fourierdlem102 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem102 46246

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 12499	. . . . . 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 9205	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ∈ Fin
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ∈ Fin)
12		pire 26388	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
13	12	renegcli 11417	. . . . . . . . . . . . . 14 ⊢ -π ∈ ℝ
14	13	rexri 11165	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ^*
15	12	rexri 11165	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ^*
16		negpilt0 45322	. . . . . . . . . . . . . . 15 ⊢ -π < 0
17		pipos 26390	. . . . . . . . . . . . . . 15 ⊢ 0 < π
18		0re 11109	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
19	13, 18, 12	lttri 11234	. . . . . . . . . . . . . . 15 ⊢ ((-π < 0 ∧ 0 < π) → -π < π)
20	16, 17, 19	mp2an 692	. . . . . . . . . . . . . 14 ⊢ -π < π
21	13, 12, 20	ltleii 11231	. . . . . . . . . . . . 13 ⊢ -π ≤ π
22		prunioo 13376	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → ((-π(,)π) ∪ {-π, π}) = (-π[,]π))
23	14, 15, 21, 22	mp3an 1463	. . . . . . . . . . . 12 ⊢ ((-π(,)π) ∪ {-π, π}) = (-π[,]π)
24	23	difeq1i 4067	. . . . . . . . . . 11 ⊢ (((-π(,)π) ∪ {-π, π}) ∖ dom 𝐺) = ((-π[,]π) ∖ dom 𝐺)
25		difundir 4236	. . . . . . . . . . 11 ⊢ (((-π(,)π) ∪ {-π, π}) ∖ dom 𝐺) = (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺))
26	24, 25	eqtr3i 2756	. . . . . . . . . 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 9203	. . . . . . . . . . . 12 ⊢ {-π, π} ∈ Fin
29		diffi 9079	. . . . . . . . . . . 12 ⊢ ({-π, π} ∈ Fin → ({-π, π} ∖ dom 𝐺) ∈ Fin)
30	28, 29	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → ({-π, π} ∖ dom 𝐺) ∈ Fin)
31		unfi 9075	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (𝜑 → (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺)) ∈ Fin)
33	26, 32	eqeltrid 2835	. . . . . . . . 9 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ∈ Fin)
34		unfi 9075	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ Fin)
36	9, 35	eqeltrid 2835	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Fin)
37		hashcl 14258	. . . . . . 7 ⊢ (𝐻 ∈ Fin → (♯‘𝐻) ∈ ℕ₀)
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) ∈ ℕ₀)
39	38	nn0zd 12489	. . . . 5 ⊢ (𝜑 → (♯‘𝐻) ∈ ℤ)
40	13, 20	ltneii 11221	. . . . . . 7 ⊢ -π ≠ π
41		hashprg 14297	. . . . . . . 8 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π ≠ π ↔ (♯‘{-π, π}) = 2))
42	13, 12, 41	mp2an 692	. . . . . . 7 ⊢ (-π ≠ π ↔ (♯‘{-π, π}) = 2)
43	40, 42	mpbi 230	. . . . . 6 ⊢ (♯‘{-π, π}) = 2
44	10	elexi 3459	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ∈ V
45		ovex 7374	. . . . . . . . . . 11 ⊢ (-π[,]π) ∈ V
46		difexg 5262	. . . . . . . . . . 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 7672	. . . . . . . . 9 ⊢ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ V
49	9, 48	eqeltri 2827	. . . . . . . 8 ⊢ 𝐻 ∈ V
50		negex 11353	. . . . . . . . . . 11 ⊢ -π ∈ V
51	50	tpid1 4716	. . . . . . . . . 10 ⊢ -π ∈ {-π, π, (𝐸‘𝑋)}
52	12	elexi 3459	. . . . . . . . . . 11 ⊢ π ∈ V
53	52	tpid2 4718	. . . . . . . . . 10 ⊢ π ∈ {-π, π, (𝐸‘𝑋)}
54		prssi 4768	. . . . . . . . . 10 ⊢ ((-π ∈ {-π, π, (𝐸‘𝑋)} ∧ π ∈ {-π, π, (𝐸‘𝑋)}) → {-π, π} ⊆ {-π, π, (𝐸‘𝑋)})
55	51, 53, 54	mp2an 692	. . . . . . . . 9 ⊢ {-π, π} ⊆ {-π, π, (𝐸‘𝑋)}
56		ssun1 4123	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ⊆ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))
57	56, 9	sseqtrri 3979	. . . . . . . . 9 ⊢ {-π, π, (𝐸‘𝑋)} ⊆ 𝐻
58	55, 57	sstri 3939	. . . . . . . 8 ⊢ {-π, π} ⊆ 𝐻
59		hashss 14311	. . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ {-π, π} ⊆ 𝐻) → (♯‘{-π, π}) ≤ (♯‘𝐻))
60	49, 58, 59	mp2an 692	. . . . . . 7 ⊢ (♯‘{-π, π}) ≤ (♯‘𝐻)
61	60	a1i 11	. . . . . 6 ⊢ (𝜑 → (♯‘{-π, π}) ≤ (♯‘𝐻))
62	43, 61	eqbrtrrid 5122	. . . . 5 ⊢ (𝜑 → 2 ≤ (♯‘𝐻))
63		eluz2 12733	. . . . 5 ⊢ ((♯‘𝐻) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (♯‘𝐻) ∈ ℤ ∧ 2 ≤ (♯‘𝐻)))
64	8, 39, 62, 63	syl3anbrc 1344	. . . 4 ⊢ (𝜑 → (♯‘𝐻) ∈ (ℤ_≥‘2))
65		uz2m1nn 12816	. . . 4 ⊢ ((♯‘𝐻) ∈ (ℤ_≥‘2) → ((♯‘𝐻) − 1) ∈ ℕ)
66	64, 65	syl 17	. . 3 ⊢ (𝜑 → ((♯‘𝐻) − 1) ∈ ℕ)
67	6, 66	eqeltrid 2835	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
68	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → -π ∈ ℝ)
69	12	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → π ∈ ℝ)
70		negpitopissre 26471	. . . . . . . . . . . 12 ⊢ (-π(,]π) ⊆ ℝ
71	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → -π < π)
72		picn 26389	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℂ
73	72	2timesi 12253	. . . . . . . . . . . . . . 15 ⊢ (2 · π) = (π + π)
74	72, 72	subnegi 11435	. . . . . . . . . . . . . . 15 ⊢ (π − -π) = (π + π)
75	73, 2, 74	3eqtr4i 2764	. . . . . . . . . . . . . 14 ⊢ 𝑇 = (π − -π)
76		fourierdlem102.e	. . . . . . . . . . . . . 14 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
77	68, 69, 71, 75, 76	fourierdlem4 46149	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸:ℝ⟶(-π(,]π))
78	77, 4	ffvelcdmd 7013	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (-π(,]π))
79	70, 78	sselid 3927	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ)
80	68, 69, 79	3jca 1128	. . . . . . . . . 10 ⊢ (𝜑 → (-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝐸‘𝑋) ∈ ℝ))
81		fvex 6830	. . . . . . . . . . 11 ⊢ (𝐸‘𝑋) ∈ V
82	50, 52, 81	tpss 4784	. . . . . . . . . 10 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝐸‘𝑋) ∈ ℝ) ↔ {-π, π, (𝐸‘𝑋)} ⊆ ℝ)
83	80, 82	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ⊆ ℝ)
84		iccssre 13324	. . . . . . . . . . 11 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
85	13, 12, 84	mp2an 692	. . . . . . . . . 10 ⊢ (-π[,]π) ⊆ ℝ
86		ssdifss 4085	. . . . . . . . . 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 4137	. . . . . . . 8 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ⊆ ℝ)
89	9, 88	eqsstrid 3968	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
90		fourierdlem102.q	. . . . . . 7 ⊢ 𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻))
91	36, 89, 90, 6	fourierdlem36 46181	. . . . . 6 ⊢ (𝜑 → 𝑄 Isom < , < ((0...𝑀), 𝐻))
92		isof1o 7252	. . . . . 6 ⊢ (𝑄 Isom < , < ((0...𝑀), 𝐻) → 𝑄:(0...𝑀)–1-1-onto→𝐻)
93		f1of 6758	. . . . . 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 6663	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
96		reex 11092	. . . . 5 ⊢ ℝ ∈ V
97		ovex 7374	. . . . 5 ⊢ (0...𝑀) ∈ V
98	96, 97	elmap 8790	. . . 4 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
99	95, 98	sylibr 234	. . 3 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
100		fveq2 6817	. . . . . . . . . . 11 ⊢ (0 = 𝑖 → (𝑄‘0) = (𝑄‘𝑖))
101	100	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘0) = (𝑄‘𝑖))
102	95	ffvelcdmda 7012	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
103	102	leidd 11678	. . . . . . . . . . 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 5108	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
106		elfzelz 13419	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
107	106	zred 12572	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
108	107	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 𝑖 ∈ ℝ)
109		elfzle1 13422	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
110	109	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 0 ≤ 𝑖)
111		neqne 2936	. . . . . . . . . . . . 13 ⊢ (¬ 0 = 𝑖 → 0 ≠ 𝑖)
112	111	necomd 2983	. . . . . . . . . . . 12 ⊢ (¬ 0 = 𝑖 → 𝑖 ≠ 0)
113	112	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 𝑖 ≠ 0)
114	108, 110, 113	ne0gt0d 11245	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 0 < 𝑖)
115		nnssnn0 12379	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ⊆ ℕ₀
116		nn0uz 12769	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
117	115, 116	sseqtri 3978	. . . . . . . . . . . . . . . 16 ⊢ ℕ ⊆ (ℤ_≥‘0)
118	117, 67	sselid 3927	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
119		eluzfz1 13426	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
120	118, 119	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (0...𝑀))
121	94, 120	ffvelcdmd 7013	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘0) ∈ 𝐻)
122	89, 121	sseldd 3930	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
123	122	ad2antrr 726	. . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
127	120	anim1i 615	. . . . . . . . . . . . . 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 7255	. . . . . . . . . . . . 13 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀))) → (0 < 𝑖 ↔ (𝑄‘0) < (𝑄‘𝑖)))
130	126, 128, 129	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (0 < 𝑖 ↔ (𝑄‘0) < (𝑄‘𝑖)))
131	125, 130	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) < (𝑄‘𝑖))
132	123, 124, 131	ltled 11256	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
133	114, 132	syldan 591	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
134	105, 133	pm2.61dan 812	. . . . . . . 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 5112	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ≤ -π)
138	68	rexrd 11157	. . . . . . . 8 ⊢ (𝜑 → -π ∈ ℝ^*)
139	69	rexrd 11157	. . . . . . . 8 ⊢ (𝜑 → π ∈ ℝ^*)
140		lbicc2 13359	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → -π ∈ (-π[,]π))
141	14, 15, 21, 140	mp3an 1463	. . . . . . . . . . . . 13 ⊢ -π ∈ (-π[,]π)
142	141	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -π ∈ (-π[,]π))
143		ubicc2 13360	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → π ∈ (-π[,]π))
144	14, 15, 21, 143	mp3an 1463	. . . . . . . . . . . . 13 ⊢ π ∈ (-π[,]π)
145	144	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → π ∈ (-π[,]π))
146		iocssicc 13332	. . . . . . . . . . . . 13 ⊢ (-π(,]π) ⊆ (-π[,]π)
147	146, 78	sselid 3927	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (-π[,]π))
148		tpssi 4785	. . . . . . . . . . . 12 ⊢ ((-π ∈ (-π[,]π) ∧ π ∈ (-π[,]π) ∧ (𝐸‘𝑋) ∈ (-π[,]π)) → {-π, π, (𝐸‘𝑋)} ⊆ (-π[,]π))
149	142, 145, 147, 148	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ⊆ (-π[,]π))
150		difssd 4082	. . . . . . . . . . 11 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ⊆ (-π[,]π))
151	149, 150	unssd 4137	. . . . . . . . . 10 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ⊆ (-π[,]π))
152	9, 151	eqsstrid 3968	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (-π[,]π))
153	152, 121	sseldd 3930	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘0) ∈ (-π[,]π))
154		iccgelb 13297	. . . . . . . 8 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑄‘0) ∈ (-π[,]π)) → -π ≤ (𝑄‘0))
155	138, 139, 153, 154	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → -π ≤ (𝑄‘0))
156	155	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → -π ≤ (𝑄‘0))
157	122	ad2antrr 726	. . . . . . 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 11250	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → ((𝑄‘0) = -π ↔ ((𝑄‘0) ≤ -π ∧ -π ≤ (𝑄‘0))))
160	137, 156, 159	mpbir2and 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) = -π)
161	57, 51	sselii 3926	. . . . . . 7 ⊢ -π ∈ 𝐻
162		f1ofo 6765	. . . . . . . . 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 6733	. . . . . . . 8 ⊢ (𝑄:(0...𝑀)–onto→𝐻 → ran 𝑄 = 𝐻)
165	91, 163, 164	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝑄 = 𝐻)
166	161, 165	eleqtrrid 2838	. . . . . 6 ⊢ (𝜑 → -π ∈ ran 𝑄)
167		ffn 6646	. . . . . . 7 ⊢ (𝑄:(0...𝑀)⟶𝐻 → 𝑄 Fn (0...𝑀))
168		fvelrnb 6877	. . . . . . 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 232	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π)
171	160, 170	r19.29a 3140	. . . 4 ⊢ (𝜑 → (𝑄‘0) = -π)
172	57, 53	sselii 3926	. . . . . . 7 ⊢ π ∈ 𝐻
173	172, 165	eleqtrrid 2838	. . . . . 6 ⊢ (𝜑 → π ∈ ran 𝑄)
174		fvelrnb 6877	. . . . . . 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 232	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π)
177	94, 152	fssd 6663	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
178		eluzfz2 13427	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
179	118, 178	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
180	177, 179	ffvelcdmd 7013	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (-π[,]π))
181		iccleub 13296	. . . . . . . . 9 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑄‘𝑀) ∈ (-π[,]π)) → (𝑄‘𝑀) ≤ π)
182	138, 139, 180, 181	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝑀) ≤ π)
183	182	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) ≤ π)
184		id 22	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖) = π → (𝑄‘𝑖) = π)
185	184	eqcomd 2737	. . . . . . . . 9 ⊢ ((𝑄‘𝑖) = π → π = (𝑄‘𝑖))
186	185	3ad2ant3 1135	. . . . . . . 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 6817	. . . . . . . . . . . 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 5112	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
191	107	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 ∈ ℝ)
192		elfzel2 13417	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
193	192	zred 12572	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
194	193	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑀 ∈ ℝ)
195		elfzle2 13423	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ≤ 𝑀)
196	195	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 ≤ 𝑀)
197		neqne 2936	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑖 = 𝑀 → 𝑖 ≠ 𝑀)
198	197	necomd 2983	. . . . . . . . . . . . 13 ⊢ (¬ 𝑖 = 𝑀 → 𝑀 ≠ 𝑖)
199	198	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑀 ≠ 𝑖)
200	191, 194, 196, 199	leneltd 11262	. . . . . . . . . . 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 3927	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
203	202	ad2antrr 726	. . . . . . . . . . . 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 726	. . . . . . . . . . . . . 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 7255	. . . . . . . . . . . . . 14 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀))) → (𝑖 < 𝑀 ↔ (𝑄‘𝑖) < (𝑄‘𝑀)))
211	205, 209, 210	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑖 < 𝑀 ↔ (𝑄‘𝑖) < (𝑄‘𝑀)))
212	204, 211	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) < (𝑄‘𝑀))
213	201, 203, 212	ltled 11256	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
214	200, 213	syldan 591	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
215	190, 214	pm2.61dan 812	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
216	215	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
217	186, 216	eqbrtrd 5108	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π ≤ (𝑄‘𝑀))
218	202	3ad2ant1 1133	. . . . . . . 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 11250	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → ((𝑄‘𝑀) = π ↔ ((𝑄‘𝑀) ≤ π ∧ π ≤ (𝑄‘𝑀))))
221	183, 217, 220	mpbir2and 713	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) = π)
222	221	rexlimdv3a 3137	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π → (𝑄‘𝑀) = π))
223	176, 222	mpd 15	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = π)
224		elfzoelz 13554	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
225	224	zred 12572	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
226	225	ltp1d 12047	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
227	226	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 < (𝑖 + 1))
228		elfzofz 13570	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
229		fzofzp1 13659	. . . . . . . 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 7255	. . . . . . 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 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 < (𝑖 + 1) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
233	227, 232	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
234	233	ralrimiva 3124	. . . 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 46147	. . . 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 713	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
239		fourierdlem102.g	. . . . 5 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
240	239	reseq1i 5919	. . . 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 46172	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π(,)π))
246	245	resabs1d 5952	. . . 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 2779	. . 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 46183	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
250	247, 249	eqeltrd 2831	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
251	247	oveq1d 7356	. . 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 715	. . . . 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 715	. . . . 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 46190	. . . 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 2995	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
264	247	oveq1d 7356	. . 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 2995	. 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 46238	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {crab 3395 Vcvv 3436 ∖ cdif 3894 ∪ cun 3895 ⊆ wss 3897 ∅c0 4278 {cpr 4573 {ctp 4575 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5611 ran crn 5612 ↾ cres 5613 ℩cio 6430 Fn wfn 6471 ⟶wf 6472 –onto→wfo 6474 –1-1-onto→wf1o 6475 ‘cfv 6476 Isom wiso 6477 (class class class)co 7341 ↑_m cmap 8745 Fincfn 8864 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 + caddc 11004 · cmul 11006 +∞cpnf 11138 -∞cmnf 11139 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 − cmin 11339 -cneg 11340 / cdiv 11769 ℕcn 12120 2c2 12175 ℕ₀cn0 12376 ℤcz 12463 ℤ_≥cuz 12727 (,)cioo 13240 (,]cioc 13241 [,)cico 13242 [,]cicc 13243 ...cfz 13402 ..^cfzo 13549 ⌊cfl 13689 ♯chash 14232 πcpi 15968 –cn→ccncf 24791 lim_ℂ climc 25785 D cdv 25786

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9789 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-xnn0 12450 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-seq 13904 df-exp 13964 df-fac 14176 df-bc 14205 df-hash 14233 df-shft 14969 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-limsup 15373 df-clim 15390 df-rlim 15391 df-sum 15589 df-ef 15969 df-sin 15971 df-cos 15972 df-pi 15974 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-perf 23047 df-cn 23137 df-cnp 23138 df-haus 23225 df-cmp 23297 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-limc 25789 df-dv 25790

This theorem is referenced by: fourierdlem106 46250

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