
31		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 − 1) → (𝑄‘𝑖) = (𝑄‘(𝑗 − 1)))
32		fvoveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 − 1) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝑗 − 1) + 1)))
33	31, 32	oveq12d 7414	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 − 1) → ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1))))
34	33	eleq2d 2848	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 − 1) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) ↔ (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))))
35		nnuz 12878	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
36		1zzd 12602	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 1 ∈ ℤ)
37		elfzelz 13529	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
38	37	ad2antlr 737	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℤ)
39		1e0p1 12735	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
40	38	zred 12677	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℝ)
41		elfzle1 13532	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗)
42	41	ad2antlr 737	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ≤ 𝑗)
43		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑄‘𝑗) = (𝐸‘𝑋) → (𝑄‘𝑗) = (𝐸‘𝑋))
44	43	eqcomd 2768	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑄‘𝑗) = (𝐸‘𝑋) → (𝐸‘𝑋) = (𝑄‘𝑗))
45	44	ad2antlr 737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = (𝑄‘𝑗))
46		fveq2 6867	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → (𝑄‘𝑗) = (𝑄‘0))
47	46	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝑄‘𝑗) = (𝑄‘0))
48	22	simprld 781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
49	48	simpld 498	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
50	49	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝑄‘0) = 𝐴)
51	45, 47, 50	3eqtrd 2801	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴)
52	51	adantllr 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴)
53	52	adantllr 729	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴)
54	1	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 ∈ ℝ)
55	1	rexrd 11232	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
56	55	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 ∈ ℝ^*)
57	2	rexrd 11232	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
58	57	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐵 ∈ ℝ^*)
59		simpr 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ (𝐴(,]𝐵))
60	56, 58, 59	iocgtlbd 46145	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 < (𝐸‘𝑋))
61	54, 60	gtned 11318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ≠ 𝐴)
62	61	neneqd 2962	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → ¬ (𝐸‘𝑋) = 𝐴)
63	62	ad3antrrr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → ¬ (𝐸‘𝑋) = 𝐴)
64	53, 63	pm2.65da 826	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ¬ 𝑗 = 0)
65	64	neqned 2964	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ≠ 0)
66	40, 42, 65	ne0gt0d 11320	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 < 𝑗)
67		0zd 12580	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ∈ ℤ)
68	67, 38	zltp1led 12626	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 < 𝑗 ↔ (0 + 1) ≤ 𝑗))
69	66, 68	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 + 1) ≤ 𝑗)
70	39, 69	eqbrtrid 5135	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 1 ≤ 𝑗)
71	35, 36, 38, 70	eluzd 45983	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℕ)
72		nnm1nn0 12522	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → (𝑗 − 1) ∈ ℕ₀)
73	71, 72	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ ℕ₀)
74		nn0uz 12877	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
75	73, 74	eleqtrdi 2872	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (ℤ_≥‘0))
76	18	nnzd 12594	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
77	76	ad3antrrr 740	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑀 ∈ ℤ)
78		peano2zm 12614	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈ ℤ)
79	37, 78	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℤ)
80	79	zred 12677	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ)
81	37	zred 12677	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
82		elfzel2 13527	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
83	82	zred 12677	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
84	81	ltm1d 12124	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑗)
85		elfzle2 13533	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ≤ 𝑀)
86	80, 81, 83, 84, 85	ltletrd 11343	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀)
87	86	ad2antlr 737	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) < 𝑀)
88	75, 77, 87	elfzod 13668	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (0..^𝑀))
89	25	ad3antrrr 740	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑄:(0...𝑀)⟶ℝ)
90	38, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ ℤ)
91	73	nn0ge0d 12545	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ≤ (𝑗 − 1))
92	80, 83, 86	ltled 11331	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ≤ 𝑀)
93	92	ad2antlr 737	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ≤ 𝑀)
94	67, 77, 90, 91, 93	elfzd 13520	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (0...𝑀))
95	89, 94	ffvelcdmd 7066	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) ∈ ℝ)
96	95	rexrd 11232	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) ∈ ℝ^*)
97	25	ffvelcdmda 7065	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
98	97	rexrd 11232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ^*)
99	98	adantlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ^*)
100	99	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) ∈ ℝ^*)
101		iocssre 13431	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
102	55, 2, 101	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
103	102	sselda 3936	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ ℝ)
104	103	rexrd 11232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ ℝ^*)
105	104	ad2antrr 736	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ^*)
106		simplll 784	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝜑)
107		ovex 7429	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 − 1) ∈ V
108		eleq1 2850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑗 − 1) → (𝑖 ∈ (0..^𝑀) ↔ (𝑗 − 1) ∈ (0..^𝑀)))
109	108	anbi2d 639	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑗 − 1) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀))))
110	31, 32	breq12d 5113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑗 − 1) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1))))
111	109, 110	imbi12d 346	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 − 1) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1)))))
112	22	simprrd 783	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
113	112	r19.21bi 3254	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
114	107, 111, 113	vtocl 3525	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1)))
115	106, 88, 114	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1)))
116	37	zcnd 12678	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ)
117		1cnd 11175	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → 1 ∈ ℂ)
118	116, 117	npcand 11546	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 − 1) + 1) = 𝑗)
119	118	eqcomd 2768	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 = ((𝑗 − 1) + 1))
120	119	fveq2d 6871	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → (𝑄‘𝑗) = (𝑄‘((𝑗 − 1) + 1)))
121	120	eqcomd 2768	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗))
122	121	ad2antlr 737	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗))
123	115, 122	breqtrd 5126	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝑄‘𝑗))
124		simpr 488	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) = (𝐸‘𝑋))
125	123, 124	breqtrd 5126	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝐸‘𝑋))
126	102, 15	sseldd 3937	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ)
127	126	leidd 11753	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸‘𝑋) ≤ (𝐸‘𝑋))
128	127	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝐸‘𝑋))
129	44	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) = (𝑄‘𝑗))
130	128, 129	breqtrd 5126	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘𝑗))
131	130	adantllr 729	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘𝑗))
132	96, 100, 105, 125, 131	eliocd 46083	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘𝑗)))
133	120	oveq2d 7412	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑄‘(𝑗 − 1))(,](𝑄‘𝑗)) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1))))
134	133	ad2antlr 737	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ((𝑄‘(𝑗 − 1))(,](𝑄‘𝑗)) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1))))
135	132, 134	eleqtrd 2864	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1))))
136	34, 88, 135	rspcedvdw 3584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
137	136	ex 416	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
138	137	adantlr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
139	138	rexlimdva 3163	. . . . . 6 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
140	30, 139	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
141	18	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑀 ∈ ℕ)
142	25	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
143		iocssicc 13441	. . . . . . . . . 10 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
144	49	eqcomd 2768	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
145	48	simprd 499	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
146	145	eqcomd 2768	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
147	144, 146	oveq12d 7414	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
148	143, 147	sseqtrid 3978	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ((𝑄‘0)[,](𝑄‘𝑀)))
149	148	sselda 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
150	149	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → (𝐸‘𝑋) ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
151		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ¬ (𝐸‘𝑋) ∈ ran 𝑄)
152		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
153	152	breq1d 5110	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < (𝐸‘𝑋) ↔ (𝑄‘𝑗) < (𝐸‘𝑋)))
154	153	cbvrabv 3424	. . . . . . . 8 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < (𝐸‘𝑋)} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < (𝐸‘𝑋)}
155	154	supeq1i 9393	. . . . . . 7 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < (𝐸‘𝑋)}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < (𝐸‘𝑋)}, ℝ, < )
156	141, 142, 150, 151, 155	fourierdlem25 46706	. . . . . 6 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
157		ioossioc 46068	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))
158	157	sseli 3932	. . . . . . . 8 ⊢ ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
159	158	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
160	159	reximdva 3175	. . . . . 6 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
161	156, 160	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
162	140, 161	pm2.61dan 822	. . . 4 ⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
163	15, 162	mpdan 697	. . 3 ⊢ (𝜑 → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
164		resindm 6016	. . . . . . . . 9 ⊢ (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹)) = (𝐹 ↾ (-∞(,)(𝐸‘𝑋)))
165		fourierdlem49.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
166	165	fdmd 6702	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐷)
167	166	ineq2d 4172	. . . . . . . . . 10 ⊢ (𝜑 → ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹) = ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))
168	167	reseq2d 5965	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹)) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)))
169	164, 168	eqtr3id 2811	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)))
170	169	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)))
171	170	oveq1d 7411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) lim_ℂ (𝐸‘𝑋)))
172		ax-resscn 11130	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
173	172	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
174	165, 173	fssd 6709	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
175	174	3ad2ant1 1146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐹:𝐷⟶ℂ)
176		inss2 4189	. . . . . . . . 9 ⊢ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ 𝐷
177	176	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ 𝐷)
178	175, 177	fssresd 6731	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)):((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)⟶ℂ)
179		mnfxr 11239	. . . . . . . . . 10 ⊢ -∞ ∈ ℝ^*
180	25	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
181		elfzofz 13681	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
182	181	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
183	180, 182	ffvelcdmd 7066	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
184	183	rexrd 11232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
185	184	mnfled 13138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ ≤ (𝑄‘𝑖))
186		iooss1 13384	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ -∞ ≤ (𝑄‘𝑖)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋)))
187	179, 185, 186	sylancr 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋)))
188	187	3adant3 1145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋)))
189		fzofzp1 13770	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
190	189	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
191	180, 190	ffvelcdmd 7066	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
192	191	3adant3 1145	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
193	192	rexrd 11232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
194	183	3adant3 1145	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
195	194	rexrd 11232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
196		simp3 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))
197	195, 193, 196	iocleubd 46134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1)))
198		iooss2 13385	. . . . . . . . . 10 ⊢ (((𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
199	193, 197, 198	syl2anc 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
200		fourierdlem49.cn	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
201		cncff 24955	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
202		fdm 6701	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
203	200, 201, 202	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
204		ssdmres 5999	. . . . . . . . . . . 12 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
205	203, 204	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
206	166	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = 𝐷)
207	205, 206	sseqtrd 3972	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)
208	207	3adant3 1145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)
209	199, 208	sstrd 3946	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ 𝐷)
210	188, 209	ssind 4192	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))
211		fourierdlem49.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
212	211, 173	sstrd 3946	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
213	212	3ad2ant1 1146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐷 ⊆ ℂ)
214	213	ssinss2d 45640	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℂ)
215		eqid 2762	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
216		eqid 2762	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) = ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))
217	126	3ad2ant1 1146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ℝ)
218	217	rexrd 11232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ℝ^*)
219	195, 193, 196	iocgtlbd 46145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝐸‘𝑋))
220	217	leidd 11753	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝐸‘𝑋))
221	195, 218, 218, 219, 220	eliocd 46083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋)))
222		ioounsn 13481	. . . . . . . . . . 11 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝐸‘𝑋) ∈ ℝ^* ∧ (𝑄‘𝑖) < (𝐸‘𝑋)) → (((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)}) = ((𝑄‘𝑖)(,](𝐸‘𝑋)))
223	195, 218, 219, 222	syl3anc 1390	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)}) = ((𝑄‘𝑖)(,](𝐸‘𝑋)))
224	223	fveq2d 6871	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘(((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)})) = ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘((𝑄‘𝑖)(,](𝐸‘𝑋))))
225	215	cnfldtop 24843	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
226		ovex 7429	. . . . . . . . . . . . 13 ⊢ (-∞(,)(𝐸‘𝑋)) ∈ V
227	226	inex1 5273	. . . . . . . . . . . 12 ⊢ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∈ V
228		snex 5396	. . . . . . . . . . . 12 ⊢ {(𝐸‘𝑋)} ∈ V
229	227, 228	unex 7727	. . . . . . . . . . 11 ⊢ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V
230		resttop 23220	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ Top)
231	225, 229, 230	mp2an 702	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ Top
232		retop 24821	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ Top
233		iooretop 24825	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑖)(,)+∞) ∈ (topGen‘ran (,))
234	233	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)+∞) ∈ (topGen‘ran (,)))
235		elrestr 17457	. . . . . . . . . . . 12 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V ∧ ((𝑄‘𝑖)(,)+∞) ∈ (topGen‘ran (,))) → (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ ((topGen‘ran (,)) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
236	232, 229, 234, 235	mp3an12i 1486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ ((topGen‘ran (,)) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
237		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → 𝑥 = (𝐸‘𝑋))
238		pnfxr 11236	. . . . . . . . . . . . . . . . . . . 20 ⊢ +∞ ∈ ℝ^*
239	238	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → +∞ ∈ ℝ^*)
240	217	ltpnfd 13123	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) < +∞)
241	195, 239, 217, 219, 240	eliood 46074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)+∞))
242		snidg 4619	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸‘𝑋) ∈ ℝ → (𝐸‘𝑋) ∈ {(𝐸‘𝑋)})
243		elun2 4135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸‘𝑋) ∈ {(𝐸‘𝑋)} → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))
244	126, 242, 243	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))
245	244	3ad2ant1 1146	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))
246	241, 245	elind 4152	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
247	246	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
248	237, 247	eqeltrd 2862	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
249	248	adantlr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
250	195	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) ∈ ℝ^*)
251	238	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → +∞ ∈ ℝ^*)
252	184	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) ∈ ℝ^*)
253	126	ad2antrr 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝐸‘𝑋) ∈ ℝ)
254		iocssre 13431	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝐸‘𝑋) ∈ ℝ) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ⊆ ℝ)
255	252, 253, 254	syl2anc 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ⊆ ℝ)
256		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋)))
257	255, 256	sseldd 3937	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ℝ)
258	257	3adantl3 1182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ℝ)
259	253	rexrd 11232	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝐸‘𝑋) ∈ ℝ^*)
260	252, 259, 256	iocgtlbd 46145	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) < 𝑥)
261	260	3adantl3 1182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) < 𝑥)
262	258	ltpnfd 13123	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 < +∞)
263	250, 251, 258, 261, 262	eliood 46074	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞))
264	263	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞))
265	179	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → -∞ ∈ ℝ^*)
266	259	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ^*)
267	257	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ℝ)
268	267	mnfltd 13126	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → -∞ < 𝑥)
269	126	ad3antrrr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ)
270	252, 259, 256	iocleubd 46134	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ≤ (𝐸‘𝑋))
271	270	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ≤ (𝐸‘𝑋))
272		neqne 2965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑥 = (𝐸‘𝑋) → 𝑥 ≠ (𝐸‘𝑋))
273	272	necomd 3012	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑥 = (𝐸‘𝑋) → (𝐸‘𝑋) ≠ 𝑥)
274	273	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ≠ 𝑥)
275	267, 269, 271, 274	leneltd 11337	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝐸‘𝑋))
276	265, 266, 267, 268, 275	eliood 46074	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋)))
277	276	3adantll3 45622	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋)))
278	208	ad2antrr 736	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)
279	195	ad2antrr 736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘𝑖) ∈ ℝ^*)
280	193	ad2antrr 736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
281	258	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ℝ)
282	261	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘𝑖) < 𝑥)
283	217	ad2antrr 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ)
284	192	ad2antrr 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
285	275	3adantll3 45622	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝐸‘𝑋))
286	197	ad2antrr 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1)))
287	281, 283, 284, 285, 286	ltletrd 11343	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝑄‘(𝑖 + 1)))
288	279, 280, 281, 282, 287	eliood 46074	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
289	278, 288	sseldd 3937	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ 𝐷)
290	277, 289	elind 4152	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))
291		elun1 4134	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) → 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))
292	290, 291	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))
293	264, 292	elind 4152	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
294	249, 293	pm2.61dan 822	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
295	195	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) ∈ ℝ^*)
296	218	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝐸‘𝑋) ∈ ℝ^*)
297		elinel1 4153	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞))
298		elioore 13379	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)+∞) → 𝑥 ∈ ℝ)
299	298	rexrd 11232	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)+∞) → 𝑥 ∈ ℝ^*)
300	297, 299	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ ℝ^*)
301	300	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ∈ ℝ^*)
302	184	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) ∈ ℝ^*)
303	238	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → +∞ ∈ ℝ^*)
304	297	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞))
305	302, 303, 304	ioogtlbd 46126	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) < 𝑥)
306	305	3adantl3 1182	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) < 𝑥)
307		elinel2 4154	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))
308		elsni 4599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {(𝐸‘𝑋)} → 𝑥 = (𝐸‘𝑋))
309	308	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 = (𝐸‘𝑋))
310	127	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → (𝐸‘𝑋) ≤ (𝐸‘𝑋))
311	309, 310	eqbrtrd 5122	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋))
312	311	adantlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋))
313		simpll 776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝜑)
314		elunnel2 4108	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))
315	314	adantll 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))
316		elinel1 4153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋)))
317		elioore 13379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (-∞(,)(𝐸‘𝑋)) → 𝑥 ∈ ℝ)
318	317	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ∈ ℝ)
319	126	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → (𝐸‘𝑋) ∈ ℝ)
320	179	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → -∞ ∈ ℝ^*)
321	319	rexrd 11232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → (𝐸‘𝑋) ∈ ℝ^*)
322		simpr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋)))
323	320, 321, 322	iooltubd 46120	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 < (𝐸‘𝑋))
324	318, 319, 323	ltled 11331	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ≤ (𝐸‘𝑋))
325	316, 324	sylan2 602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) → 𝑥 ≤ (𝐸‘𝑋))
326	313, 315, 325	syl2anc 593	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋))
327	312, 326	pm2.61dan 822	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ≤ (𝐸‘𝑋))
328	327	adantlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ≤ (𝐸‘𝑋))
329	307, 328	sylan2 602	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ≤ (𝐸‘𝑋))
330	329	3adantl3 1182	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ≤ (𝐸‘𝑋))
331	295, 296, 301, 306, 330	eliocd 46083	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋)))
332	294, 331	impbida 810	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋)) ↔ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))))
333	332	eqrdv 2760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) = (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
334		ioossre 13411	. . . . . . . . . . . . . 14 ⊢ (-∞(,)(𝐸‘𝑋)) ⊆ ℝ
335		ssinss1 4197	. . . . . . . . . . . . . 14 ⊢ ((-∞(,)(𝐸‘𝑋)) ⊆ ℝ → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℝ)
336	334, 335	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℝ)
337	217	snssd 4745	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {(𝐸‘𝑋)} ⊆ ℝ)
338	336, 337	unssd 4144	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ⊆ ℝ)
339		eqid 2762	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
340	215, 339	rerest 24864	. . . . . . . . . . . 12 ⊢ ((((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ⊆ ℝ → ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) = ((topGen‘ran (,)) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
341	338, 340	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) = ((topGen‘ran (,)) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
342	236, 333, 341	3eltr4d 2877	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ∈ ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))
343		isopn3i 23142	. . . . . . . . . 10 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ Top ∧ ((𝑄‘𝑖)(,](𝐸‘𝑋)) ∈ ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘((𝑄‘𝑖)(,](𝐸‘𝑋))) = ((𝑄‘𝑖)(,](𝐸‘𝑋)))
344	231, 342, 343	sylancr 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘((𝑄‘𝑖)(,](𝐸‘𝑋))) = ((𝑄‘𝑖)(,](𝐸‘𝑋)))
345	224, 344	eqtr2d 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) = ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘(((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)})))
346	221, 345	eleqtrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘(((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)})))
347	178, 210, 214, 215, 216, 346	limcres 25948	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) lim_ℂ (𝐸‘𝑋)))
348	210	resabs1d 5994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))))
349	348	oveq1d 7411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
350	171, 347, 349	3eqtr2d 2803	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
351	174	ffdmd 6722	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
352	351	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → 𝐹:dom 𝐹⟶ℂ)
353	352	3ad2antl1 1199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → 𝐹:dom 𝐹⟶ℂ)
354		ioosscn 13412	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ℂ
355	354	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ℂ)
356	166	eqcomd 2768	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 = dom 𝐹)
357	356	3ad2ant1 1146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐷 = dom 𝐹)
358	209, 357	sseqtrd 3972	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ dom 𝐹)
359	358	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ dom 𝐹)
360		oveq2 7404	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → (𝐵 − 𝑥) = (𝐵 − 𝑋))
361	360	fvoveq1d 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑋) / 𝑇)))
362	361	oveq1d 7411	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
363	2, 14	resubcld 11615	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
364	2, 1	resubcld 11615	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
365	4, 364	eqeltrid 2866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑇 ∈ ℝ)
366	1, 2	posdifd 11774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
367	3, 366	mpbid 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
368	367, 4	breqtrrdi 5142	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 < 𝑇)
369	368	gt0ne0d 11751	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑇 ≠ 0)
370	363, 365, 369	redivcld 12019	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐵 − 𝑋) / 𝑇) ∈ ℝ)
371	370	flcld 13808	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
372	371	zred 12677	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℝ)
373	372, 365	remulcld 11212	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℝ)
374	7, 362, 14, 373	fvmptd3 6999	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑍‘𝑋) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
375	374, 373	eqeltrd 2862	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍‘𝑋) ∈ ℝ)
376	375	recnd 11210	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍‘𝑋) ∈ ℂ)
377	376	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → (𝑍‘𝑋) ∈ ℂ)
378	377	3ad2antl1 1199	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → (𝑍‘𝑋) ∈ ℂ)
379	378	negcld 11529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → -(𝑍‘𝑋) ∈ ℂ)
380		eqid 2762	. . . . . . . . 9 ⊢ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}
381		ioosscn 13412	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ ℂ
382	381	sseli 3932	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) → 𝑦 ∈ ℂ)
383	382	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℂ)
384	376	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℂ)
385	383, 384	pncand 11543	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = 𝑦)
386	385	eqcomd 2768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)))
387	386	3ad2antl1 1199	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)))
388	374	oveq2d 7412	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
389	388	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
390	383, 384	addcld 11201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℂ)
391	373	recnd 11210	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℂ)
392	391	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℂ)
393	390, 392	negsubd 11548	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
394	371	zcnd 12678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℂ)
395	365	recnd 11210	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑇 ∈ ℂ)
396	394, 395	mulneg1d 11640	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
397	396	eqcomd 2768	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
398	397	oveq2d 7412	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
399	398	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
400	389, 393, 399	3eqtr2d 2803	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
401	400	3ad2antl1 1199	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
402	371	znegcld 12679	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
403	402	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
404	403	3ad2antl1 1199	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
405		simpl1 1205	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝜑)
406	209	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ 𝐷)
407	184	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) ∈ ℝ^*)
408	126	rexrd 11232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ^*)
409	408	ad2antrr 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐸‘𝑋) ∈ ℝ^*)
410		elioore 13379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) → 𝑦 ∈ ℝ)
411	410	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℝ)
412	375	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℝ)
413	411, 412	readdcld 11211	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ)
414	413	adantlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ)
415	375	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑍‘𝑋) ∈ ℝ)
416	183, 415	resubcld 11615	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ)
417	416	rexrd 11232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
418	417	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
419	14	rexrd 11232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
420	419	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑋 ∈ ℝ^*)
421		simpr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))
422	418, 420, 421	ioogtlbd 46126	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦)
423	183	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) ∈ ℝ)
424	375	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℝ)
425	410	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℝ)
426	423, 424, 425	ltsubaddd 11783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦 ↔ (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋))))
427	422, 426	mpbid 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋)))
428	14	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑋 ∈ ℝ)
429	418, 420, 421	iooltubd 46120	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 < 𝑋)
430	425, 428, 424, 429	ltadd1dd 11798	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝑋 + (𝑍‘𝑋)))
431		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
432		fveq2 6867	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑋 → (𝑍‘𝑥) = (𝑍‘𝑋))
433	431, 432	oveq12d 7414	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝑍‘𝑥)) = (𝑋 + (𝑍‘𝑋)))
434	14, 375	readdcld 11211	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) ∈ ℝ)
435	5, 433, 14, 434	fvmptd3 6999	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋)))
436	435	eqcomd 2768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋))
437	436	ad2antrr 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋))
438	430, 437	breqtrd 5126	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝐸‘𝑋))
439	407, 409, 414, 427, 438	eliood 46074	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋)))
440	439	3adantl3 1182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋)))
441	406, 440	sseldd 3937	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ 𝐷)
442	405, 441, 404	3jca 1141	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))
443		eleq1 2850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 ∈ ℤ ↔ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))
444	443	3anbi3d 1463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)))
445		oveq1 7403	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 · 𝑇) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
446	445	oveq2d 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
447	446	eleq1d 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷))
448	444, 447	imbi12d 346	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷) ↔ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)))
449		ovex 7429	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 + (𝑍‘𝑋)) ∈ V
450		eleq1 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (𝑥 ∈ 𝐷 ↔ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷))
451	450	3anbi2d 1462	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ)))
452		oveq1 7403	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (𝑥 + (𝑘 · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)))
453	452	eleq1d 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → ((𝑥 + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷))
454	451, 453	imbi12d 346	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) ↔ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷)))
455		fourierdlem49.dper	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)
456	449, 454, 455	vtocl 3525	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷)
457	448, 456	vtoclg 3522	. . . . . . . . . . . . . . 15 ⊢ (-(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷))
458	404, 442, 457	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)
459	401, 458	eqeltrd 2862	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) ∈ 𝐷)
460	387, 459	eqeltrd 2862	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ 𝐷)
461	460	ssd 45660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷)
462	183	recnd 11210	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
463	376	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑍‘𝑋) ∈ ℂ)
464	462, 463	negsubd 11548	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + -(𝑍‘𝑋)) = ((𝑄‘𝑖) − (𝑍‘𝑋)))
465	464	eqcomd 2768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) = ((𝑄‘𝑖) + -(𝑍‘𝑋)))
466	435	oveq1d 7411	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸‘𝑋) + -(𝑍‘𝑋)) = ((𝑋 + (𝑍‘𝑋)) + -(𝑍‘𝑋)))
467	434	recnd 11210	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) ∈ ℂ)
468	467, 376	negsubd 11548	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑋 + (𝑍‘𝑋)) + -(𝑍‘𝑋)) = ((𝑋 + (𝑍‘𝑋)) − (𝑍‘𝑋)))
469	14	recnd 11210	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℂ)
470	469, 376	pncand 11543	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑋 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = 𝑋)
471	466, 468, 470	3eqtrrd 2802	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ((𝐸‘𝑋) + -(𝑍‘𝑋)))
472	471	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 = ((𝐸‘𝑋) + -(𝑍‘𝑋)))
473	465, 472	oveq12d 7414	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) = (((𝑄‘𝑖) + -(𝑍‘𝑋))(,)((𝐸‘𝑋) + -(𝑍‘𝑋))))
474	126	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) ∈ ℝ)
475	415	renegcld 11614	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -(𝑍‘𝑋) ∈ ℝ)
476	183, 474, 475	iooshift 46098	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + -(𝑍‘𝑋))(,)((𝐸‘𝑋) + -(𝑍‘𝑋))) = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))})
477	473, 476	eqtr2d 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} = (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))
478	477	3adant3 1145	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} = (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))
479	166	3ad2ant1 1146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → dom 𝐹 = 𝐷)
480	461, 478, 479	3sstr4d 3991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} ⊆ dom 𝐹)
481	480	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} ⊆ dom 𝐹)
482	374	negeqd 11424	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → -(𝑍‘𝑋) = -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
483	482, 397	eqtrd 2797	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → -(𝑍‘𝑋) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
484	483	oveq2d 7412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 + -(𝑍‘𝑋)) = (𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
485	484	fveq2d 6871	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))))
486	485	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))))
487	486	3ad2antl1 1199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))))
488	402	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
489	488	3ad2antl1 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
490		simpl1 1205	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → 𝜑)
491	209	sselda 3936	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → 𝑥 ∈ 𝐷)
492	490, 491, 489	3jca 1141	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))
493	443	3anbi3d 1463	. . . . . . . . . . . . . 14 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)))
494	445	oveq2d 7412	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
495	494	fveqeq2d 6875	. . . . . . . . . . . . . 14 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))
496	493, 495	imbi12d 346	. . . . . . . . . . . . 13 ⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))))
497		fourierdlem49.per	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥))
498	496, 497	vtoclg 3522	. . . . . . . . . . . 12 ⊢ (-(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))
499	489, 492, 498	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))
500	487, 499	eqtrd 2797	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘𝑥))
501	500	adantlr 725	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘𝑥))
502		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
503	353, 355, 359, 379, 380, 481, 501, 502	limcperiod 46204	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → 𝑦 ∈ ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) lim_ℂ ((𝐸‘𝑋) + -(𝑍‘𝑋))))
504	477	reseq2d 5965	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) = (𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)))
505	472	eqcomd 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐸‘𝑋) + -(𝑍‘𝑋)) = 𝑋)
506	504, 505	oveq12d 7414	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) lim_ℂ ((𝐸‘𝑋) + -(𝑍‘𝑋))) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋))
507	506	3adant3 1145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) lim_ℂ ((𝐸‘𝑋) + -(𝑍‘𝑋))) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋))
508	507	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) lim_ℂ ((𝐸‘𝑋) + -(𝑍‘𝑋))) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋))
509	503, 508	eleqtrd 2864	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))) → 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋))
510	351	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → 𝐹:dom 𝐹⟶ℂ)
511	510	3ad2antl1 1199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → 𝐹:dom 𝐹⟶ℂ)
512	381	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ ℂ)
513	461, 479	sseqtrrd 3973	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ dom 𝐹)
514	513	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ dom 𝐹)
515	376	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → (𝑍‘𝑋) ∈ ℂ)
516	515	3ad2antl1 1199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → (𝑍‘𝑋) ∈ ℂ)
517		eqid 2762	. . . . . . . . 9 ⊢ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}
518	462, 463	npcand 11546	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋)) = (𝑄‘𝑖))
519	518	eqcomd 2768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋)))
520	435	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋)))
521	519, 520	oveq12d 7414	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) = ((((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋))(,)(𝑋 + (𝑍‘𝑋))))
522	14	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
523	416, 522, 415	iooshift 46098	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋))(,)(𝑋 + (𝑍‘𝑋))) = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))})
524	521, 523	eqtr2d 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} = ((𝑄‘𝑖)(,)(𝐸‘𝑋)))
525	524	3adant3 1145	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} = ((𝑄‘𝑖)(,)(𝐸‘𝑋)))
526	209, 525, 479	3sstr4d 3991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} ⊆ dom 𝐹)
527	526	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} ⊆ dom 𝐹)
528	374	oveq2d 7412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 + (𝑍‘𝑋)) = (𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
529	528	fveq2d 6871	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))))
530	529	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))))
531	530	3ad2antl1 1199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))))
532	371	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
533	532	3ad2antl1 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)
534		simpl1 1205	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝜑)
535	461	sselda 3936	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ 𝐷)
536	534, 535, 533	3jca 1141	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))
537		eleq1 2850	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))
538	537	3anbi3d 1463	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)))
539		oveq1 7403	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))
540	539	oveq2d 7412	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))
541	540	fveqeq2d 6875	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))
542	538, 541	imbi12d 346	. . . . . . . . . . . . 13 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))))
543	542, 497	vtoclg 3522	. . . . . . . . . . . 12 ⊢ ((⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))
544	533, 536, 543	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))
545	531, 544	eqtrd 2797	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘𝑥))
546	545	adantlr 725	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘𝑥))
547		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋))
548	511, 512, 514, 516, 517, 527, 546, 547	limcperiod 46204	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → 𝑦 ∈ ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) lim_ℂ (𝑋 + (𝑍‘𝑋))))
549	524	reseq2d 5965	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))))
550	436	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋))
551	549, 550	oveq12d 7414	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) lim_ℂ (𝑋 + (𝑍‘𝑋))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
552	551	3adant3 1145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) lim_ℂ (𝑋 + (𝑍‘𝑋))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
553	552	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) lim_ℂ (𝑋 + (𝑍‘𝑋))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
554	548, 553	eleqtrd 2864	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)) → 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
555	509, 554	impbida 810	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) ↔ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋)))
556	555	eqrdv 2760	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋))
557		resindm 6016	. . . . . . . . 9 ⊢ (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹)) = (𝐹 ↾ (-∞(,)𝑋))
558	166	ineq2d 4172	. . . . . . . . . 10 ⊢ (𝜑 → ((-∞(,)𝑋) ∩ dom 𝐹) = ((-∞(,)𝑋) ∩ 𝐷))
559	558	reseq2d 5965	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)))
560	557, 559	eqtr3id 2811	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)))
561	560	oveq1d 7411	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) = ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) lim_ℂ 𝑋))
562	561	3ad2ant1 1146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) = ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) lim_ℂ 𝑋))
563		inss2 4189	. . . . . . . . 9 ⊢ ((-∞(,)𝑋) ∩ 𝐷) ⊆ 𝐷
564	563	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)𝑋) ∩ 𝐷) ⊆ 𝐷)
565	175, 564	fssresd 6731	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)):((-∞(,)𝑋) ∩ 𝐷)⟶ℂ)
566	417	mnfled 13138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ ≤ ((𝑄‘𝑖) − (𝑍‘𝑋)))
567		iooss1 13384	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ -∞ ≤ ((𝑄‘𝑖) − (𝑍‘𝑋))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋))
568	179, 566, 567	sylancr 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋))
569	568	3adant3 1145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋))
570	569, 461	ssind 4192	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ 𝐷))
571	213	ssinss2d 45640	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)𝑋) ∩ 𝐷) ⊆ ℂ)
572		eqid 2762	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) = ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
573	417	3adant3 1145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
574	419	3ad2ant1 1146	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ^*)
575	435	3ad2ant1 1146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋)))
576	219, 575	breqtrd 5126	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑋 + (𝑍‘𝑋)))
577	375	3ad2ant1 1146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑍‘𝑋) ∈ ℝ)
578	14	3ad2ant1 1146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
579	194, 577, 578	ltsubaddd 11783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋 ↔ (𝑄‘𝑖) < (𝑋 + (𝑍‘𝑋))))
580	576, 579	mpbird 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋)
581	14	leidd 11753	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝑋)
582	581	3ad2ant1 1146	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ≤ 𝑋)
583	573, 574, 574, 580, 582	eliocd 46083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋))
584		ioounsn 13481	. . . . . . . . . . 11 ⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^* ∧ 𝑋 ∈ ℝ^* ∧ ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋}) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋))
585	573, 574, 580, 584	syl3anc 1390	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋}) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋))
586	585	fveq2d 6871	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))‘((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋})) = ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))‘(((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)))
587		ovex 7429	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝑋) ∈ V
588	587	inex1 5273	. . . . . . . . . . . 12 ⊢ ((-∞(,)𝑋) ∩ 𝐷) ∈ V
589		snex 5396	. . . . . . . . . . . 12 ⊢ {𝑋} ∈ V
590	588, 589	unex 7727	. . . . . . . . . . 11 ⊢ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∈ V
591		resttop 23220	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ Top)
592	225, 590, 591	mp2an 702	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ Top
593		iooretop 24825	. . . . . . . . . . . . 13 ⊢ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∈ (topGen‘ran (,))
594	593	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∈ (topGen‘ran (,)))
595		elrestr 17457	. . . . . . . . . . . 12 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∈ V ∧ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∈ (topGen‘ran (,))) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ ((topGen‘ran (,)) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))
596	232, 590, 594, 595	mp3an12i 1486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ ((topGen‘ran (,)) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))
597	417	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
598	238	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → +∞ ∈ ℝ^*)
599	14	ad2antrr 736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑋 ∈ ℝ)
600		iocssre 13431	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^* ∧ 𝑋 ∈ ℝ) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ⊆ ℝ)
601	597, 599, 600	syl2anc 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ⊆ ℝ)
602		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋))
603	601, 602	sseldd 3937	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ ℝ)
604	419	ad2antrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑋 ∈ ℝ^*)
605	597, 604, 602	iocgtlbd 46145	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥)
606	603	ltpnfd 13123	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 < +∞)
607	597, 598, 603, 605, 606	eliood 46074	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞))
608	607	3adantl3 1182	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞))
609		eqvisset 3474	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑋 → 𝑋 ∈ V)
610		snidg 4619	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ V → 𝑋 ∈ {𝑋})
611	609, 610	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑋 → 𝑋 ∈ {𝑋})
612	431, 611	eqeltrd 2862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → 𝑥 ∈ {𝑋})
613		elun2 4135	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑋} → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
614	612, 613	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
615	614	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
616		simpll 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))))
617	417	ad2antrr 736	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
618	419	ad3antrrr 740	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ∈ ℝ^*)
619	603	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ ℝ)
620	605	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥)
621	14	ad3antrrr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ∈ ℝ)
622	597, 604, 602	iocleubd 46134	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ≤ 𝑋)
623	622	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ≤ 𝑋)
624	431	eqcoms 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 = 𝑥 → 𝑥 = 𝑋)
625	624	necon3bi 2983	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = 𝑋 → 𝑋 ≠ 𝑥)
626	625	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ≠ 𝑥)
627	619, 621, 623, 626	leneltd 11337	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 < 𝑋)
628	617, 618, 619, 620, 627	eliood 46074	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))
629	628	3adantll3 45622	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))
630	570	sselda 3936	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷))
631		elun1 4134	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
632	630, 631	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
633	616, 629, 632	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
634	615, 633	pm2.61dan 822	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
635	608, 634	elind 4152	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))
636	573	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
637	574	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑋 ∈ ℝ^*)
638		elinel1 4153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞))
639		elioore 13379	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) → 𝑥 ∈ ℝ)
640	638, 639	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ ℝ)
641	640	rexrd 11232	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ ℝ^*)
642	641	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ∈ ℝ^*)
643	417	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ^*)
644	238	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → +∞ ∈ ℝ^*)
645	638	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞))
646	643, 644, 645	ioogtlbd 46126	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥)
647	646	3adantl3 1182	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥)
648		elinel2 4154	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))
649		elsni 4599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑋} → 𝑥 = 𝑋)
650	649	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑥 = 𝑋)
651	581	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑋 ≤ 𝑋)
652	650, 651	eqbrtrd 5122	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋)
653	652	adantlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋)
654		simpll 776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝜑)
655		elunnel2 4108	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷))
656	655	adantll 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷))
657	656	elin1d 4156	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ (-∞(,)𝑋))
658		elioore 13379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (-∞(,)𝑋) → 𝑥 ∈ ℝ)
659	658	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ∈ ℝ)
660	14	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑋 ∈ ℝ)
661	179	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → -∞ ∈ ℝ^*)
662	419	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑋 ∈ ℝ^*)
663		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ∈ (-∞(,)𝑋))
664	661, 662, 663	iooltubd 46120	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 < 𝑋)
665	659, 660, 664	ltled 11331	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ≤ 𝑋)
666	654, 657, 665	syl2anc 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋)
667	653, 666	pm2.61dan 822	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ≤ 𝑋)
668	648, 667	sylan2 602	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ≤ 𝑋)
669	668	3ad2antl1 1199	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ≤ 𝑋)
670	636, 637, 642, 647, 669	eliocd 46083	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋))
671	635, 670	impbida 810	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ↔ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))))
672	671	eqrdv 2760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) = ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))
673	211	ssinss2d 45640	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((-∞(,)𝑋) ∩ 𝐷) ⊆ ℝ)
674	14	snssd 4745	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑋} ⊆ ℝ)
675	673, 674	unssd 4144	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ⊆ ℝ)
676	675	3ad2ant1 1146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ⊆ ℝ)
677	215, 339	rerest 24864	. . . . . . . . . . . 12 ⊢ ((((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ⊆ ℝ → ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) = ((topGen‘ran (,)) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))
678	676, 677	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) = ((topGen‘ran (,)) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))
679	596, 672, 678	3eltr4d 2877	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ∈ ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))
680		isopn3i 23142	. . . . . . . . . 10 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ Top ∧ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ∈ ((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))‘(((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋))
681	592, 679, 680	sylancr 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))‘(((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋))
682	586, 681	eqtr2d 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) = ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))‘((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋})))
683	583, 682	eleqtrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))‘((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋})))
684	565, 570, 571, 215, 572, 683	limcres 25948	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋) = ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) lim_ℂ 𝑋))
685	570	resabs1d 5994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) = (𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)))
686	685	oveq1d 7411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋))
687	562, 684, 686	3eqtr2rd 2804	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) lim_ℂ 𝑋) = ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
688	350, 556, 687	3eqtrrd 2802	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
689	688	rexlimdv3a 3167	. . 3 ⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋))))
690	163, 689	mpd 15	. 2 ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
691	113	3adant3 1145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
692	200	3adant3 1145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
693		fourierdlem49.l	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
694	693	3adant3 1145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
695		eqid 2762	. . . . . . . 8 ⊢ if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋))) = if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋)))
696		eqid 2762	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) = ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}))
697	194, 192, 691, 692, 694, 194, 217, 219, 199, 695, 696	fourierdlem33 46714	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋))) ∈ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
698	199	resabs1d 5994	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))))
699	698	oveq1d 7411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
700	697, 699	eleqtrd 2864	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋))) ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)))
701	700	ne0d 4294	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) ≠ ∅)
702	350, 701	eqnetrd 3024	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) ≠ ∅)
703	702	rexlimdv3a 3167	. . 3 ⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) ≠ ∅))
704	163, 703	mpd 15	. 2 ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) lim_ℂ (𝐸‘𝑋)) ≠ ∅)
705	690, 704	eqnetrd 3024	1 ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∀wral 3076 ∃wrex 3086 {crab 3414 Vcvv 3454 ∪ cun 3902 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 ifcif 4480 {csn 4582 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5647 ran crn 5648 ↾ cres 5649 Fn wfn 6516 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ↑_m cmap 8808 supcsup 9386 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 + caddc 11076 · cmul 11078 +∞cpnf 11213 -∞cmnf 11214 ℝ^*cxr 11215 < clt 11216 ≤ cle 11217 − cmin 11414 -cneg 11415 / cdiv 11844 ℕcn 12210 ℕ₀cn0 12481 ℤcz 12568 ℤ_≥cuz 12839 (,)cioo 13349 (,]cioc 13350 [,]cicc 13352 ...cfz 13512 ..^cfzo 13659 ⌊cfl 13800 ↾_t crest 17449 TopOpenctopn 17450 topGenctg 17466 ℂ_fldccnfld 21424 Topctop 22953 intcnt 23077 –cn→ccncf 24938 lim_ℂ climc 25924

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fi 9357 df-sup 9388 df-inf 9389 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-seq 14015 df-exp 14075 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-struct 17183 df-slot 17218 df-ndx 17230 df-base 17246 df-plusg 17299 df-mulr 17300 df-starv 17301 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-rest 17451 df-topn 17452 df-topgen 17472 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-cnfld 21425 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-ntr 23080 df-cn 23287 df-cnp 23288 df-xms 24380 df-ms 24381 df-cncf 24940 df-limc 25928

This theorem is referenced by: fourierdlem94 46774 fourierdlem113 46793

W3C validator