fourierdlem102 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem102 43639

Description: For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem102.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem102.t	⊢ 𝑇 = (2 · π)
fourierdlem102.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem102.g	⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
fourierdlem102.dmdv	⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
fourierdlem102.gcn	⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
fourierdlem102.rlim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fourierdlem102.llim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
fourierdlem102.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem102.p	⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem102.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem102.h	⊢ 𝐻 = ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))
fourierdlem102.m	⊢ 𝑀 = ((♯‘𝐻) − 1)
fourierdlem102.q	⊢ 𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻))

**Assertion**
Ref	Expression
fourierdlem102	⊢ (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅))

Distinct variable groups: 𝑥,𝐸 𝑖,𝐹,𝑛,𝑥 𝑖,𝐺,𝑥 𝑔,𝐻 𝑔,𝑀 𝑖,𝑀,𝑛,𝑝 𝑥,𝑀 𝑄,𝑔 𝑄,𝑖,𝑛,𝑝 𝑥,𝑄 𝑇,𝑖,𝑛,𝑝 𝑥,𝑇 𝑖,𝑋,𝑛,𝑝 𝑥,𝑋 𝜑,𝑔 𝜑,𝑖,𝑛,𝑥 Allowed substitution hints: 𝜑(𝑝) 𝑃(𝑥,𝑔,𝑖,𝑛,𝑝) 𝑇(𝑔) 𝐸(𝑔,𝑖,𝑛,𝑝) 𝐹(𝑔,𝑝) 𝐺(𝑔,𝑛,𝑝) 𝐻(𝑥,𝑖,𝑛,𝑝) 𝑋(𝑔)

**Proof of Theorem fourierdlem102**
Step	Hyp	Ref	Expression
1		fourierdlem102.f	. 2 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		fourierdlem102.t	. 2 ⊢ 𝑇 = (2 · π)
3		fourierdlem102.per	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
4		fourierdlem102.x	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5		fourierdlem102.p	. 2 ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
6		fourierdlem102.m	. . 3 ⊢ 𝑀 = ((♯‘𝐻) − 1)
7		2z 12282	. . . . . 6 ⊢ 2 ∈ ℤ
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
9		fourierdlem102.h	. . . . . . . 8 ⊢ 𝐻 = ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))
10		tpfi 9020	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ∈ Fin
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ∈ Fin)
12		pire 25520	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
13	12	renegcli 11212	. . . . . . . . . . . . . 14 ⊢ -π ∈ ℝ
14	13	rexri 10964	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ^*
15	12	rexri 10964	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ^*
16		negpilt0 42708	. . . . . . . . . . . . . . 15 ⊢ -π < 0
17		pipos 25522	. . . . . . . . . . . . . . 15 ⊢ 0 < π
18		0re 10908	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
19	13, 18, 12	lttri 11031	. . . . . . . . . . . . . . 15 ⊢ ((-π < 0 ∧ 0 < π) → -π < π)
20	16, 17, 19	mp2an 688	. . . . . . . . . . . . . 14 ⊢ -π < π
21	13, 12, 20	ltleii 11028	. . . . . . . . . . . . 13 ⊢ -π ≤ π
22		prunioo 13142	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → ((-π(,)π) ∪ {-π, π}) = (-π[,]π))
23	14, 15, 21, 22	mp3an 1459	. . . . . . . . . . . 12 ⊢ ((-π(,)π) ∪ {-π, π}) = (-π[,]π)
24	23	difeq1i 4049	. . . . . . . . . . 11 ⊢ (((-π(,)π) ∪ {-π, π}) ∖ dom 𝐺) = ((-π[,]π) ∖ dom 𝐺)
25		difundir 4211	. . . . . . . . . . 11 ⊢ (((-π(,)π) ∪ {-π, π}) ∖ dom 𝐺) = (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺))
26	24, 25	eqtr3i 2768	. . . . . . . . . 10 ⊢ ((-π[,]π) ∖ dom 𝐺) = (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺))
27		fourierdlem102.dmdv	. . . . . . . . . . 11 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
28		prfi 9019	. . . . . . . . . . . 12 ⊢ {-π, π} ∈ Fin
29		diffi 8979	. . . . . . . . . . . 12 ⊢ ({-π, π} ∈ Fin → ({-π, π} ∖ dom 𝐺) ∈ Fin)
30	28, 29	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → ({-π, π} ∖ dom 𝐺) ∈ Fin)
31		unfi 8917	. . . . . . . . . . 11 ⊢ ((((-π(,)π) ∖ dom 𝐺) ∈ Fin ∧ ({-π, π} ∖ dom 𝐺) ∈ Fin) → (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺)) ∈ Fin)
32	27, 30, 31	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺)) ∈ Fin)
33	26, 32	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ∈ Fin)
34		unfi 8917	. . . . . . . . 9 ⊢ (({-π, π, (𝐸‘𝑋)} ∈ Fin ∧ ((-π[,]π) ∖ dom 𝐺) ∈ Fin) → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ Fin)
35	11, 33, 34	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ Fin)
36	9, 35	eqeltrid 2843	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Fin)
37		hashcl 13999	. . . . . . 7 ⊢ (𝐻 ∈ Fin → (♯‘𝐻) ∈ ℕ₀)
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) ∈ ℕ₀)
39	38	nn0zd 12353	. . . . 5 ⊢ (𝜑 → (♯‘𝐻) ∈ ℤ)
40	13, 20	ltneii 11018	. . . . . . 7 ⊢ -π ≠ π
41		hashprg 14038	. . . . . . . 8 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π ≠ π ↔ (♯‘{-π, π}) = 2))
42	13, 12, 41	mp2an 688	. . . . . . 7 ⊢ (-π ≠ π ↔ (♯‘{-π, π}) = 2)
43	40, 42	mpbi 229	. . . . . 6 ⊢ (♯‘{-π, π}) = 2
44	10	elexi 3441	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ∈ V
45		ovex 7288	. . . . . . . . . . 11 ⊢ (-π[,]π) ∈ V
46		difexg 5246	. . . . . . . . . . 11 ⊢ ((-π[,]π) ∈ V → ((-π[,]π) ∖ dom 𝐺) ∈ V)
47	45, 46	ax-mp 5	. . . . . . . . . 10 ⊢ ((-π[,]π) ∖ dom 𝐺) ∈ V
48	44, 47	unex 7574	. . . . . . . . 9 ⊢ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ V
49	9, 48	eqeltri 2835	. . . . . . . 8 ⊢ 𝐻 ∈ V
50		negex 11149	. . . . . . . . . . 11 ⊢ -π ∈ V
51	50	tpid1 4701	. . . . . . . . . 10 ⊢ -π ∈ {-π, π, (𝐸‘𝑋)}
52	12	elexi 3441	. . . . . . . . . . 11 ⊢ π ∈ V
53	52	tpid2 4703	. . . . . . . . . 10 ⊢ π ∈ {-π, π, (𝐸‘𝑋)}
54		prssi 4751	. . . . . . . . . 10 ⊢ ((-π ∈ {-π, π, (𝐸‘𝑋)} ∧ π ∈ {-π, π, (𝐸‘𝑋)}) → {-π, π} ⊆ {-π, π, (𝐸‘𝑋)})
55	51, 53, 54	mp2an 688	. . . . . . . . 9 ⊢ {-π, π} ⊆ {-π, π, (𝐸‘𝑋)}
56		ssun1 4102	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ⊆ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))
57	56, 9	sseqtrri 3954	. . . . . . . . 9 ⊢ {-π, π, (𝐸‘𝑋)} ⊆ 𝐻
58	55, 57	sstri 3926	. . . . . . . 8 ⊢ {-π, π} ⊆ 𝐻
59		hashss 14052	. . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ {-π, π} ⊆ 𝐻) → (♯‘{-π, π}) ≤ (♯‘𝐻))
60	49, 58, 59	mp2an 688	. . . . . . 7 ⊢ (♯‘{-π, π}) ≤ (♯‘𝐻)
61	60	a1i 11	. . . . . 6 ⊢ (𝜑 → (♯‘{-π, π}) ≤ (♯‘𝐻))
62	43, 61	eqbrtrrid 5106	. . . . 5 ⊢ (𝜑 → 2 ≤ (♯‘𝐻))
63		eluz2 12517	. . . . 5 ⊢ ((♯‘𝐻) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (♯‘𝐻) ∈ ℤ ∧ 2 ≤ (♯‘𝐻)))
64	8, 39, 62, 63	syl3anbrc 1341	. . . 4 ⊢ (𝜑 → (♯‘𝐻) ∈ (ℤ_≥‘2))
65		uz2m1nn 12592	. . . 4 ⊢ ((♯‘𝐻) ∈ (ℤ_≥‘2) → ((♯‘𝐻) − 1) ∈ ℕ)
66	64, 65	syl 17	. . 3 ⊢ (𝜑 → ((♯‘𝐻) − 1) ∈ ℕ)
67	6, 66	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
68	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → -π ∈ ℝ)
69	12	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → π ∈ ℝ)
70		negpitopissre 25601	. . . . . . . . . . . 12 ⊢ (-π(,]π) ⊆ ℝ
71	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → -π < π)
72		picn 25521	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℂ
73	72	2timesi 12041	. . . . . . . . . . . . . . 15 ⊢ (2 · π) = (π + π)
74	72, 72	subnegi 11230	. . . . . . . . . . . . . . 15 ⊢ (π − -π) = (π + π)
75	73, 2, 74	3eqtr4i 2776	. . . . . . . . . . . . . 14 ⊢ 𝑇 = (π − -π)
76		fourierdlem102.e	. . . . . . . . . . . . . 14 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
77	68, 69, 71, 75, 76	fourierdlem4 43542	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸:ℝ⟶(-π(,]π))
78	77, 4	ffvelrnd 6944	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (-π(,]π))
79	70, 78	sselid 3915	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ)
80	68, 69, 79	3jca 1126	. . . . . . . . . 10 ⊢ (𝜑 → (-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝐸‘𝑋) ∈ ℝ))
81		fvex 6769	. . . . . . . . . . 11 ⊢ (𝐸‘𝑋) ∈ V
82	50, 52, 81	tpss 4765	. . . . . . . . . 10 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝐸‘𝑋) ∈ ℝ) ↔ {-π, π, (𝐸‘𝑋)} ⊆ ℝ)
83	80, 82	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ⊆ ℝ)
84		iccssre 13090	. . . . . . . . . . 11 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
85	13, 12, 84	mp2an 688	. . . . . . . . . 10 ⊢ (-π[,]π) ⊆ ℝ
86		ssdifss 4066	. . . . . . . . . 10 ⊢ ((-π[,]π) ⊆ ℝ → ((-π[,]π) ∖ dom 𝐺) ⊆ ℝ)
87	85, 86	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ⊆ ℝ)
88	83, 87	unssd 4116	. . . . . . . 8 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ⊆ ℝ)
89	9, 88	eqsstrid 3965	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
90		fourierdlem102.q	. . . . . . 7 ⊢ 𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻))
91	36, 89, 90, 6	fourierdlem36 43574	. . . . . 6 ⊢ (𝜑 → 𝑄 Isom < , < ((0...𝑀), 𝐻))
92		isof1o 7174	. . . . . 6 ⊢ (𝑄 Isom < , < ((0...𝑀), 𝐻) → 𝑄:(0...𝑀)–1-1-onto→𝐻)
93		f1of 6700	. . . . . 6 ⊢ (𝑄:(0...𝑀)–1-1-onto→𝐻 → 𝑄:(0...𝑀)⟶𝐻)
94	91, 92, 93	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶𝐻)
95	94, 89	fssd 6602	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
96		reex 10893	. . . . 5 ⊢ ℝ ∈ V
97		ovex 7288	. . . . 5 ⊢ (0...𝑀) ∈ V
98	96, 97	elmap 8617	. . . 4 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
99	95, 98	sylibr 233	. . 3 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
100		fveq2 6756	. . . . . . . . . . 11 ⊢ (0 = 𝑖 → (𝑄‘0) = (𝑄‘𝑖))
101	100	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘0) = (𝑄‘𝑖))
102	95	ffvelrnda 6943	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
103	102	leidd 11471	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
104	103	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
105	101, 104	eqbrtrd 5092	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
106		elfzelz 13185	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
107	106	zred 12355	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
108	107	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 𝑖 ∈ ℝ)
109		elfzle1 13188	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
110	109	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 0 ≤ 𝑖)
111		neqne 2950	. . . . . . . . . . . . 13 ⊢ (¬ 0 = 𝑖 → 0 ≠ 𝑖)
112	111	necomd 2998	. . . . . . . . . . . 12 ⊢ (¬ 0 = 𝑖 → 𝑖 ≠ 0)
113	112	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 𝑖 ≠ 0)
114	108, 110, 113	ne0gt0d 11042	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 0 < 𝑖)
115		nnssnn0 12166	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ⊆ ℕ₀
116		nn0uz 12549	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
117	115, 116	sseqtri 3953	. . . . . . . . . . . . . . . 16 ⊢ ℕ ⊆ (ℤ_≥‘0)
118	117, 67	sselid 3915	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
119		eluzfz1 13192	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
120	118, 119	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (0...𝑀))
121	94, 120	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘0) ∈ 𝐻)
122	89, 121	sseldd 3918	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
123	122	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) ∈ ℝ)
124	102	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘𝑖) ∈ ℝ)
125		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → 0 < 𝑖)
126	91	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
127	120	anim1i 614	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀)))
128	127	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀)))
129		isorel 7177	. . . . . . . . . . . . 13 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀))) → (0 < 𝑖 ↔ (𝑄‘0) < (𝑄‘𝑖)))
130	126, 128, 129	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (0 < 𝑖 ↔ (𝑄‘0) < (𝑄‘𝑖)))
131	125, 130	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) < (𝑄‘𝑖))
132	123, 124, 131	ltled 11053	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
133	114, 132	syldan 590	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
134	105, 133	pm2.61dan 809	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄‘𝑖))
135	134	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ≤ (𝑄‘𝑖))
136		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘𝑖) = -π)
137	135, 136	breqtrd 5096	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ≤ -π)
138	68	rexrd 10956	. . . . . . . 8 ⊢ (𝜑 → -π ∈ ℝ^*)
139	69	rexrd 10956	. . . . . . . 8 ⊢ (𝜑 → π ∈ ℝ^*)
140		lbicc2 13125	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → -π ∈ (-π[,]π))
141	14, 15, 21, 140	mp3an 1459	. . . . . . . . . . . . 13 ⊢ -π ∈ (-π[,]π)
142	141	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -π ∈ (-π[,]π))
143		ubicc2 13126	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → π ∈ (-π[,]π))
144	14, 15, 21, 143	mp3an 1459	. . . . . . . . . . . . 13 ⊢ π ∈ (-π[,]π)
145	144	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → π ∈ (-π[,]π))
146		iocssicc 13098	. . . . . . . . . . . . 13 ⊢ (-π(,]π) ⊆ (-π[,]π)
147	146, 78	sselid 3915	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (-π[,]π))
148		tpssi 4766	. . . . . . . . . . . 12 ⊢ ((-π ∈ (-π[,]π) ∧ π ∈ (-π[,]π) ∧ (𝐸‘𝑋) ∈ (-π[,]π)) → {-π, π, (𝐸‘𝑋)} ⊆ (-π[,]π))
149	142, 145, 147, 148	syl3anc 1369	. . . . . . . . . . 11 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ⊆ (-π[,]π))
150		difssd 4063	. . . . . . . . . . 11 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ⊆ (-π[,]π))
151	149, 150	unssd 4116	. . . . . . . . . 10 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ⊆ (-π[,]π))
152	9, 151	eqsstrid 3965	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (-π[,]π))
153	152, 121	sseldd 3918	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘0) ∈ (-π[,]π))
154		iccgelb 13064	. . . . . . . 8 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑄‘0) ∈ (-π[,]π)) → -π ≤ (𝑄‘0))
155	138, 139, 153, 154	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → -π ≤ (𝑄‘0))
156	155	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → -π ≤ (𝑄‘0))
157	122	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ∈ ℝ)
158	13	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → -π ∈ ℝ)
159	157, 158	letri3d 11047	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → ((𝑄‘0) = -π ↔ ((𝑄‘0) ≤ -π ∧ -π ≤ (𝑄‘0))))
160	137, 156, 159	mpbir2and 709	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) = -π)
161	57, 51	sselii 3914	. . . . . . 7 ⊢ -π ∈ 𝐻
162		f1ofo 6707	. . . . . . . . 9 ⊢ (𝑄:(0...𝑀)–1-1-onto→𝐻 → 𝑄:(0...𝑀)–onto→𝐻)
163	92, 162	syl 17	. . . . . . . 8 ⊢ (𝑄 Isom < , < ((0...𝑀), 𝐻) → 𝑄:(0...𝑀)–onto→𝐻)
164		forn 6675	. . . . . . . 8 ⊢ (𝑄:(0...𝑀)–onto→𝐻 → ran 𝑄 = 𝐻)
165	91, 163, 164	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝑄 = 𝐻)
166	161, 165	eleqtrrid 2846	. . . . . 6 ⊢ (𝜑 → -π ∈ ran 𝑄)
167		ffn 6584	. . . . . . 7 ⊢ (𝑄:(0...𝑀)⟶𝐻 → 𝑄 Fn (0...𝑀))
168		fvelrnb 6812	. . . . . . 7 ⊢ (𝑄 Fn (0...𝑀) → (-π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π))
169	94, 167, 168	3syl 18	. . . . . 6 ⊢ (𝜑 → (-π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π))
170	166, 169	mpbid 231	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π)
171	160, 170	r19.29a 3217	. . . 4 ⊢ (𝜑 → (𝑄‘0) = -π)
172	57, 53	sselii 3914	. . . . . . 7 ⊢ π ∈ 𝐻
173	172, 165	eleqtrrid 2846	. . . . . 6 ⊢ (𝜑 → π ∈ ran 𝑄)
174		fvelrnb 6812	. . . . . . 7 ⊢ (𝑄 Fn (0...𝑀) → (π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π))
175	94, 167, 174	3syl 18	. . . . . 6 ⊢ (𝜑 → (π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π))
176	173, 175	mpbid 231	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π)
177	94, 152	fssd 6602	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
178		eluzfz2 13193	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
179	118, 178	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
180	177, 179	ffvelrnd 6944	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (-π[,]π))
181		iccleub 13063	. . . . . . . . 9 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑄‘𝑀) ∈ (-π[,]π)) → (𝑄‘𝑀) ≤ π)
182	138, 139, 180, 181	syl3anc 1369	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝑀) ≤ π)
183	182	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) ≤ π)
184		id 22	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖) = π → (𝑄‘𝑖) = π)
185	184	eqcomd 2744	. . . . . . . . 9 ⊢ ((𝑄‘𝑖) = π → π = (𝑄‘𝑖))
186	185	3ad2ant3 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π = (𝑄‘𝑖))
187	103	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
188		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
189	188	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) = (𝑄‘𝑀))
190	187, 189	breqtrd 5096	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
191	107	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 ∈ ℝ)
192		elfzel2 13183	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
193	192	zred 12355	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
194	193	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑀 ∈ ℝ)
195		elfzle2 13189	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ≤ 𝑀)
196	195	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 ≤ 𝑀)
197		neqne 2950	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑖 = 𝑀 → 𝑖 ≠ 𝑀)
198	197	necomd 2998	. . . . . . . . . . . . 13 ⊢ (¬ 𝑖 = 𝑀 → 𝑀 ≠ 𝑖)
199	198	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑀 ≠ 𝑖)
200	191, 194, 196, 199	leneltd 11059	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 < 𝑀)
201	102	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) ∈ ℝ)
202	85, 180	sselid 3915	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
203	202	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑀) ∈ ℝ)
204		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
205	91	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
206		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
207	179	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (0...𝑀))
208	206, 207	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)))
209	208	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)))
210		isorel 7177	. . . . . . . . . . . . . 14 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀))) → (𝑖 < 𝑀 ↔ (𝑄‘𝑖) < (𝑄‘𝑀)))
211	205, 209, 210	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑖 < 𝑀 ↔ (𝑄‘𝑖) < (𝑄‘𝑀)))
212	204, 211	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) < (𝑄‘𝑀))
213	201, 203, 212	ltled 11053	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
214	200, 213	syldan 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
215	190, 214	pm2.61dan 809	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
216	215	3adant3 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
217	186, 216	eqbrtrd 5092	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π ≤ (𝑄‘𝑀))
218	202	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) ∈ ℝ)
219	12	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π ∈ ℝ)
220	218, 219	letri3d 11047	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → ((𝑄‘𝑀) = π ↔ ((𝑄‘𝑀) ≤ π ∧ π ≤ (𝑄‘𝑀))))
221	183, 217, 220	mpbir2and 709	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) = π)
222	221	rexlimdv3a 3214	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π → (𝑄‘𝑀) = π))
223	176, 222	mpd 15	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = π)
224		elfzoelz 13316	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
225	224	zred 12355	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
226	225	ltp1d 11835	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
227	226	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 < (𝑖 + 1))
228		elfzofz 13331	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
229		fzofzp1 13412	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
230	228, 229	jca 511	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 ∈ (0...𝑀) ∧ (𝑖 + 1) ∈ (0...𝑀)))
231		isorel 7177	. . . . . . 7 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝑖 ∈ (0...𝑀) ∧ (𝑖 + 1) ∈ (0...𝑀))) → (𝑖 < (𝑖 + 1) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
232	91, 230, 231	syl2an 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 < (𝑖 + 1) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
233	227, 232	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
234	233	ralrimiva 3107	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
235	171, 223, 234	jca31 514	. . 3 ⊢ (𝜑 → (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
236	5	fourierdlem2 43540	. . . 4 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
237	67, 236	syl 17	. . 3 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
238	99, 235, 237	mpbir2and 709	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
239		fourierdlem102.g	. . . . 5 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
240	239	reseq1i 5876	. . . 4 ⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
241	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
242	15	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
243	177	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
244		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
245	241, 242, 243, 244	fourierdlem27 43565	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π(,)π))
246	245	resabs1d 5911	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
247	240, 246	eqtr2id 2792	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
248		fourierdlem102.gcn	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
249	248, 5, 67, 238, 9, 165	fourierdlem38 43576	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
250	247, 249	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
251	247	oveq1d 7270	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
252	248	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 ∈ (dom 𝐺–cn→ℂ))
253		fourierdlem102.rlim	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
254	253	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
255		fourierdlem102.llim	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
256	255	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
257	91	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
258	257, 92, 93	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶𝐻)
259	79	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) ∈ ℝ)
260	257, 163, 164	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ran 𝑄 = 𝐻)
261	252, 254, 256, 257, 258, 244, 233, 245, 259, 9, 260	fourierdlem46 43583	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅ ∧ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅))
262	261	simpld 494	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
263	251, 262	eqnetrd 3010	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
264	247	oveq1d 7270	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
265	261	simprd 495	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
266	264, 265	eqnetrd 3010	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
267	1, 2, 3, 4, 5, 67, 238, 250, 263, 266	fourierdlem94 43631	1 ⊢ (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ∅c0 4253 {cpr 4560 {ctp 4562 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ran crn 5581 ↾ cres 5582 ℩cio 6374 Fn wfn 6413 ⟶wf 6414 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 Isom wiso 6419 (class class class)co 7255 ↑_m cmap 8573 Fincfn 8691 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 (,)cioo 13008 (,]cioc 13009 [,)cico 13010 [,]cicc 13011 ...cfz 13168 ..^cfzo 13311 ⌊cfl 13438 ♯chash 13972 πcpi 15704 –cn→ccncf 23945 lim_ℂ climc 24931 D cdv 24932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936

This theorem is referenced by: fourierdlem106 43643

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