fourierdlem114 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem114 41937

Description: Fourier series convergence for periodic, piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem114.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem114.t	⊢ 𝑇 = (2 · π)
fourierdlem114.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem114.g	⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
fourierdlem114.dmdv	⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
fourierdlem114.gcn	⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
fourierdlem114.rlim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fourierdlem114.llim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
fourierdlem114.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem114.l	⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierdlem114.r	⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem114.a	⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem114.b	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem114.s	⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))
fourierdlem114.p	⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem114.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem114.h	⊢ 𝐻 = ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))
fourierdlem114.m	⊢ 𝑀 = ((♯‘𝐻) − 1)
fourierdlem114.q	⊢ 𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻))

**Assertion**
Ref	Expression
fourierdlem114	⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))

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

**Proof of Theorem fourierdlem114**
Step	Hyp	Ref	Expression
1		fourierdlem114.f	. 2 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		fourierdlem114.t	. 2 ⊢ 𝑇 = (2 · π)
3		fourierdlem114.per	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
4		fourierdlem114.x	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5		fourierdlem114.l	. 2 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
6		fourierdlem114.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
7		fourierdlem114.p	. 2 ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
8		fourierdlem114.m	. . 3 ⊢ 𝑀 = ((♯‘𝐻) − 1)
9		2z 11830	. . . . . 6 ⊢ 2 ∈ ℤ
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
11		fourierdlem114.h	. . . . . . . 8 ⊢ 𝐻 = ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))
12		tpfi 8591	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ∈ Fin
13	12	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ∈ Fin)
14		pire 24750	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
15	14	renegcli 10750	. . . . . . . . . . . . . 14 ⊢ -π ∈ ℝ
16	15	rexri 10501	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ^*
17	14	rexri 10501	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ^*
18		negpilt0 40976	. . . . . . . . . . . . . . 15 ⊢ -π < 0
19		pipos 24752	. . . . . . . . . . . . . . 15 ⊢ 0 < π
20		0re 10443	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
21	15, 20, 14	lttri 10568	. . . . . . . . . . . . . . 15 ⊢ ((-π < 0 ∧ 0 < π) → -π < π)
22	18, 19, 21	mp2an 679	. . . . . . . . . . . . . 14 ⊢ -π < π
23	15, 14, 22	ltleii 10565	. . . . . . . . . . . . 13 ⊢ -π ≤ π
24		prunioo 12686	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → ((-π(,)π) ∪ {-π, π}) = (-π[,]π))
25	16, 17, 23, 24	mp3an 1440	. . . . . . . . . . . 12 ⊢ ((-π(,)π) ∪ {-π, π}) = (-π[,]π)
26	25	difeq1i 3987	. . . . . . . . . . 11 ⊢ (((-π(,)π) ∪ {-π, π}) ∖ dom 𝐺) = ((-π[,]π) ∖ dom 𝐺)
27		difundir 4146	. . . . . . . . . . 11 ⊢ (((-π(,)π) ∪ {-π, π}) ∖ dom 𝐺) = (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺))
28	26, 27	eqtr3i 2804	. . . . . . . . . 10 ⊢ ((-π[,]π) ∖ dom 𝐺) = (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺))
29		fourierdlem114.dmdv	. . . . . . . . . . 11 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
30		prfi 8590	. . . . . . . . . . . 12 ⊢ {-π, π} ∈ Fin
31		diffi 8547	. . . . . . . . . . . 12 ⊢ ({-π, π} ∈ Fin → ({-π, π} ∖ dom 𝐺) ∈ Fin)
32	30, 31	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → ({-π, π} ∖ dom 𝐺) ∈ Fin)
33		unfi 8582	. . . . . . . . . . 11 ⊢ ((((-π(,)π) ∖ dom 𝐺) ∈ Fin ∧ ({-π, π} ∖ dom 𝐺) ∈ Fin) → (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺)) ∈ Fin)
34	29, 32, 33	syl2anc 576	. . . . . . . . . 10 ⊢ (𝜑 → (((-π(,)π) ∖ dom 𝐺) ∪ ({-π, π} ∖ dom 𝐺)) ∈ Fin)
35	28, 34	syl5eqel 2870	. . . . . . . . 9 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ∈ Fin)
36		unfi 8582	. . . . . . . . 9 ⊢ (({-π, π, (𝐸‘𝑋)} ∈ Fin ∧ ((-π[,]π) ∖ dom 𝐺) ∈ Fin) → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ Fin)
37	13, 35, 36	syl2anc 576	. . . . . . . 8 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ Fin)
38	11, 37	syl5eqel 2870	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Fin)
39		hashcl 13535	. . . . . . 7 ⊢ (𝐻 ∈ Fin → (♯‘𝐻) ∈ ℕ₀)
40	38, 39	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) ∈ ℕ₀)
41	40	nn0zd 11901	. . . . 5 ⊢ (𝜑 → (♯‘𝐻) ∈ ℤ)
42	15, 22	ltneii 10555	. . . . . . 7 ⊢ -π ≠ π
43		hashprg 13572	. . . . . . . 8 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π ≠ π ↔ (♯‘{-π, π}) = 2))
44	15, 14, 43	mp2an 679	. . . . . . 7 ⊢ (-π ≠ π ↔ (♯‘{-π, π}) = 2)
45	42, 44	mpbi 222	. . . . . 6 ⊢ (♯‘{-π, π}) = 2
46	12	elexi 3434	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ∈ V
47		ovex 7010	. . . . . . . . . . 11 ⊢ (-π[,]π) ∈ V
48		difexg 5088	. . . . . . . . . . 11 ⊢ ((-π[,]π) ∈ V → ((-π[,]π) ∖ dom 𝐺) ∈ V)
49	47, 48	ax-mp 5	. . . . . . . . . 10 ⊢ ((-π[,]π) ∖ dom 𝐺) ∈ V
50	46, 49	unex 7288	. . . . . . . . 9 ⊢ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ∈ V
51	11, 50	eqeltri 2862	. . . . . . . 8 ⊢ 𝐻 ∈ V
52		negex 10686	. . . . . . . . . . 11 ⊢ -π ∈ V
53	52	tpid1 4579	. . . . . . . . . 10 ⊢ -π ∈ {-π, π, (𝐸‘𝑋)}
54	14	elexi 3434	. . . . . . . . . . 11 ⊢ π ∈ V
55	54	tpid2 4581	. . . . . . . . . 10 ⊢ π ∈ {-π, π, (𝐸‘𝑋)}
56		prssi 4629	. . . . . . . . . 10 ⊢ ((-π ∈ {-π, π, (𝐸‘𝑋)} ∧ π ∈ {-π, π, (𝐸‘𝑋)}) → {-π, π} ⊆ {-π, π, (𝐸‘𝑋)})
57	53, 55, 56	mp2an 679	. . . . . . . . 9 ⊢ {-π, π} ⊆ {-π, π, (𝐸‘𝑋)}
58		ssun1 4039	. . . . . . . . . 10 ⊢ {-π, π, (𝐸‘𝑋)} ⊆ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))
59	58, 11	sseqtr4i 3896	. . . . . . . . 9 ⊢ {-π, π, (𝐸‘𝑋)} ⊆ 𝐻
60	57, 59	sstri 3869	. . . . . . . 8 ⊢ {-π, π} ⊆ 𝐻
61		hashss 13586	. . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ {-π, π} ⊆ 𝐻) → (♯‘{-π, π}) ≤ (♯‘𝐻))
62	51, 60, 61	mp2an 679	. . . . . . 7 ⊢ (♯‘{-π, π}) ≤ (♯‘𝐻)
63	62	a1i 11	. . . . . 6 ⊢ (𝜑 → (♯‘{-π, π}) ≤ (♯‘𝐻))
64	45, 63	syl5eqbrr 4966	. . . . 5 ⊢ (𝜑 → 2 ≤ (♯‘𝐻))
65		eluz2 12067	. . . . 5 ⊢ ((♯‘𝐻) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (♯‘𝐻) ∈ ℤ ∧ 2 ≤ (♯‘𝐻)))
66	10, 41, 64, 65	syl3anbrc 1323	. . . 4 ⊢ (𝜑 → (♯‘𝐻) ∈ (ℤ_≥‘2))
67		uz2m1nn 12140	. . . 4 ⊢ ((♯‘𝐻) ∈ (ℤ_≥‘2) → ((♯‘𝐻) − 1) ∈ ℕ)
68	66, 67	syl 17	. . 3 ⊢ (𝜑 → ((♯‘𝐻) − 1) ∈ ℕ)
69	8, 68	syl5eqel 2870	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
70	15	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → -π ∈ ℝ)
71	14	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → π ∈ ℝ)
72		negpitopissre 24828	. . . . . . . . . . . 12 ⊢ (-π(,]π) ⊆ ℝ
73	22	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → -π < π)
74		picn 24751	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℂ
75	74	2timesi 11588	. . . . . . . . . . . . . . 15 ⊢ (2 · π) = (π + π)
76	74, 74	subnegi 10768	. . . . . . . . . . . . . . 15 ⊢ (π − -π) = (π + π)
77	75, 2, 76	3eqtr4i 2812	. . . . . . . . . . . . . 14 ⊢ 𝑇 = (π − -π)
78		fourierdlem114.e	. . . . . . . . . . . . . 14 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
79	70, 71, 73, 77, 78	fourierdlem4 41828	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸:ℝ⟶(-π(,]π))
80	79, 4	ffvelrnd 6679	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (-π(,]π))
81	72, 80	sseldi 3858	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ)
82	70, 71, 81	3jca 1108	. . . . . . . . . 10 ⊢ (𝜑 → (-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝐸‘𝑋) ∈ ℝ))
83		fvex 6514	. . . . . . . . . . 11 ⊢ (𝐸‘𝑋) ∈ V
84	52, 54, 83	tpss 4643	. . . . . . . . . 10 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝐸‘𝑋) ∈ ℝ) ↔ {-π, π, (𝐸‘𝑋)} ⊆ ℝ)
85	82, 84	sylib 210	. . . . . . . . 9 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ⊆ ℝ)
86		iccssre 12637	. . . . . . . . . . 11 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
87	15, 14, 86	mp2an 679	. . . . . . . . . 10 ⊢ (-π[,]π) ⊆ ℝ
88		ssdifss 4004	. . . . . . . . . 10 ⊢ ((-π[,]π) ⊆ ℝ → ((-π[,]π) ∖ dom 𝐺) ⊆ ℝ)
89	87, 88	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ⊆ ℝ)
90	85, 89	unssd 4052	. . . . . . . 8 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ⊆ ℝ)
91	11, 90	syl5eqss 3907	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
92		fourierdlem114.q	. . . . . . 7 ⊢ 𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻))
93	38, 91, 92, 8	fourierdlem36 41860	. . . . . 6 ⊢ (𝜑 → 𝑄 Isom < , < ((0...𝑀), 𝐻))
94		isof1o 6901	. . . . . 6 ⊢ (𝑄 Isom < , < ((0...𝑀), 𝐻) → 𝑄:(0...𝑀)–1-1-onto→𝐻)
95		f1of 6446	. . . . . 6 ⊢ (𝑄:(0...𝑀)–1-1-onto→𝐻 → 𝑄:(0...𝑀)⟶𝐻)
96	93, 94, 95	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶𝐻)
97	96, 91	fssd 6360	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
98		reex 10428	. . . . 5 ⊢ ℝ ∈ V
99		ovex 7010	. . . . 5 ⊢ (0...𝑀) ∈ V
100	98, 99	elmap 8237	. . . 4 ⊢ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
101	97, 100	sylibr 226	. . 3 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)))
102		fveq2 6501	. . . . . . . . . . 11 ⊢ (0 = 𝑖 → (𝑄‘0) = (𝑄‘𝑖))
103	102	adantl 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘0) = (𝑄‘𝑖))
104	97	ffvelrnda 6678	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
105	104	leidd 11009	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
106	105	adantr 473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
107	103, 106	eqbrtrd 4952	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 = 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
108		elfzelz 12727	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
109	108	zred 11903	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
110	109	ad2antlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 𝑖 ∈ ℝ)
111		elfzle1 12729	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
112	111	ad2antlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 0 ≤ 𝑖)
113		neqne 2975	. . . . . . . . . . . . 13 ⊢ (¬ 0 = 𝑖 → 0 ≠ 𝑖)
114	113	necomd 3022	. . . . . . . . . . . 12 ⊢ (¬ 0 = 𝑖 → 𝑖 ≠ 0)
115	114	adantl 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 𝑖 ≠ 0)
116	110, 112, 115	ne0gt0d 10579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → 0 < 𝑖)
117		nnssnn0 11713	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ⊆ ℕ₀
118		nn0uz 12097	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
119	117, 118	sseqtri 3895	. . . . . . . . . . . . . . . 16 ⊢ ℕ ⊆ (ℤ_≥‘0)
120	119, 69	sseldi 3858	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
121		eluzfz1 12733	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
122	120, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (0...𝑀))
123	96, 122	ffvelrnd 6679	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘0) ∈ 𝐻)
124	91, 123	sseldd 3861	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
125	124	ad2antrr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) ∈ ℝ)
126	104	adantr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘𝑖) ∈ ℝ)
127		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → 0 < 𝑖)
128	93	ad2antrr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
129	122	anim1i 605	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀)))
130	129	adantr 473	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀)))
131		isorel 6904	. . . . . . . . . . . . 13 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (0 ∈ (0...𝑀) ∧ 𝑖 ∈ (0...𝑀))) → (0 < 𝑖 ↔ (𝑄‘0) < (𝑄‘𝑖)))
132	128, 130, 131	syl2anc 576	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (0 < 𝑖 ↔ (𝑄‘0) < (𝑄‘𝑖)))
133	127, 132	mpbid 224	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) < (𝑄‘𝑖))
134	125, 126, 133	ltled 10590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 0 < 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
135	116, 134	syldan 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 0 = 𝑖) → (𝑄‘0) ≤ (𝑄‘𝑖))
136	107, 135	pm2.61dan 800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄‘𝑖))
137	136	adantr 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ≤ (𝑄‘𝑖))
138		simpr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘𝑖) = -π)
139	137, 138	breqtrd 4956	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ≤ -π)
140	70	rexrd 10492	. . . . . . . 8 ⊢ (𝜑 → -π ∈ ℝ^*)
141	71	rexrd 10492	. . . . . . . 8 ⊢ (𝜑 → π ∈ ℝ^*)
142		lbicc2 12671	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → -π ∈ (-π[,]π))
143	16, 17, 23, 142	mp3an 1440	. . . . . . . . . . . . 13 ⊢ -π ∈ (-π[,]π)
144	143	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -π ∈ (-π[,]π))
145		ubicc2 12672	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ -π ≤ π) → π ∈ (-π[,]π))
146	16, 17, 23, 145	mp3an 1440	. . . . . . . . . . . . 13 ⊢ π ∈ (-π[,]π)
147	146	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → π ∈ (-π[,]π))
148		iocssicc 12644	. . . . . . . . . . . . 13 ⊢ (-π(,]π) ⊆ (-π[,]π)
149	148, 80	sseldi 3858	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (-π[,]π))
150		tpssi 4644	. . . . . . . . . . . 12 ⊢ ((-π ∈ (-π[,]π) ∧ π ∈ (-π[,]π) ∧ (𝐸‘𝑋) ∈ (-π[,]π)) → {-π, π, (𝐸‘𝑋)} ⊆ (-π[,]π))
151	144, 147, 149, 150	syl3anc 1351	. . . . . . . . . . 11 ⊢ (𝜑 → {-π, π, (𝐸‘𝑋)} ⊆ (-π[,]π))
152		difssd 4001	. . . . . . . . . . 11 ⊢ (𝜑 → ((-π[,]π) ∖ dom 𝐺) ⊆ (-π[,]π))
153	151, 152	unssd 4052	. . . . . . . . . 10 ⊢ (𝜑 → ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) ⊆ (-π[,]π))
154	11, 153	syl5eqss 3907	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (-π[,]π))
155	154, 123	sseldd 3861	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘0) ∈ (-π[,]π))
156		iccgelb 12612	. . . . . . . 8 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑄‘0) ∈ (-π[,]π)) → -π ≤ (𝑄‘0))
157	140, 141, 155, 156	syl3anc 1351	. . . . . . 7 ⊢ (𝜑 → -π ≤ (𝑄‘0))
158	157	ad2antrr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → -π ≤ (𝑄‘0))
159	124	ad2antrr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) ∈ ℝ)
160	15	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → -π ∈ ℝ)
161	159, 160	letri3d 10584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → ((𝑄‘0) = -π ↔ ((𝑄‘0) ≤ -π ∧ -π ≤ (𝑄‘0))))
162	139, 158, 161	mpbir2and 700	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = -π) → (𝑄‘0) = -π)
163	59, 53	sselii 3857	. . . . . . 7 ⊢ -π ∈ 𝐻
164		f1ofo 6453	. . . . . . . . 9 ⊢ (𝑄:(0...𝑀)–1-1-onto→𝐻 → 𝑄:(0...𝑀)–onto→𝐻)
165	94, 164	syl 17	. . . . . . . 8 ⊢ (𝑄 Isom < , < ((0...𝑀), 𝐻) → 𝑄:(0...𝑀)–onto→𝐻)
166		forn 6424	. . . . . . . 8 ⊢ (𝑄:(0...𝑀)–onto→𝐻 → ran 𝑄 = 𝐻)
167	93, 165, 166	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝑄 = 𝐻)
168	163, 167	syl5eleqr 2873	. . . . . 6 ⊢ (𝜑 → -π ∈ ran 𝑄)
169		ffn 6346	. . . . . . 7 ⊢ (𝑄:(0...𝑀)⟶𝐻 → 𝑄 Fn (0...𝑀))
170		fvelrnb 6558	. . . . . . 7 ⊢ (𝑄 Fn (0...𝑀) → (-π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π))
171	96, 169, 170	3syl 18	. . . . . 6 ⊢ (𝜑 → (-π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π))
172	168, 171	mpbid 224	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = -π)
173	162, 172	r19.29a 3234	. . . 4 ⊢ (𝜑 → (𝑄‘0) = -π)
174	59, 55	sselii 3857	. . . . . . 7 ⊢ π ∈ 𝐻
175	174, 167	syl5eleqr 2873	. . . . . 6 ⊢ (𝜑 → π ∈ ran 𝑄)
176		fvelrnb 6558	. . . . . . 7 ⊢ (𝑄 Fn (0...𝑀) → (π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π))
177	96, 169, 176	3syl 18	. . . . . 6 ⊢ (𝜑 → (π ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π))
178	175, 177	mpbid 224	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π)
179	96, 154	fssd 6360	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
180		eluzfz2 12734	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
181	120, 180	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
182	179, 181	ffvelrnd 6679	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (-π[,]π))
183		iccleub 12611	. . . . . . . . 9 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑄‘𝑀) ∈ (-π[,]π)) → (𝑄‘𝑀) ≤ π)
184	140, 141, 182, 183	syl3anc 1351	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝑀) ≤ π)
185	184	3ad2ant1 1113	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) ≤ π)
186		id 22	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖) = π → (𝑄‘𝑖) = π)
187	186	eqcomd 2784	. . . . . . . . 9 ⊢ ((𝑄‘𝑖) = π → π = (𝑄‘𝑖))
188	187	3ad2ant3 1115	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π = (𝑄‘𝑖))
189	105	adantr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑖))
190		fveq2 6501	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
191	190	adantl 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) = (𝑄‘𝑀))
192	189, 191	breqtrd 4956	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
193	109	ad2antlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 ∈ ℝ)
194		elfzel2 12725	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
195	194	zred 11903	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
196	195	ad2antlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑀 ∈ ℝ)
197		elfzle2 12730	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ≤ 𝑀)
198	197	ad2antlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 ≤ 𝑀)
199		neqne 2975	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑖 = 𝑀 → 𝑖 ≠ 𝑀)
200	199	necomd 3022	. . . . . . . . . . . . 13 ⊢ (¬ 𝑖 = 𝑀 → 𝑀 ≠ 𝑖)
201	200	adantl 474	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑀 ≠ 𝑖)
202	193, 196, 198, 201	leneltd 10596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → 𝑖 < 𝑀)
203	104	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) ∈ ℝ)
204	87, 182	sseldi 3858	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
205	204	ad2antrr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑀) ∈ ℝ)
206		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
207	93	ad2antrr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
208		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
209	181	adantr 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (0...𝑀))
210	208, 209	jca 504	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)))
211	210	adantr 473	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)))
212		isorel 6904	. . . . . . . . . . . . . 14 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑀 ∈ (0...𝑀))) → (𝑖 < 𝑀 ↔ (𝑄‘𝑖) < (𝑄‘𝑀)))
213	207, 211, 212	syl2anc 576	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑖 < 𝑀 ↔ (𝑄‘𝑖) < (𝑄‘𝑀)))
214	206, 213	mpbid 224	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) < (𝑄‘𝑀))
215	203, 205, 214	ltled 10590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
216	202, 215	syldan 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑀) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
217	192, 216	pm2.61dan 800	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
218	217	3adant3 1112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
219	188, 218	eqbrtrd 4952	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π ≤ (𝑄‘𝑀))
220	204	3ad2ant1 1113	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) ∈ ℝ)
221	14	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → π ∈ ℝ)
222	220, 221	letri3d 10584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → ((𝑄‘𝑀) = π ↔ ((𝑄‘𝑀) ≤ π ∧ π ≤ (𝑄‘𝑀))))
223	185, 219, 222	mpbir2and 700	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀) ∧ (𝑄‘𝑖) = π) → (𝑄‘𝑀) = π)
224	223	rexlimdv3a 3231	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = π → (𝑄‘𝑀) = π))
225	178, 224	mpd 15	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = π)
226		elfzoelz 12857	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
227	226	zred 11903	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
228	227	ltp1d 11373	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 < (𝑖 + 1))
229	228	adantl 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 < (𝑖 + 1))
230		elfzofz 12872	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
231		fzofzp1 12952	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
232	230, 231	jca 504	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 ∈ (0...𝑀) ∧ (𝑖 + 1) ∈ (0...𝑀)))
233		isorel 6904	. . . . . . 7 ⊢ ((𝑄 Isom < , < ((0...𝑀), 𝐻) ∧ (𝑖 ∈ (0...𝑀) ∧ (𝑖 + 1) ∈ (0...𝑀))) → (𝑖 < (𝑖 + 1) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
234	93, 232, 233	syl2an 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 < (𝑖 + 1) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
235	229, 234	mpbid 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
236	235	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
237	173, 225, 236	jca31 507	. . 3 ⊢ (𝜑 → (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
238	7	fourierdlem2 41826	. . . 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))))))
239	69, 238	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))))))
240	101, 237, 239	mpbir2and 700	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
241		fourierdlem114.g	. . . . 5 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
242	241	reseq1i 5692	. . . 4 ⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
243	16	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
244	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
245	179	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
246		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
247	243, 244, 245, 246	fourierdlem27 41851	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π(,)π))
248	247	resabs1d 5731	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
249	242, 248	syl5req 2827	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
250		fourierdlem114.gcn	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
251	250, 7, 69, 240, 11, 167	fourierdlem38 41862	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
252	249, 251	eqeltrd 2866	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
253	249	oveq1d 6993	. . 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)))
254	250	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 ∈ (dom 𝐺–cn→ℂ))
255		fourierdlem114.rlim	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
256	255	adantlr 702	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
257		fourierdlem114.llim	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
258	257	adantlr 702	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
259	93	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 Isom < , < ((0...𝑀), 𝐻))
260	259, 94, 95	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶𝐻)
261	81	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) ∈ ℝ)
262	259, 165, 166	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ran 𝑄 = 𝐻)
263	254, 256, 258, 259, 260, 246, 235, 247, 261, 11, 262	fourierdlem46 41869	. . . 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))) ≠ ∅))
264	263	simpld 487	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
265	253, 264	eqnetrd 3034	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
266	249	oveq1d 6993	. . 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))))
267	263	simprd 488	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
268	266, 267	eqnetrd 3034	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
269		fourierdlem114.a	. 2 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
270		fourierdlem114.b	. 2 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
271		fourierdlem114.s	. 2 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))
272	83	tpid3 4584	. . . . 5 ⊢ (𝐸‘𝑋) ∈ {-π, π, (𝐸‘𝑋)}
273		elun1 4043	. . . . 5 ⊢ ((𝐸‘𝑋) ∈ {-π, π, (𝐸‘𝑋)} → (𝐸‘𝑋) ∈ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)))
274	272, 273	mp1i 13	. . . 4 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)))
275	274, 11	syl6eleqr 2877	. . 3 ⊢ (𝜑 → (𝐸‘𝑋) ∈ 𝐻)
276	275, 167	eleqtrrd 2869	. 2 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ran 𝑄)
277	1, 2, 3, 4, 5, 6, 7, 69, 240, 252, 265, 268, 269, 270, 271, 78, 276	fourierdlem113 41936	1 ⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 ∃wrex 3089 {crab 3092 Vcvv 3415 ∖ cdif 3828 ∪ cun 3829 ⊆ wss 3831 ∅c0 4180 {cpr 4444 {ctp 4446 class class class wbr 4930 ↦ cmpt 5009 dom cdm 5408 ran crn 5409 ↾ cres 5410 ℩cio 6152 Fn wfn 6185 ⟶wf 6186 –onto→wfo 6188 –1-1-onto→wf1o 6189 ‘cfv 6190 Isom wiso 6191 (class class class)co 6978 ↑_𝑚 cmap 8208 Fincfn 8308 ℂcc 10335 ℝcr 10336 0cc0 10337 1c1 10338 + caddc 10340 · cmul 10342 +∞cpnf 10473 -∞cmnf 10474 ℝ^*cxr 10475 < clt 10476 ≤ cle 10477 − cmin 10672 -cneg 10673 / cdiv 11100 ℕcn 11441 2c2 11498 ℕ₀cn0 11710 ℤcz 11796 ℤ_≥cuz 12061 (,)cioo 12557 (,]cioc 12558 [,)cico 12559 [,]cicc 12560 ...cfz 12711 ..^cfzo 12852 ⌊cfl 12978 seqcseq 13187 ♯chash 13508 ⇝ cli 14705 Σcsu 14906 sincsin 15280 cosccos 15281 πcpi 15283 –cn→ccncf 23190 ∫citg 23925 lim_ℂ climc 24166 D cdv 24167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cc 9657 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-symdif 4108 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-disj 4899 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-ofr 7230 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-omul 7912 df-er 8091 df-map 8210 df-pm 8211 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-dju 9126 df-card 9164 df-acn 9167 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-xnn0 11783 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ioc 12562 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-fl 12980 df-mod 13056 df-seq 13188 df-exp 13248 df-fac 13452 df-bc 13481 df-hash 13509 df-shft 14290 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-limsup 14692 df-clim 14709 df-rlim 14710 df-sum 14907 df-ef 15284 df-sin 15286 df-cos 15287 df-pi 15289 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-mulg 18015 df-cntz 18221 df-cmn 18671 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-lp 21451 df-perf 21452 df-cn 21542 df-cnp 21543 df-t1 21629 df-haus 21630 df-cmp 21702 df-tx 21877 df-hmeo 22070 df-fil 22161 df-fm 22253 df-flim 22254 df-flf 22255 df-xms 22636 df-ms 22637 df-tms 22638 df-cncf 23192 df-ovol 23771 df-vol 23772 df-mbf 23926 df-itg1 23927 df-itg2 23928 df-ibl 23929 df-itg 23930 df-0p 23977 df-ditg 24151 df-limc 24170 df-dv 24171

This theorem is referenced by: fourierdlem115 41938

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