fourierdlem46 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem46 44383

Description: The function 𝐹 has a limit at the bounds of every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem46.cn	⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
fourierdlem46.rlim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐹)) → ((𝐹 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fourierdlem46.llim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐹)) → ((𝐹 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
fourierdlem46.qiso	⊢ (𝜑 → 𝑄 Isom < , < ((0...𝑀), 𝐻))
fourierdlem46.qf	⊢ (𝜑 → 𝑄:(0...𝑀)⟶𝐻)
fourierdlem46.i	⊢ (𝜑 → 𝐼 ∈ (0..^𝑀))
fourierdlem46.10	⊢ (𝜑 → (𝑄‘𝐼) < (𝑄‘(𝐼 + 1)))
fourierdlem46.qiss	⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-π(,)π))
fourierdlem46.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem46.h	⊢ 𝐻 = ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹))
fourierdlem46.ranq	⊢ (𝜑 → ran 𝑄 = 𝐻)

**Assertion**
Ref	Expression
fourierdlem46	⊢ (𝜑 → (((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅ ∧ ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐼 𝑥,𝑄 𝜑,𝑥 Allowed substitution hints: 𝐶(𝑥) 𝐻(𝑥) 𝑀(𝑥)

Proof of Theorem fourierdlem46 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem46.h	. . . . . . . . 9 ⊢ 𝐻 = ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹))
2		pire 25815	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
3	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → π ∈ ℝ)
4	3	renegcld 11582	. . . . . . . . . . 11 ⊢ (𝜑 → -π ∈ ℝ)
5		fourierdlem46.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6		tpssi 4796	. . . . . . . . . . 11 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ 𝐶 ∈ ℝ) → {-π, π, 𝐶} ⊆ ℝ)
7	4, 3, 5, 6	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → {-π, π, 𝐶} ⊆ ℝ)
8	4, 3	iccssred 13351	. . . . . . . . . . 11 ⊢ (𝜑 → (-π[,]π) ⊆ ℝ)
9	8	ssdifssd 4102	. . . . . . . . . 10 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐹) ⊆ ℝ)
10	7, 9	unssd 4146	. . . . . . . . 9 ⊢ (𝜑 → ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹)) ⊆ ℝ)
11	1, 10	eqsstrid 3992	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
12		fourierdlem46.qf	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶𝐻)
13		fourierdlem46.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀))
14		elfzofz 13588	. . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))
16	12, 15	ffvelcdmd 7036	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝐼) ∈ 𝐻)
17	11, 16	sseldd 3945	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ)
18	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ ℝ)
19		fzofzp1 13669	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀))
20	13, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀))
21	12, 20	ffvelcdmd 7036	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ 𝐻)
22	11, 21	sseldd 3945	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ)
23	22	rexrd 11205	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
24	23	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
25		fourierdlem46.10	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐼) < (𝑄‘(𝐼 + 1)))
26	25	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) < (𝑄‘(𝐼 + 1)))
27		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑄‘𝐼) ∈ dom 𝐹 ∧ 𝑥 = (𝑄‘𝐼)) → 𝑥 = (𝑄‘𝐼))
28		simpl 483	. . . . . . . . . . . . 13 ⊢ (((𝑄‘𝐼) ∈ dom 𝐹 ∧ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) ∈ dom 𝐹)
29	27, 28	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ (((𝑄‘𝐼) ∈ dom 𝐹 ∧ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
30	29	adantll 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) ∧ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
31	30	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
32		ssun2 4133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-π[,]π) ∖ dom 𝐹) ⊆ ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹))
33	32, 1	sseqtrri 3981	. . . . . . . . . . . . . . . . . 18 ⊢ ((-π[,]π) ∖ dom 𝐹) ⊆ 𝐻
34		fourierdlem46.qiss	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-π(,)π))
35		ioossicc 13350	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (-π(,)π) ⊆ (-π[,]π)
36	34, 35	sstrdi 3956	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-π[,]π))
37	36	sselda 3944	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ (-π[,]π))
38	37	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (-π[,]π))
39		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ dom 𝐹)
40	38, 39	eldifd 3921	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((-π[,]π) ∖ dom 𝐹))
41	33, 40	sselid 3942	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ 𝐻)
42		fourierdlem46.ranq	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝑄 = 𝐻)
43	42	eqcomd 2742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐻 = ran 𝑄)
44	43	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝐻 = ran 𝑄)
45	41, 44	eleqtrd 2840	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ran 𝑄)
46		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
47		ffn 6668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑄:(0...𝑀)⟶𝐻 → 𝑄 Fn (0...𝑀))
48	12, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
49	48	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑄) → 𝑄 Fn (0...𝑀))
50		fvelrnb 6903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑄 Fn (0...𝑀) → (𝑥 ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑥))
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑄) → (𝑥 ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑥))
52	46, 51	mpbid 231	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑥)
53	52	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑥)
54		elfzelz 13441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
55	54	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → 𝑗 ∈ ℤ)
56		simplll 773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → 𝜑)
57		simplr 767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → 𝑗 ∈ (0...𝑀))
58		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ (𝑄‘𝑗) = 𝑥) → (𝑄‘𝑗) = 𝑥)
59		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ (𝑄‘𝑗) = 𝑥) → 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
60	58, 59	eqeltrd 2838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ (𝑄‘𝑗) = 𝑥) → (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
61	60	adantlr 713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
62		elfzoelz 13572	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ ℤ)
63	13, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐼 ∈ ℤ)
64	63	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐼 ∈ ℤ)
65	17	rexrd 11205	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ^*)
66	65	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ^*)
67	23	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
68		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
69		ioogtlb 43723	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < (𝑄‘𝑗))
70	66, 67, 68, 69	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < (𝑄‘𝑗))
71		fourierdlem46.qiso	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑄 Isom < , < ((0...𝑀), 𝐻))
72	71	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
73	15	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐼 ∈ (0...𝑀))
74		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑗 ∈ (0...𝑀))
75		isorel 7271	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝐼 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑀))) → (𝐼 < 𝑗 ↔ (𝑄‘𝐼) < (𝑄‘𝑗)))
76	72, 73, 74, 75	syl12anc 835	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝐼 < 𝑗 ↔ (𝑄‘𝐼) < (𝑄‘𝑗)))
77	70, 76	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐼 < 𝑗)
78		iooltub 43738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝑗) < (𝑄‘(𝐼 + 1)))
79	66, 67, 68, 78	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝑗) < (𝑄‘(𝐼 + 1)))
80	20	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝐼 + 1) ∈ (0...𝑀))
81		isorel 7271	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝑗 ∈ (0...𝑀) ∧ (𝐼 + 1) ∈ (0...𝑀))) → (𝑗 < (𝐼 + 1) ↔ (𝑄‘𝑗) < (𝑄‘(𝐼 + 1))))
82	72, 74, 80, 81	syl12anc 835	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑗 < (𝐼 + 1) ↔ (𝑄‘𝑗) < (𝑄‘(𝐼 + 1))))
83	79, 82	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑗 < (𝐼 + 1))
84		btwnnz 12579	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼 ∈ ℤ ∧ 𝐼 < 𝑗 ∧ 𝑗 < (𝐼 + 1)) → ¬ 𝑗 ∈ ℤ)
85	64, 77, 83, 84	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → ¬ 𝑗 ∈ ℤ)
86	56, 57, 61, 85	syl21anc 836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → ¬ 𝑗 ∈ ℤ)
87	86	adantllr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → ¬ 𝑗 ∈ ℤ)
88	55, 87	pm2.65da 815	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) ∧ 𝑗 ∈ (0...𝑀)) → ¬ (𝑄‘𝑗) = 𝑥)
89	88	nrexdv 3146	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) → ¬ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑥)
90	53, 89	pm2.65da 815	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → ¬ 𝑥 ∈ ran 𝑄)
91	90	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝑄)
92	45, 91	condan 816	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ dom 𝐹)
93	92	ralrimiva 3143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
94		dfss3 3932	. . . . . . . . . . . . . 14 ⊢ (((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
95	93, 94	sylibr 233	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
96	95	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
97	65	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) ∈ ℝ^*)
98	23	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
99		icossre 13345	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝐼) ∈ ℝ ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^*) → ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) ⊆ ℝ)
100	17, 23, 99	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) ⊆ ℝ)
101	100	sselda 3944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ)
102	101	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ ℝ)
103	17	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) ∈ ℝ)
104	65	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ^*)
105	23	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
106		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))
107		icogelb 13315	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ≤ 𝑥)
108	104, 105, 106, 107	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ≤ 𝑥)
109	108	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) ≤ 𝑥)
110		neqne 2951	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 = (𝑄‘𝐼) → 𝑥 ≠ (𝑄‘𝐼))
111	110	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ≠ (𝑄‘𝐼))
112	103, 102, 109, 111	leneltd 11309	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) < 𝑥)
113		icoltub 43736	. . . . . . . . . . . . . . 15 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1)))
114	104, 105, 106, 113	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1)))
115	114	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 < (𝑄‘(𝐼 + 1)))
116	97, 98, 102, 112, 115	eliood 43726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
117	96, 116	sseldd 3945	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
118	117	adantllr 717	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
119	31, 118	pm2.61dan 811	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ dom 𝐹)
120	119	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ∀𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
121		dfss3 3932	. . . . . . . 8 ⊢ (((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
122	120, 121	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
123		fourierdlem46.cn	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
124	123	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → 𝐹 ∈ (dom 𝐹–cn→ℂ))
125		rescncf 24260	. . . . . . 7 ⊢ (((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹 → (𝐹 ∈ (dom 𝐹–cn→ℂ) → (𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∈ (((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))–cn→ℂ)))
126	122, 124, 125	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∈ (((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))–cn→ℂ))
127	18, 24, 26, 126	icocncflimc 44120	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))‘(𝑄‘𝐼)) ∈ (((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)))
128	17	leidd 11721	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝐼) ≤ (𝑄‘𝐼))
129	65, 23, 65, 128, 25	elicod 13314	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))
130		fvres 6861	. . . . . . . 8 ⊢ ((𝑄‘𝐼) ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) → ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))‘(𝑄‘𝐼)) = (𝐹‘(𝑄‘𝐼)))
131	129, 130	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))‘(𝑄‘𝐼)) = (𝐹‘(𝑄‘𝐼)))
132	131	eqcomd 2742	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝑄‘𝐼)) = ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))‘(𝑄‘𝐼)))
133	132	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝐹‘(𝑄‘𝐼)) = ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))‘(𝑄‘𝐼)))
134		ioossico 13355	. . . . . . . . 9 ⊢ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))
135	134	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))
136	135	resabs1d 5968	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
137	136	eqcomd 2742	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
138	137	oveq1d 7372	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) = (((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)))
139	127, 133, 138	3eltr4d 2853	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝐹‘(𝑄‘𝐼)) ∈ ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)))
140	139	ne0d 4295	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
141		pnfxr 11209	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
142	141	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → +∞ ∈ ℝ^*)
143	22	ltpnfd 13042	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) < +∞)
144	23, 142, 143	xrltled 13069	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ +∞)
145		iooss2 13300	. . . . . . . . 9 ⊢ ((+∞ ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ≤ +∞) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)(,)+∞))
146	141, 144, 145	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)(,)+∞))
147	146	resabs1d 5968	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
148	147	oveq1d 7372	. . . . . 6 ⊢ (𝜑 → (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) = ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)))
149	148	eqcomd 2742	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) = (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)))
150	149	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) = (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)))
151		limcresi 25249	. . . . 5 ⊢ ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ⊆ (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼))
152	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ ℝ)
153		simpl 483	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → 𝜑)
154	2	renegcli 11462	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
155	154	rexri 11213	. . . . . . . . . . 11 ⊢ -π ∈ ℝ^*
156	155	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → -π ∈ ℝ^*)
157	2	rexri 11213	. . . . . . . . . . 11 ⊢ π ∈ ℝ^*
158	157	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → π ∈ ℝ^*)
159	4, 3, 17, 22, 25, 34	fourierdlem10 44348	. . . . . . . . . . 11 ⊢ (𝜑 → (-π ≤ (𝑄‘𝐼) ∧ (𝑄‘(𝐼 + 1)) ≤ π))
160	159	simpld 495	. . . . . . . . . 10 ⊢ (𝜑 → -π ≤ (𝑄‘𝐼))
161	159	simprd 496	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ π)
162	17, 22, 3, 25, 161	ltletrd 11315	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘𝐼) < π)
163	156, 158, 65, 160, 162	elicod 13314	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (-π[,)π))
164	163	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ (-π[,)π))
165		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → ¬ (𝑄‘𝐼) ∈ dom 𝐹)
166	164, 165	eldifd 3921	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ ((-π[,)π) ∖ dom 𝐹))
167	153, 166	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (𝜑 ∧ (𝑄‘𝐼) ∈ ((-π[,)π) ∖ dom 𝐹)))
168		eleq1 2825	. . . . . . . . 9 ⊢ (𝑥 = (𝑄‘𝐼) → (𝑥 ∈ ((-π[,)π) ∖ dom 𝐹) ↔ (𝑄‘𝐼) ∈ ((-π[,)π) ∖ dom 𝐹)))
169	168	anbi2d 629	. . . . . . . 8 ⊢ (𝑥 = (𝑄‘𝐼) → ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐹)) ↔ (𝜑 ∧ (𝑄‘𝐼) ∈ ((-π[,)π) ∖ dom 𝐹))))
170		oveq1 7364	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑄‘𝐼) → (𝑥(,)+∞) = ((𝑄‘𝐼)(,)+∞))
171	170	reseq2d 5937	. . . . . . . . . 10 ⊢ (𝑥 = (𝑄‘𝐼) → (𝐹 ↾ (𝑥(,)+∞)) = (𝐹 ↾ ((𝑄‘𝐼)(,)+∞)))
172		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (𝑄‘𝐼) → 𝑥 = (𝑄‘𝐼))
173	171, 172	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑥 = (𝑄‘𝐼) → ((𝐹 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) = ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)))
174	173	neeq1d 3003	. . . . . . . 8 ⊢ (𝑥 = (𝑄‘𝐼) → (((𝐹 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅ ↔ ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ≠ ∅))
175	169, 174	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = (𝑄‘𝐼) → (((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐹)) → ((𝐹 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) ↔ ((𝜑 ∧ (𝑄‘𝐼) ∈ ((-π[,)π) ∖ dom 𝐹)) → ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ≠ ∅)))
176		fourierdlem46.rlim	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐹)) → ((𝐹 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
177	175, 176	vtoclg 3525	. . . . . 6 ⊢ ((𝑄‘𝐼) ∈ ℝ → ((𝜑 ∧ (𝑄‘𝐼) ∈ ((-π[,)π) ∖ dom 𝐹)) → ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ≠ ∅))
178	152, 167, 177	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
179		ssn0 4360	. . . . 5 ⊢ ((((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ⊆ (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ∧ ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ≠ ∅) → (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
180	151, 178, 179	sylancr 587	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
181	150, 180	eqnetrd 3011	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
182	140, 181	pm2.61dan 811	. 2 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
183	65	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ ℝ^*)
184	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
185	25	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘𝐼) < (𝑄‘(𝐼 + 1)))
186		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑄‘(𝐼 + 1)) ∈ dom 𝐹 ∧ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 = (𝑄‘(𝐼 + 1)))
187		simpl 483	. . . . . . . . . . . . 13 ⊢ (((𝑄‘(𝐼 + 1)) ∈ dom 𝐹 ∧ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘(𝐼 + 1)) ∈ dom 𝐹)
188	186, 187	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ (((𝑄‘(𝐼 + 1)) ∈ dom 𝐹 ∧ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ dom 𝐹)
189	188	adantll 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) ∧ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ dom 𝐹)
190	189	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ dom 𝐹)
191	95	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
192	65	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘𝐼) ∈ ℝ^*)
193	23	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
194	65	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ^*)
195	22	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
196		iocssre 13344	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ) → ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ ℝ)
197	194, 195, 196	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ ℝ)
198		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))
199	197, 198	sseldd 3945	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ)
200	199	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ ℝ)
201	23	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
202		iocgtlb 43730	. . . . . . . . . . . . . . 15 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥)
203	194, 201, 198, 202	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥)
204	203	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘𝐼) < 𝑥)
205	22	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
206		iocleub 43731	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ≤ (𝑄‘(𝐼 + 1)))
207	194, 201, 198, 206	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ≤ (𝑄‘(𝐼 + 1)))
208	207	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ≤ (𝑄‘(𝐼 + 1)))
209		neqne 2951	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 = (𝑄‘(𝐼 + 1)) → 𝑥 ≠ (𝑄‘(𝐼 + 1)))
210	209	necomd 2999	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 = (𝑄‘(𝐼 + 1)) → (𝑄‘(𝐼 + 1)) ≠ 𝑥)
211	210	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘(𝐼 + 1)) ≠ 𝑥)
212	200, 205, 208, 211	leneltd 11309	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 < (𝑄‘(𝐼 + 1)))
213	192, 193, 200, 204, 212	eliood 43726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
214	191, 213	sseldd 3945	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ dom 𝐹)
215	214	adantllr 717	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ dom 𝐹)
216	190, 215	pm2.61dan 811	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ∈ dom 𝐹)
217	216	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ∀𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
218		dfss3 3932	. . . . . . . 8 ⊢ (((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
219	217, 218	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
220	123	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → 𝐹 ∈ (dom 𝐹–cn→ℂ))
221		rescncf 24260	. . . . . . 7 ⊢ (((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ dom 𝐹 → (𝐹 ∈ (dom 𝐹–cn→ℂ) → (𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∈ (((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))–cn→ℂ)))
222	219, 220, 221	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∈ (((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))–cn→ℂ))
223	183, 184, 185, 222	ioccncflimc 44116	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))‘(𝑄‘(𝐼 + 1))) ∈ (((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))))
224	22	leidd 11721	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ (𝑄‘(𝐼 + 1)))
225	65, 23, 23, 25, 224	eliocd 43735	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))
226		fvres 6861	. . . . . . . . 9 ⊢ ((𝑄‘(𝐼 + 1)) ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) → ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))‘(𝑄‘(𝐼 + 1))) = (𝐹‘(𝑄‘(𝐼 + 1))))
227	225, 226	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))‘(𝑄‘(𝐼 + 1))) = (𝐹‘(𝑄‘(𝐼 + 1))))
228	227	eqcomd 2742	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑄‘(𝐼 + 1))) = ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))‘(𝑄‘(𝐼 + 1))))
229		ioossioc 43720	. . . . . . . . . . 11 ⊢ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))
230		resabs1 5967	. . . . . . . . . . 11 ⊢ (((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) → ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
231	229, 230	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
232	231	eqcomi 2745	. . . . . . . . 9 ⊢ (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
233	232	oveq1i 7367	. . . . . . . 8 ⊢ ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) = (((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1)))
234	233	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) = (((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))))
235	228, 234	eleq12d 2832	. . . . . 6 ⊢ (𝜑 → ((𝐹‘(𝑄‘(𝐼 + 1))) ∈ ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ↔ ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))‘(𝑄‘(𝐼 + 1))) ∈ (((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1)))))
236	235	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝐹‘(𝑄‘(𝐼 + 1))) ∈ ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ↔ ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))‘(𝑄‘(𝐼 + 1))) ∈ (((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1)))))
237	223, 236	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝐹‘(𝑄‘(𝐼 + 1))) ∈ ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))))
238	237	ne0d 4295	. . 3 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
239		mnfxr 11212	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
240	239	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → -∞ ∈ ℝ^*)
241	17	mnfltd 13045	. . . . . . . . . 10 ⊢ (𝜑 → -∞ < (𝑄‘𝐼))
242	240, 65, 241	xrltled 13069	. . . . . . . . 9 ⊢ (𝜑 → -∞ ≤ (𝑄‘𝐼))
243		iooss1 13299	. . . . . . . . 9 ⊢ ((-∞ ∈ ℝ^* ∧ -∞ ≤ (𝑄‘𝐼)) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-∞(,)(𝑄‘(𝐼 + 1))))
244	239, 242, 243	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-∞(,)(𝑄‘(𝐼 + 1))))
245	244	resabs1d 5968	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
246	245	eqcomd 2742	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
247	246	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
248	247	oveq1d 7372	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) = (((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))))
249		limcresi 25249	. . . . 5 ⊢ ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ⊆ (((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1)))
250	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
251		simpl 483	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → 𝜑)
252	155	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → -π ∈ ℝ^*)
253	157	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → π ∈ ℝ^*)
254	23	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
255	4, 17, 22, 160, 25	lelttrd 11313	. . . . . . . . . 10 ⊢ (𝜑 → -π < (𝑄‘(𝐼 + 1)))
256	255	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → -π < (𝑄‘(𝐼 + 1)))
257	161	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ≤ π)
258	252, 253, 254, 256, 257	eliocd 43735	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ (-π(,]π))
259		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹)
260	258, 259	eldifd 3921	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ((-π(,]π) ∖ dom 𝐹))
261	251, 260	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ ((-π(,]π) ∖ dom 𝐹)))
262		eleq1 2825	. . . . . . . . 9 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → (𝑥 ∈ ((-π(,]π) ∖ dom 𝐹) ↔ (𝑄‘(𝐼 + 1)) ∈ ((-π(,]π) ∖ dom 𝐹)))
263	262	anbi2d 629	. . . . . . . 8 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐹)) ↔ (𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ ((-π(,]π) ∖ dom 𝐹))))
264		oveq2 7365	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → (-∞(,)𝑥) = (-∞(,)(𝑄‘(𝐼 + 1))))
265	264	reseq2d 5937	. . . . . . . . . 10 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → (𝐹 ↾ (-∞(,)𝑥)) = (𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))))
266		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → 𝑥 = (𝑄‘(𝐼 + 1)))
267	265, 266	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → ((𝐹 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) = ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))))
268	267	neeq1d 3003	. . . . . . . 8 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → (((𝐹 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅ ↔ ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅))
269	263, 268	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → (((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐹)) → ((𝐹 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) ↔ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ ((-π(,]π) ∖ dom 𝐹)) → ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)))
270		fourierdlem46.llim	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐹)) → ((𝐹 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
271	269, 270	vtoclg 3525	. . . . . 6 ⊢ ((𝑄‘(𝐼 + 1)) ∈ ℝ → ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ ((-π(,]π) ∖ dom 𝐹)) → ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅))
272	250, 261, 271	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
273		ssn0 4360	. . . . 5 ⊢ ((((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ⊆ (((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ∧ ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅) → (((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
274	249, 272, 273	sylancr 587	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
275	248, 274	eqnetrd 3011	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
276	238, 275	pm2.61dan 811	. 2 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
277	182, 276	jca 512	1 ⊢ (𝜑 → (((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅ ∧ ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 {ctp 4590 class class class wbr 5105 dom cdm 5633 ran crn 5634 ↾ cres 5635 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 Isom wiso 6497 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 +∞cpnf 11186 -∞cmnf 11187 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 -cneg 11386 ℤcz 12499 (,)cioo 13264 (,]cioc 13265 [,)cico 13266 [,]cicc 13267 ...cfz 13424 ..^cfzo 13567 πcpi 15949 –cn→ccncf 24239 lim_ℂ climc 25226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231

This theorem is referenced by: fourierdlem102 44439 fourierdlem114 44451

W3C validator