fourierdlem102 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem102 46807

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 12622	. . . . . 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 9281	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ∈ Fin
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ∈ Fin)
12		pire 26581	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
13	12	renegcli 11515	. . . . . . . . . . . . . 14 ⊢ -π ∈ ℝ
14	13	rexri 11263	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ^*
15	12	rexri 11263	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ^*
16		negpilt0 45885	. . . . . . . . . . . . . . 15 ⊢ -π < 0
17		pipos 26585	. . . . . . . . . . . . . . 15 ⊢ 0 < π
18		0re 11206	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
19	13, 18, 12	lttri 11332	. . . . . . . . . . . . . . 15 ⊢ ((-π < 0 ∧ 0 < π) → -π < π)
20	16, 17, 19	mp2an 704	. . . . . . . . . . . . . 14 ⊢ -π < π
21	13, 12, 20	ltleii 11329	. . . . . . . . . . . . 13 ⊢ -π ≤ π
22		prunioo 13504	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → ((-π(,)π) ∪ {-π, π}) = (-π[,]π))
23	14, 15, 21, 22	mp3an 1487	. . . . . . . . . . . 12 ⊢ ((-π(,)π) ∪ {-π, π}) = (-π[,]π)
24	23	difeq1i 4085	. . . . . . . . . . 11 ⊢ (((-π(,)π) ∪ {-π, π}) ∖ dom 𝐺) = ((-π[,]π) ∖ dom 𝐺)
25		difundir 4252	. . . . . . . . . . 11 ⊢ (((-π(,)π) ∪ {-π, π}) ∖ dom 𝐺) = (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺))
26	24, 25	eqtr3i 2794	. . . . . . . . . 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 9279	. . . . . . . . . . . 12 ⊢ {-π, π} ∈ Fin
29		diffi 9155	. . . . . . . . . . . 12 ⊢ ({-π, π} ∈ Fin → ({-π, π} ∖ dom 𝐺) ∈ Fin)
30	28, 29	mp1i 14	. . . . . . . . . . 11 ⊢ (𝜑 → ({-π, π} ∖ dom 𝐺) ∈ Fin)
31		unfi 9151	. . . . . . . . . . 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 595	. . . . . . . . . 10 ⊢ (𝜑 → (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺)) ∈ Fin)
33	26, 32	eqeltrid 2873	. . . . . . . . 9 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ∈ Fin)
34		unfi 9151	. . . . . . . . 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 595	. . . . . . . 8 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ Fin)
36	9, 35	eqeltrid 2873	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Fin)
37		hashcl 14388	. . . . . . 7 ⊢ (𝐻 ∈ Fin → (♯‘𝐻) ∈ ℕ₀)
38	36, 37	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) ∈ ℕ₀)
39	38	nn0zd 12612	. . . . 5 ⊢ (𝜑 → (♯‘𝐻) ∈ ℤ)
40	13, 20	ltneii 11319	. . . . . . 7 ⊢ -π ≠ π
41		hashprg 14427	. . . . . . . 8 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π ≠ π ↔ (♯‘{-π, π}) = 2))
42	13, 12, 41	mp2an 704	. . . . . . 7 ⊢ (-π ≠ π ↔ (♯‘{-π, π}) = 2)
43	40, 42	mpbi 233	. . . . . 6 ⊢ (♯‘{-π, π}) = 2
44	10	elexi 3485	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ∈ V
45		ovex 7441	. . . . . . . . . . 11 ⊢ (-π[,]π) ∈ V
46		difexg 5297	. . . . . . . . . . 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 7739	. . . . . . . . 9 ⊢ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ V
49	9, 48	eqeltri 2865	. . . . . . . 8 ⊢ 𝐻 ∈ V
50		negex 11451	. . . . . . . . . . 11 ⊢ -π ∈ V
51	50	tpid1 4736	. . . . . . . . . 10 ⊢ -π ∈ {-π, π, (𝐸‘𝑋)}
52	12	elexi 3485	. . . . . . . . . . 11 ⊢ π ∈ V
53	52	tpid2 4738	. . . . . . . . . 10 ⊢ π ∈ {-π, π, (𝐸‘𝑋)}
54		prssi 4788	. . . . . . . . . 10 ⊢ ((-π ∈ {-π, π, (𝐸‘𝑋)} ∧ π ∈ {-π, π, (𝐸‘𝑋)}) → {-π, π} ⊆ {-π, π, (𝐸‘𝑋)})
55	51, 53, 54	mp2an 704	. . . . . . . . 9 ⊢ {-π, π} ⊆ {-π, π, (𝐸‘𝑋)}
56		ssun1 4139	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ⊆ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))
57	56, 9	sseqtrri 3994	. . . . . . . . 9 ⊢ {-π, π, (𝐸‘𝑋)} ⊆ 𝐻
58	55, 57	sstri 3954	. . . . . . . 8 ⊢ {-π, π} ⊆ 𝐻
59		hashss 14441	. . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ {-π, π} ⊆ 𝐻) → (♯‘{-π, π}) ≤ (♯‘𝐻))
60	49, 58, 59	mp2an 704	. . . . . . 7 ⊢ (♯‘{-π, π}) ≤ (♯‘𝐻)
61	60	a1i 11	. . . . . 6 ⊢ (𝜑 → (♯‘{-π, π}) ≤ (♯‘𝐻))
62	43, 61	eqbrtrrid 5148	. . . . 5 ⊢ (𝜑 → 2 ≤ (♯‘𝐻))
63		eluz2 12864	. . . . 5 ⊢ ((♯‘𝐻) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (♯‘𝐻) ∈ ℤ ∧ 2 ≤ (♯‘𝐻)))
64	8, 39, 62, 63	syl3anbrc 1360	. . . 4 ⊢ (𝜑 → (♯‘𝐻) ∈ (ℤ_≥‘2))
65		uz2m1nn 12943	. . . 4 ⊢ ((♯‘𝐻) ∈ (ℤ_≥‘2) → ((♯‘𝐻) − 1) ∈ ℕ)
66	64, 65	syl 18	. . 3 ⊢ (𝜑 → ((♯‘𝐻) − 1) ∈ ℕ)
67	6, 66	eqeltrid 2873	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
68	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → -π ∈ ℝ)
69	12	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → π ∈ ℝ)
70		negpitopissre 26667	. . . . . . . . . . . 12 ⊢ (-π(,]π) ⊆ ℝ
71	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → -π < π)
72		picn 26583	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℂ
73	72	2timesi 12374	. . . . . . . . . . . . . . 15 ⊢ (2 · π) = (π + π)
74	72, 72	subnegi 11533	. . . . . . . . . . . . . . 15 ⊢ (π − -π) = (π + π)
75	73, 2, 74	3eqtr4i 2802	. . . . . . . . . . . . . 14 ⊢ 𝑇 = (π − -π)
76		fourierdlem102.e	. . . . . . . . . . . . . 14 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
77	68, 69, 71, 75, 76	fourierdlem4 46710	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸:ℝ⟶(-π(,]π))
78	77, 4	ffvelcdmd 7078	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (-π(,]π))
79	70, 78	sselid 3943	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ)
80	68, 69, 79	3jca 1144	. . . . . . . . . 10 ⊢ (𝜑 → (-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝐸‘𝑋) ∈ ℝ))
81		fvex 6892	. . . . . . . . . . 11 ⊢ (𝐸‘𝑋) ∈ V
82	50, 52, 81	tpss 4803	. . . . . . . . . 10 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝐸‘𝑋) ∈ ℝ) ↔ {-π, π, (𝐸‘𝑋)} ⊆ ℝ)
83	80, 82	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ⊆ ℝ)
84		iccssre 13452	. . . . . . . . . . 11 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
85	13, 12, 84	mp2an 704	. . . . . . . . . 10 ⊢ (-π[,]π) ⊆ ℝ
86		ssdifss 4102	. . . . . . . . . 10 ⊢ ((-π[,]π) ⊆ ℝ → ((-π[,]π) ∖ dom 𝐺) ⊆ ℝ)
87	85, 86	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ⊆ ℝ)
88	83, 87	unssd 4153	. . . . . . . 8 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ⊆ ℝ)
89	9, 88	eqsstrid 3983	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
90		fourierdlem102.q	. . . . . . 7 ⊢ 𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻))
91	36, 89, 90, 6	fourierdlem36 46742	. . . . . 6 ⊢ (𝜑 → 𝑄 Isom < , < ((0...𝑀), 𝐻))
92		isof1o 7319	. . . . . 6 ⊢ (𝑄 Isom < , < ((0...𝑀), 𝐻) → 𝑄:(0...𝑀)–1-1-onto→𝐻)
93		f1of 6818	. . . . . 6 ⊢ (𝑄:(0...𝑀)–1-1-onto→𝐻 → 𝑄:(0...𝑀)⟶𝐻)
94	91, 92, 93	3syl 19	. . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶𝐻)
95	94, 89	fssd 6721	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
96		reex 11187	. . . . 5 ⊢ ℝ ∈ V
97		ovex 7441	. . . . 5 ⊢ (0...𝑀) ∈ V
98	96, 97	elmap 8865	. . . 4 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
99	95, 98	sylibr 237	. . 3 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
100		fveq2 6879	. . . . . . . . . . 11 ⊢ (0 = 𝑖 → (𝑄‘0) = (𝑄‘𝑖))
101	100	adantl 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘0) = (𝑄‘𝑖))
102	95	ffvelcdmda 7077	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
103	102	leidd 11776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
104	103	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
105	101, 104	eqbrtrd 5134	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
106		elfzelz 13548	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
107	106	zred 12696	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
108	107	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 𝑖 ∈ ℝ)
109		elfzle1 13551	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
110	109	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 0 ≤ 𝑖)
111		neqne 2972	. . . . . . . . . . . . 13 ⊢ (¬ 0 = 𝑖 → 0 ≠ 𝑖)
112	111	necomd 3019	. . . . . . . . . . . 12 ⊢ (¬ 0 = 𝑖 → 𝑖 ≠ 0)
113	112	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 𝑖 ≠ 0)
114	108, 110, 113	ne0gt0d 11343	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 0 < 𝑖)
115		nnssnn0 12503	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ⊆ ℕ₀
116		nn0uz 12896	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
117	115, 116	sseqtri 3993	. . . . . . . . . . . . . . . 16 ⊢ ℕ ⊆ (ℤ_≥‘0)
118	117, 67	sselid 3943	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
119		eluzfz1 13555	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
120	118, 119	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (0...𝑀))
121	94, 120	ffvelcdmd 7078	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘0) ∈ 𝐻)
122	89, 121	sseldd 3946	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
123	122	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) ∈ ℝ)
124	102	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘𝑖) ∈ ℝ)
125		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → 0 < 𝑖)
126	91	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
127	120	anim1i 626	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀)))
128	127	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀)))
129		isorel 7322	. . . . . . . . . . . . 13 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀))) → (0 < 𝑖 ↔ (𝑄‘0) < (𝑄‘𝑖)))
130	126, 128, 129	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (0 < 𝑖 ↔ (𝑄‘0) < (𝑄‘𝑖)))
131	125, 130	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) < (𝑄‘𝑖))
132	123, 124, 131	ltled 11354	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
133	114, 132	syldan 602	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
134	105, 133	pm2.61dan 824	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄‘𝑖))
135	134	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ≤ (𝑄‘𝑖))
136		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘𝑖) = -π)
137	135, 136	breqtrd 5138	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ≤ -π)
138	68	rexrd 11255	. . . . . . . 8 ⊢ (𝜑 → -π ∈ ℝ^*)
139	69	rexrd 11255	. . . . . . . 8 ⊢ (𝜑 → π ∈ ℝ^*)
140		lbicc2 13487	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → -π ∈ (-π[,]π))
141	14, 15, 21, 140	mp3an 1487	. . . . . . . . . . . . 13 ⊢ -π ∈ (-π[,]π)
142	141	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -π ∈ (-π[,]π))
143		ubicc2 13488	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → π ∈ (-π[,]π))
144	14, 15, 21, 143	mp3an 1487	. . . . . . . . . . . . 13 ⊢ π ∈ (-π[,]π)
145	144	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → π ∈ (-π[,]π))
146		iocssicc 13460	. . . . . . . . . . . . 13 ⊢ (-π(,]π) ⊆ (-π[,]π)
147	146, 78	sselid 3943	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (-π[,]π))
148		tpssi 4804	. . . . . . . . . . . 12 ⊢ ((-π ∈ (-π[,]π) ∧ π ∈ (-π[,]π) ∧ (𝐸‘𝑋) ∈ (-π[,]π)) → {-π, π, (𝐸‘𝑋)} ⊆ (-π[,]π))
149	142, 145, 147, 148	syl3anc 1396	. . . . . . . . . . 11 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ⊆ (-π[,]π))
150		difssd 4099	. . . . . . . . . . 11 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ⊆ (-π[,]π))
151	149, 150	unssd 4153	. . . . . . . . . 10 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ⊆ (-π[,]π))
152	9, 151	eqsstrid 3983	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (-π[,]π))
153	152, 121	sseldd 3946	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘0) ∈ (-π[,]π))
154		iccgelb 13425	. . . . . . . 8 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑄‘0) ∈ (-π[,]π)) → -π ≤ (𝑄‘0))
155	138, 139, 153, 154	syl3anc 1396	. . . . . . 7 ⊢ (𝜑 → -π ≤ (𝑄‘0))
156	155	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → -π ≤ (𝑄‘0))
157	122	ad2antrr 738	. . . . . . 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 11348	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → ((𝑄‘0) = -π ↔ ((𝑄‘0) ≤ -π ∧ -π ≤ (𝑄‘0))))
160	137, 156, 159	mpbir2and 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) = -π)
161	57, 51	sselii 3942	. . . . . . 7 ⊢ -π ∈ 𝐻
162		f1ofo 6826	. . . . . . . . 9 ⊢ (𝑄:(0...𝑀)–1-1-onto→𝐻 → 𝑄:(0...𝑀)–onto→𝐻)
163	92, 162	syl 18	. . . . . . . 8 ⊢ (𝑄 Isom < , < ((0...𝑀), 𝐻) → 𝑄:(0...𝑀)–onto→𝐻)
164		forn 6793	. . . . . . . 8 ⊢ (𝑄:(0...𝑀)–onto→𝐻 → ran 𝑄 = 𝐻)
165	91, 163, 164	3syl 19	. . . . . . 7 ⊢ (𝜑 → ran 𝑄 = 𝐻)
166	161, 165	eleqtrrid 2876	. . . . . 6 ⊢ (𝜑 → -π ∈ ran 𝑄)
167		ffn 6703	. . . . . . 7 ⊢ (𝑄:(0...𝑀)⟶𝐻 → 𝑄 Fn (0...𝑀))
168		fvelrnb 6939	. . . . . . 7 ⊢ (𝑄 Fn (0...𝑀) → (-π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π))
169	94, 167, 168	3syl 19	. . . . . 6 ⊢ (𝜑 → (-π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π))
170	166, 169	mpbid 235	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π)
171	160, 170	r19.29a 3179	. . . 4 ⊢ (𝜑 → (𝑄‘0) = -π)
172	57, 53	sselii 3942	. . . . . . 7 ⊢ π ∈ 𝐻
173	172, 165	eleqtrrid 2876	. . . . . 6 ⊢ (𝜑 → π ∈ ran 𝑄)
174		fvelrnb 6939	. . . . . . 7 ⊢ (𝑄 Fn (0...𝑀) → (π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π))
175	94, 167, 174	3syl 19	. . . . . 6 ⊢ (𝜑 → (π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π))
176	173, 175	mpbid 235	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π)
177	94, 152	fssd 6721	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
178		eluzfz2 13556	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
179	118, 178	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
180	177, 179	ffvelcdmd 7078	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (-π[,]π))
181		iccleub 13424	. . . . . . . . 9 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑄‘𝑀) ∈ (-π[,]π)) → (𝑄‘𝑀) ≤ π)
182	138, 139, 180, 181	syl3anc 1396	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝑀) ≤ π)
183	182	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) ≤ π)
184		id 23	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖) = π → (𝑄‘𝑖) = π)
185	184	eqcomd 2775	. . . . . . . . 9 ⊢ ((𝑄‘𝑖) = π → π = (𝑄‘𝑖))
186	185	3ad2ant3 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π = (𝑄‘𝑖))
187	103	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
188		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
189	188	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) = (𝑄‘𝑀))
190	187, 189	breqtrd 5138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
191	107	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 ∈ ℝ)
192		elfzel2 13546	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
193	192	zred 12696	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
194	193	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑀 ∈ ℝ)
195		elfzle2 13552	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ≤ 𝑀)
196	195	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 ≤ 𝑀)
197		neqne 2972	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑖 = 𝑀 → 𝑖 ≠ 𝑀)
198	197	necomd 3019	. . . . . . . . . . . . 13 ⊢ (¬ 𝑖 = 𝑀 → 𝑀 ≠ 𝑖)
199	198	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑀 ≠ 𝑖)
200	191, 194, 196, 199	leneltd 11360	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 < 𝑀)
201	102	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) ∈ ℝ)
202	85, 180	sselid 3943	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
203	202	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑀) ∈ ℝ)
204		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
205	91	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
206		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
207	179	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (0...𝑀))
208	206, 207	jca 520	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)))
209	208	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)))
210		isorel 7322	. . . . . . . . . . . . . 14 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀))) → (𝑖 < 𝑀 ↔ (𝑄‘𝑖) < (𝑄‘𝑀)))
211	205, 209, 210	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑖 < 𝑀 ↔ (𝑄‘𝑖) < (𝑄‘𝑀)))
212	204, 211	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) < (𝑄‘𝑀))
213	201, 203, 212	ltled 11354	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
214	200, 213	syldan 602	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
215	190, 214	pm2.61dan 824	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
216	215	3adant3 1148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
217	186, 216	eqbrtrd 5134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π ≤ (𝑄‘𝑀))
218	202	3ad2ant1 1149	. . . . . . . 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 11348	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → ((𝑄‘𝑀) = π ↔ ((𝑄‘𝑀) ≤ π ∧ π ≤ (𝑄‘𝑀))))
221	183, 217, 220	mpbir2and 725	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) = π)
222	221	rexlimdv3a 3176	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π → (𝑄‘𝑀) = π))
223	176, 222	mpd 16	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = π)
224		elfzoelz 13683	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
225	224	zred 12696	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
226	225	ltp1d 12141	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
227	226	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 < (𝑖 + 1))
228		elfzofz 13700	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
229		fzofzp1 13789	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
230	228, 229	jca 520	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 ∈ (0...𝑀) ∧ (𝑖 + 1) ∈ (0...𝑀)))
231		isorel 7322	. . . . . . 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 607	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 < (𝑖 + 1) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
233	227, 232	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
234	233	ralrimiva 3163	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
235	171, 223, 234	jca31 523	. . 3 ⊢ (𝜑 → (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
236	5	fourierdlem2 46708	. . . 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 18	. . 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 725	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
239		fourierdlem102.g	. . . . 5 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
240	239	reseq1i 5972	. . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
244		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
245	241, 242, 243, 244	fourierdlem27 46733	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π(,)π))
246	245	resabs1d 6005	. . . 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 2817	. . 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 46744	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
250	247, 249	eqeltrd 2869	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
251	247	oveq1d 7423	. . 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 485	. . . . 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 727	. . . . 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 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
257	91	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
258	257, 92, 93	3syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶𝐻)
259	79	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) ∈ ℝ)
260	257, 163, 164	3syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ran 𝑄 = 𝐻)
261	252, 254, 256, 257, 258, 244, 233, 245, 259, 9, 260	fourierdlem46 46751	. . . 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 499	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
263	251, 262	eqnetrd 3031	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
264	247	oveq1d 7423	. . 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 500	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
266	264, 265	eqnetrd 3031	. 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 46799	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 {crab 3423 Vcvv 3463 ∖ cdif 3910 ∪ cun 3911 ⊆ wss 3913 ∅c0 4294 {cpr 4593 {ctp 4595 class class class wbr 5110 ↦ cmpt 5193 dom cdm 5659 ran crn 5660 ↾ cres 5661 ℩cio 6487 Fn wfn 6528 ⟶wf 6529 –onto→wfo 6531 –1-1-onto→wf1o 6532 ‘cfv 6533 Isom wiso 6534 (class class class)co 7408 ↑_m cmap 8820 Fincfn 8939 ℂcc 11094 ℝcr 11095 0cc0 11096 1c1 11097 + caddc 11099 · cmul 11101 +∞cpnf 11236 -∞cmnf 11237 ℝ^*cxr 11238 < clt 11239 ≤ cle 11240 − cmin 11437 -cneg 11438 / cdiv 11867 ℕcn 12229 2c2 12291 ℕ₀cn0 12500 ℤcz 12587 ℤ_≥cuz 12858 (,)cioo 13368 (,]cioc 13369 [,)cico 13370 [,]cicc 13371 ...cfz 13531 ..^cfzo 13678 ⌊cfl 13819 ♯chash 14362 πcpi 16116 –cn→ccncf 25000 lim_ℂ climc 25986 D cdv 25987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-inf2 9606 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174 ax-addf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-dju 9883 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-xnn0 12574 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13372 df-ioc 13373 df-ico 13374 df-icc 13375 df-fz 13532 df-fzo 13679 df-fl 13821 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16117 df-sin 16119 df-cos 16120 df-pi 16122 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17471 df-topn 17472 df-0g 17490 df-gsum 17491 df-topgen 17492 df-pt 17493 df-prds 17496 df-xrs 17552 df-qtop 17557 df-imas 17558 df-xps 17560 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-mulg 19130 df-cntz 19383 df-cmn 19848 df-psmet 21479 df-xmet 21480 df-met 21481 df-bl 21482 df-mopn 21483 df-fbas 21484 df-fg 21485 df-cnfld 21488 df-top 23016 df-topon 23033 df-topsp 23055 df-bases 23068 df-cld 23141 df-ntr 23142 df-cls 23143 df-nei 23220 df-lp 23258 df-perf 23259 df-cn 23349 df-cnp 23350 df-haus 23437 df-cmp 23509 df-tx 23684 df-hmeo 23877 df-fil 23968 df-fm 24060 df-flim 24061 df-flf 24062 df-xms 24442 df-ms 24443 df-tms 24444 df-cncf 25002 df-limc 25990 df-dv 25991

This theorem is referenced by: fourierdlem106 46811

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