fourierdlem46 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem46 43368

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 25348	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
3	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → π ∈ ℝ)
4	3	renegcld 11259	. . . . . . . . . . 11 ⊢ (𝜑 → -π ∈ ℝ)
5		fourierdlem46.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6		tpssi 4749	. . . . . . . . . . 11 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ 𝐶 ∈ ℝ) → {-π, π, 𝐶} ⊆ ℝ)
7	4, 3, 5, 6	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → {-π, π, 𝐶} ⊆ ℝ)
8	4, 3	iccssred 13022	. . . . . . . . . . 11 ⊢ (𝜑 → (-π[,]π) ⊆ ℝ)
9	8	ssdifssd 4057	. . . . . . . . . 10 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐹) ⊆ ℝ)
10	7, 9	unssd 4100	. . . . . . . . 9 ⊢ (𝜑 → ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹)) ⊆ ℝ)
11	1, 10	eqsstrid 3949	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
12		fourierdlem46.qf	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶𝐻)
13		fourierdlem46.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀))
14		elfzofz 13258	. . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))
16	12, 15	ffvelrnd 6905	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝐼) ∈ 𝐻)
17	11, 16	sseldd 3902	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ)
18	17	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ ℝ)
19		fzofzp1 13339	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀))
20	13, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀))
21	12, 20	ffvelrnd 6905	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ 𝐻)
22	11, 21	sseldd 3902	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ)
23	22	rexrd 10883	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
24	23	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
25		fourierdlem46.10	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐼) < (𝑄‘(𝐼 + 1)))
26	25	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) < (𝑄‘(𝐼 + 1)))
27		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝑄‘𝐼) ∈ dom 𝐹 ∧ 𝑥 = (𝑄‘𝐼)) → 𝑥 = (𝑄‘𝐼))
28		simpl 486	. . . . . . . . . . . . 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 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) ∧ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
31	30	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
32		ssun2 4087	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-π[,]π) ∖ dom 𝐹) ⊆ ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹))
33	32, 1	sseqtrri 3938	. . . . . . . . . . . . . . . . . 18 ⊢ ((-π[,]π) ∖ dom 𝐹) ⊆ 𝐻
34		fourierdlem46.qiss	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-π(,)π))
35		ioossicc 13021	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (-π(,)π) ⊆ (-π[,]π)
36	34, 35	sstrdi 3913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-π[,]π))
37	36	sselda 3901	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ (-π[,]π))
38	37	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (-π[,]π))
39		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ dom 𝐹)
40	38, 39	eldifd 3877	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((-π[,]π) ∖ dom 𝐹))
41	33, 40	sseldi 3899	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ 𝐻)
42		fourierdlem46.ranq	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝑄 = 𝐻)
43	42	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐻 = ran 𝑄)
44	43	ad2antrr 726	. . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
47		ffn 6545	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑄:(0...𝑀)⟶𝐻 → 𝑄 Fn (0...𝑀))
48	12, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
49	48	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑄) → 𝑄 Fn (0...𝑀))
50		fvelrnb 6773	. . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑥)
53	52	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑥)
54		elfzelz 13112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
55	54	ad2antlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → 𝑗 ∈ ℤ)
56		simplll 775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → 𝜑)
57		simplr 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → 𝑗 ∈ (0...𝑀))
58		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ (𝑄‘𝑗) = 𝑥) → (𝑄‘𝑗) = 𝑥)
59		simplr 769	. . . . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
62		elfzoelz 13243	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ ℤ)
63	13, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐼 ∈ ℤ)
64	63	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐼 ∈ ℤ)
65	17	rexrd 10883	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ^*)
66	65	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ^*)
67	23	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
68		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
69		ioogtlb 42708	. . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
73	15	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐼 ∈ (0...𝑀))
74		simplr 769	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑗 ∈ (0...𝑀))
75		isorel 7135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝐼 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑀))) → (𝐼 < 𝑗 ↔ (𝑄‘𝐼) < (𝑄‘𝑗)))
76	72, 73, 74, 75	syl12anc 837	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝐼 < 𝑗 ↔ (𝑄‘𝐼) < (𝑄‘𝑗)))
77	70, 76	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐼 < 𝑗)
78		iooltub 42723	. . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝑗) < (𝑄‘(𝐼 + 1)))
80	20	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝐼 + 1) ∈ (0...𝑀))
81		isorel 7135	. . . . . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑗 < (𝐼 + 1) ↔ (𝑄‘𝑗) < (𝑄‘(𝐼 + 1))))
83	79, 82	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑗 < (𝐼 + 1))
84		btwnnz 12253	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼 ∈ ℤ ∧ 𝐼 < 𝑗 ∧ 𝑗 < (𝐼 + 1)) → ¬ 𝑗 ∈ ℤ)
85	64, 77, 83, 84	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → ¬ 𝑗 ∈ ℤ)
86	56, 57, 61, 85	syl21anc 838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → ¬ 𝑗 ∈ ℤ)
87	86	adantllr 719	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = 𝑥) → ¬ 𝑗 ∈ ℤ)
88	55, 87	pm2.65da 817	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) ∧ 𝑗 ∈ (0...𝑀)) → ¬ (𝑄‘𝑗) = 𝑥)
89	88	nrexdv 3189	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ 𝑥 ∈ ran 𝑄) → ¬ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = 𝑥)
90	53, 89	pm2.65da 817	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → ¬ 𝑥 ∈ ran 𝑄)
91	90	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝑄)
92	45, 91	condan 818	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ dom 𝐹)
93	92	ralrimiva 3105	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
94		dfss3 3888	. . . . . . . . . . . . . 14 ⊢ (((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
95	93, 94	sylibr 237	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
96	95	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
97	65	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) ∈ ℝ^*)
98	23	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
99		icossre 13016	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝐼) ∈ ℝ ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^*) → ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) ⊆ ℝ)
100	17, 23, 99	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) ⊆ ℝ)
101	100	sselda 3901	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ)
102	101	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ ℝ)
103	17	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) ∈ ℝ)
104	65	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ^*)
105	23	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
106		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))
107		icogelb 12986	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ≤ 𝑥)
108	104, 105, 106, 107	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ≤ 𝑥)
109	108	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) ≤ 𝑥)
110		neqne 2948	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 = (𝑄‘𝐼) → 𝑥 ≠ (𝑄‘𝐼))
111	110	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ≠ (𝑄‘𝐼))
112	103, 102, 109, 111	leneltd 10986	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → (𝑄‘𝐼) < 𝑥)
113		icoltub 42721	. . . . . . . . . . . . . . 15 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1)))
114	104, 105, 106, 113	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1)))
115	114	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 < (𝑄‘(𝐼 + 1)))
116	97, 98, 102, 112, 115	eliood 42711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
117	96, 116	sseldd 3902	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
118	117	adantllr 719	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘𝐼)) → 𝑥 ∈ dom 𝐹)
119	31, 118	pm2.61dan 813	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ dom 𝐹)
120	119	ralrimiva 3105	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ∀𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
121		dfss3 3888	. . . . . . . 8 ⊢ (((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
122	120, 121	sylibr 237	. . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → 𝐹 ∈ (dom 𝐹–cn→ℂ))
125		rescncf 23794	. . . . . . 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 43105	. . . . 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 11398	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝐼) ≤ (𝑄‘𝐼))
129	65, 23, 65, 128, 25	elicod 12985	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))
130		fvres 6736	. . . . . . . 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 2743	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝑄‘𝐼)) = ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))‘(𝑄‘𝐼)))
133	132	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → (𝐹‘(𝑄‘𝐼)) = ((𝐹 ↾ ((𝑄‘𝐼)[,)(𝑄‘(𝐼 + 1))))‘(𝑄‘𝐼)))
134		ioossico 13026	. . . . . . . . 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 5882	. . . . . . 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 2743	. . . . . 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 7228	. . . . 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 4250	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
141		pnfxr 10887	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
142	141	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → +∞ ∈ ℝ^*)
143	22	ltpnfd 12713	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) < +∞)
144	23, 142, 143	xrltled 12740	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ +∞)
145		iooss2 12971	. . . . . . . . 9 ⊢ ((+∞ ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ≤ +∞) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)(,)+∞))
146	141, 144, 145	sylancr 590	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)(,)+∞))
147	146	resabs1d 5882	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
148	147	oveq1d 7228	. . . . . 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 2743	. . . . 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 484	. . . 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 24782	. . . . 5 ⊢ ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ⊆ (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼))
152	17	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ ℝ)
153		simpl 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → 𝜑)
154	2	renegcli 11139	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
155	154	rexri 10891	. . . . . . . . . . 11 ⊢ -π ∈ ℝ^*
156	155	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → -π ∈ ℝ^*)
157	2	rexri 10891	. . . . . . . . . . 11 ⊢ π ∈ ℝ^*
158	157	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → π ∈ ℝ^*)
159	4, 3, 17, 22, 25, 34	fourierdlem10 43333	. . . . . . . . . . 11 ⊢ (𝜑 → (-π ≤ (𝑄‘𝐼) ∧ (𝑄‘(𝐼 + 1)) ≤ π))
160	159	simpld 498	. . . . . . . . . 10 ⊢ (𝜑 → -π ≤ (𝑄‘𝐼))
161	159	simprd 499	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ π)
162	17, 22, 3, 25, 161	ltletrd 10992	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘𝐼) < π)
163	156, 158, 65, 160, 162	elicod 12985	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (-π[,)π))
164	163	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ (-π[,)π))
165		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → ¬ (𝑄‘𝐼) ∈ dom 𝐹)
166	164, 165	eldifd 3877	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ ((-π[,)π) ∖ dom 𝐹))
167	153, 166	jca 515	. . . . . 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 632	. . . . . . . 8 ⊢ (𝑥 = (𝑄‘𝐼) → ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐹)) ↔ (𝜑 ∧ (𝑄‘𝐼) ∈ ((-π[,)π) ∖ dom 𝐹))))
170		oveq1 7220	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑄‘𝐼) → (𝑥(,)+∞) = ((𝑄‘𝐼)(,)+∞))
171	170	reseq2d 5851	. . . . . . . . . 10 ⊢ (𝑥 = (𝑄‘𝐼) → (𝐹 ↾ (𝑥(,)+∞)) = (𝐹 ↾ ((𝑄‘𝐼)(,)+∞)))
172		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (𝑄‘𝐼) → 𝑥 = (𝑄‘𝐼))
173	171, 172	oveq12d 7231	. . . . . . . . 9 ⊢ (𝑥 = (𝑄‘𝐼) → ((𝐹 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) = ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)))
174	173	neeq1d 3000	. . . . . . . 8 ⊢ (𝑥 = (𝑄‘𝐼) → (((𝐹 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅ ↔ ((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) lim_ℂ (𝑄‘𝐼)) ≠ ∅))
175	169, 174	imbi12d 348	. . . . . . 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 3481	. . . . . 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 4315	. . . . 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 590	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → (((𝐹 ↾ ((𝑄‘𝐼)(,)+∞)) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
181	150, 180	eqnetrd 3008	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝐼) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
182	140, 181	pm2.61dan 813	. 2 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅)
183	65	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘𝐼) ∈ ℝ^*)
184	22	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
185	25	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘𝐼) < (𝑄‘(𝐼 + 1)))
186		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝑄‘(𝐼 + 1)) ∈ dom 𝐹 ∧ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 = (𝑄‘(𝐼 + 1)))
187		simpl 486	. . . . . . . . . . . . 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 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) ∧ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ dom 𝐹)
190	189	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ dom 𝐹)
191	95	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
192	65	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘𝐼) ∈ ℝ^*)
193	23	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
194	65	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ^*)
195	22	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
196		iocssre 13015	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ) → ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ ℝ)
197	194, 195, 196	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ ℝ)
198		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))
199	197, 198	sseldd 3902	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ)
200	199	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ ℝ)
201	23	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
202		iocgtlb 42715	. . . . . . . . . . . . . . 15 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥)
203	194, 201, 198, 202	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥)
204	203	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘𝐼) < 𝑥)
205	22	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
206		iocleub 42716	. . . . . . . . . . . . . . . 16 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ≤ (𝑄‘(𝐼 + 1)))
207	194, 201, 198, 206	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ≤ (𝑄‘(𝐼 + 1)))
208	207	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ≤ (𝑄‘(𝐼 + 1)))
209		neqne 2948	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 = (𝑄‘(𝐼 + 1)) → 𝑥 ≠ (𝑄‘(𝐼 + 1)))
210	209	necomd 2996	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 = (𝑄‘(𝐼 + 1)) → (𝑄‘(𝐼 + 1)) ≠ 𝑥)
211	210	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → (𝑄‘(𝐼 + 1)) ≠ 𝑥)
212	200, 205, 208, 211	leneltd 10986	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 < (𝑄‘(𝐼 + 1)))
213	192, 193, 200, 204, 212	eliood 42711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
214	191, 213	sseldd 3902	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ∧ ¬ 𝑥 = (𝑄‘(𝐼 + 1))) → 𝑥 ∈ dom 𝐹)
215	214	adantllr 719	. . . . . . . . . 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 813	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) ∧ 𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) → 𝑥 ∈ dom 𝐹)
217	216	ralrimiva 3105	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ∀𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
218		dfss3 3888	. . . . . . . 8 ⊢ (((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))𝑥 ∈ dom 𝐹)
219	217, 218	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))) ⊆ dom 𝐹)
220	123	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → 𝐹 ∈ (dom 𝐹–cn→ℂ))
221		rescncf 23794	. . . . . . 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 43101	. . . . 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 11398	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ (𝑄‘(𝐼 + 1)))
225	65, 23, 23, 25, 224	eliocd 42720	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))
226		fvres 6736	. . . . . . . . 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 2743	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑄‘(𝐼 + 1))) = ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1))))‘(𝑄‘(𝐼 + 1))))
229		ioossioc 42705	. . . . . . . . . . 11 ⊢ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))
230		resabs1 5881	. . . . . . . . . . 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 2746	. . . . . . . . 9 ⊢ (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = ((𝐹 ↾ ((𝑄‘𝐼)(,](𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
233	232	oveq1i 7223	. . . . . . . 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 484	. . . . 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 260	. . . 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 4250	. . 3 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
239		mnfxr 10890	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
240	239	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → -∞ ∈ ℝ^*)
241	17	mnfltd 12716	. . . . . . . . . 10 ⊢ (𝜑 → -∞ < (𝑄‘𝐼))
242	240, 65, 241	xrltled 12740	. . . . . . . . 9 ⊢ (𝜑 → -∞ ≤ (𝑄‘𝐼))
243		iooss1 12970	. . . . . . . . 9 ⊢ ((-∞ ∈ ℝ^* ∧ -∞ ≤ (𝑄‘𝐼)) → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-∞(,)(𝑄‘(𝐼 + 1))))
244	239, 242, 243	sylancr 590	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-∞(,)(𝑄‘(𝐼 + 1))))
245	244	resabs1d 5882	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
246	245	eqcomd 2743	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) = ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
247	246	adantr 484	. . . . 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 7228	. . . 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 24782	. . . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
251		simpl 486	. . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
255	4, 17, 22, 160, 25	lelttrd 10990	. . . . . . . . . 10 ⊢ (𝜑 → -π < (𝑄‘(𝐼 + 1)))
256	255	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → -π < (𝑄‘(𝐼 + 1)))
257	161	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ≤ π)
258	252, 253, 254, 256, 257	eliocd 42720	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ (-π(,]π))
259		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹)
260	258, 259	eldifd 3877	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → (𝑄‘(𝐼 + 1)) ∈ ((-π(,]π) ∖ dom 𝐹))
261	251, 260	jca 515	. . . . . 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 632	. . . . . . . 8 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐹)) ↔ (𝜑 ∧ (𝑄‘(𝐼 + 1)) ∈ ((-π(,]π) ∖ dom 𝐹))))
264		oveq2 7221	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → (-∞(,)𝑥) = (-∞(,)(𝑄‘(𝐼 + 1))))
265	264	reseq2d 5851	. . . . . . . . . 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 7231	. . . . . . . . 9 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → ((𝐹 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) = ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))))
268	267	neeq1d 3000	. . . . . . . 8 ⊢ (𝑥 = (𝑄‘(𝐼 + 1)) → (((𝐹 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅ ↔ ((𝐹 ↾ (-∞(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅))
269	263, 268	imbi12d 348	. . . . . . 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 3481	. . . . . 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 4315	. . . . 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 590	. . . 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 3008	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑄‘(𝐼 + 1)) ∈ dom 𝐹) → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
276	238, 275	pm2.61dan 813	. 2 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅)
277	182, 276	jca 515	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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 ∖ cdif 3863 ∪ cun 3864 ⊆ wss 3866 ∅c0 4237 {ctp 4545 class class class wbr 5053 dom cdm 5551 ran crn 5552 ↾ cres 5553 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 Isom wiso 6381 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 +∞cpnf 10864 -∞cmnf 10865 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 -cneg 11063 ℤcz 12176 (,)cioo 12935 (,]cioc 12936 [,)cico 12937 [,]cicc 12938 ...cfz 13095 ..^cfzo 13238 πcpi 15628 –cn→ccncf 23773 lim_ℂ climc 24759

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629 df-sin 15631 df-cos 15632 df-pi 15634 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764

This theorem is referenced by: fourierdlem102 43424 fourierdlem114 43436

W3C validator