fourierdlem104

Description: The half upper part of the integral equal to the fourier partial sum, converges to half the right limit of the original function. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fourierdlem104.f	$⊢ φ \to F : ℝ ⟶ ℝ$
	fourierdlem104.xre	$⊢ φ \to X \in ℝ$
	fourierdlem104.p	$⊢ P = (m \in ℕ ⟼ \{p \in (ℝ^{(0 \dots m)}) \| ((p (0) = - π + X \land p (m) = π + X) \land \forall i \in (0 ..^m) p (i) < p (i + 1))\})$
	fourierdlem104.m	$⊢ φ \to M \in ℕ$
	fourierdlem104.v	$⊢ φ \to V \in P (M)$
	fourierdlem104.x	$⊢ φ \to X \in ran V$
	fourierdlem104.fcn	$⊢ (φ \land i \in (0 ..^M)) \to ({F ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) \underset{cn}{⟶} ℂ$
	fourierdlem104.fbdioo	$⊢ (φ \land i \in (0 ..^M)) \to \exists w \in ℝ \forall t \in (V (i), V (i + 1)) \|F (t)\| \leq w$
	fourierdlem104.fdvcn	$⊢ (φ \land i \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) \underset{cn}{⟶} ℝ$
	fourierdlem104.fdvbd	$⊢ (φ \land i \in (0 ..^M)) \to \exists z \in ℝ \forall t \in (V (i), V (i + 1)) \|F_{ℝ}^{'} (t)\| \leq z$
	fourierdlem104.r	$⊢ (φ \land i \in (0 ..^M)) \to R \in (({F ↾}_{(V (i), V (i + 1))}) {lim}_{ℂ} V (i))$
	fourierdlem104.l	$⊢ (φ \land i \in (0 ..^M)) \to L \in (({F ↾}_{(V (i), V (i + 1))}) {lim}_{ℂ} V (i + 1))$
	fourierdlem104.h	$⊢ H = (s \in [- π, π] ⟼ if (s = 0, 0, \frac{F (X + s) - if (0 < s, Y, W)}{s}))$
	fourierdlem104.k	$⊢ K = (s \in [- π, π] ⟼ if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})}))$
	fourierdlem104.u	$⊢ U = (s \in [- π, π] ⟼ H (s) K (s))$
	fourierdlem104.s	$⊢ S = (s \in [- π, π] ⟼ \sin (n + (\frac{1}{2})) s)$
	fourierdlem104.g	$⊢ G = (s \in [- π, π] ⟼ U (s) S (s))$
	fourierdlem104.z	$⊢ Z = (m \in ℕ ⟼ \int_{(0, π)} F (X + s) D (m) (s) d s)$
	fourierdlem104.e	$⊢ E = (n \in ℕ ⟼ \frac{\int_{(0, π)} G (s) d s}{π})$
	fourierdlem104.y	$⊢ φ \to Y \in (({F ↾}_{(X, +∞)}) {lim}_{ℂ} X)$
	fourierdlem104.w	$⊢ φ \to W \in (({F ↾}_{(−∞, X)}) {lim}_{ℂ} X)$
	fourierdlem104.a	$⊢ φ \to A \in (({F_{ℝ}^{'} ↾}_{(−∞, X)}) {lim}_{ℂ} X)$
	fourierdlem104.b	$⊢ φ \to B \in (({F_{ℝ}^{'} ↾}_{(X, +∞)}) {lim}_{ℂ} X)$
	fourierdlem104.d	$⊢ D = (n \in ℕ ⟼ (s \in ℝ ⟼ if (s \mod 2 π = 0, \frac{2 n + 1}{2 π}, \frac{\sin (n + (\frac{1}{2})) s}{2 π \sin (\frac{s}{2})})))$
	fourierdlem104.o	$⊢ O = {U ↾}_{[d, π]}$
	fourierdlem104.t	$⊢ T = \{d, π\} \cup (ran Q \cap (d, π))$
	fourierdlem104.n	$⊢ N = \|T\| - 1$
	fourierdlem104.j	$⊢ J = (ι f \| f {Isom}_{<, <} ((0 \dots N), T))$
	fourierdlem104.q	$⊢ Q = (i \in (0 \dots M) ⟼ V (i) - X)$
	fourierdlem104.1	$⊢ C = (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))$
	fourierdlem104.ch	$⊢ χ \leftrightarrow (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
Assertion	fourierdlem104	$⊢ φ \to Z ⇝ (\frac{Y}{2})$

Proof

Step	Hyp	Ref	Expression
1		fourierdlem104.f	$⊢ φ \to F : ℝ ⟶ ℝ$
2		fourierdlem104.xre	$⊢ φ \to X \in ℝ$
3		fourierdlem104.p	$⊢ P = (m \in ℕ ⟼ \{p \in (ℝ^{(0 \dots m)}) \| ((p (0) = - π + X \land p (m) = π + X) \land \forall i \in (0 ..^m) p (i) < p (i + 1))\})$
4		fourierdlem104.m	$⊢ φ \to M \in ℕ$
5		fourierdlem104.v	$⊢ φ \to V \in P (M)$
6		fourierdlem104.x	$⊢ φ \to X \in ran V$
7		fourierdlem104.fcn	$⊢ (φ \land i \in (0 ..^M)) \to ({F ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) \underset{cn}{⟶} ℂ$
8		fourierdlem104.fbdioo	$⊢ (φ \land i \in (0 ..^M)) \to \exists w \in ℝ \forall t \in (V (i), V (i + 1)) \|F (t)\| \leq w$
9		fourierdlem104.fdvcn	$⊢ (φ \land i \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) \underset{cn}{⟶} ℝ$
10		fourierdlem104.fdvbd	$⊢ (φ \land i \in (0 ..^M)) \to \exists z \in ℝ \forall t \in (V (i), V (i + 1)) \|F_{ℝ}^{'} (t)\| \leq z$
11		fourierdlem104.r	$⊢ (φ \land i \in (0 ..^M)) \to R \in (({F ↾}_{(V (i), V (i + 1))}) {lim}_{ℂ} V (i))$
12		fourierdlem104.l	$⊢ (φ \land i \in (0 ..^M)) \to L \in (({F ↾}_{(V (i), V (i + 1))}) {lim}_{ℂ} V (i + 1))$
13		fourierdlem104.h	$⊢ H = (s \in [- π, π] ⟼ if (s = 0, 0, \frac{F (X + s) - if (0 < s, Y, W)}{s}))$
14		fourierdlem104.k	$⊢ K = (s \in [- π, π] ⟼ if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})}))$
15		fourierdlem104.u	$⊢ U = (s \in [- π, π] ⟼ H (s) K (s))$
16		fourierdlem104.s	$⊢ S = (s \in [- π, π] ⟼ \sin (n + (\frac{1}{2})) s)$
17		fourierdlem104.g	$⊢ G = (s \in [- π, π] ⟼ U (s) S (s))$
18		fourierdlem104.z	$⊢ Z = (m \in ℕ ⟼ \int_{(0, π)} F (X + s) D (m) (s) d s)$
19		fourierdlem104.e	$⊢ E = (n \in ℕ ⟼ \frac{\int_{(0, π)} G (s) d s}{π})$
20		fourierdlem104.y	$⊢ φ \to Y \in (({F ↾}_{(X, +∞)}) {lim}_{ℂ} X)$
21		fourierdlem104.w	$⊢ φ \to W \in (({F ↾}_{(−∞, X)}) {lim}_{ℂ} X)$
22		fourierdlem104.a	$⊢ φ \to A \in (({F_{ℝ}^{'} ↾}_{(−∞, X)}) {lim}_{ℂ} X)$
23		fourierdlem104.b	$⊢ φ \to B \in (({F_{ℝ}^{'} ↾}_{(X, +∞)}) {lim}_{ℂ} X)$
24		fourierdlem104.d	$⊢ D = (n \in ℕ ⟼ (s \in ℝ ⟼ if (s \mod 2 π = 0, \frac{2 n + 1}{2 π}, \frac{\sin (n + (\frac{1}{2})) s}{2 π \sin (\frac{s}{2})})))$
25		fourierdlem104.o	$⊢ O = {U ↾}_{[d, π]}$
26		fourierdlem104.t	$⊢ T = \{d, π\} \cup (ran Q \cap (d, π))$
27		fourierdlem104.n	$⊢ N = \|T\| - 1$
28		fourierdlem104.j	$⊢ J = (ι f \| f {Isom}_{<, <} ((0 \dots N), T))$
29		fourierdlem104.q	$⊢ Q = (i \in (0 \dots M) ⟼ V (i) - X)$
30		fourierdlem104.1	$⊢ C = (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))$
31		fourierdlem104.ch	$⊢ χ \leftrightarrow (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
32		eqid	$⊢ ℤ_{\geq 1} = ℤ_{\geq 1}$
33		1zzd	$⊢ φ \to 1 \in ℤ$
34		nfv	$⊢ Ⅎ n φ$
35		nfmpt1	$⊢ \underline{Ⅎ} n (n \in ℕ ⟼ \int_{(0, π)} G (s) d s)$
36		nfmpt1	$⊢ \underline{Ⅎ} n (n \in ℕ ⟼ π)$
37		nfmpt1	$⊢ \underline{Ⅎ} n (n \in ℕ ⟼ \frac{\int_{(0, π)} G (s) d s}{π})$
38	19 37	nfcxfr	$⊢ \underline{Ⅎ} n E$
39		nnuz	$⊢ ℕ = ℤ_{\geq 1}$
40		elioore	$⊢ d \in (0, π) \to d \in ℝ$
41	40	adantl	$⊢ (φ \land d \in (0, π)) \to d \in ℝ$
42		pire	$⊢ π \in ℝ$
43	42	a1i	$⊢ (φ \land d \in (0, π)) \to π \in ℝ$
44		ioossre	$⊢ (X, +∞) \subseteq ℝ$
45	44	a1i	$⊢ φ \to (X, +∞) \subseteq ℝ$
46	1 45	fssresd	$⊢ φ \to ({F ↾}_{(X, +∞)}) : (X, +∞) ⟶ ℝ$
47		ioosscn	$⊢ (X, +∞) \subseteq ℂ$
48	47	a1i	$⊢ φ \to (X, +∞) \subseteq ℂ$
49		eqid	$⊢ TopOpen (ℂ_{fld}) = TopOpen (ℂ_{fld})$
50		pnfxr	$⊢ +∞ \in ℝ^{*}$
51	50	a1i	$⊢ φ \to +∞ \in ℝ^{*}$
52	2	ltpnfd	$⊢ φ \to X < +∞$
53	49 51 2 52	lptioo1cn	$⊢ φ \to X \in limPt (TopOpen (ℂ_{fld})) ((X, +∞))$
54	46 48 53 20	limcrecl	$⊢ φ \to Y \in ℝ$
55		ioossre	$⊢ (−∞, X) \subseteq ℝ$
56	55	a1i	$⊢ φ \to (−∞, X) \subseteq ℝ$
57	1 56	fssresd	$⊢ φ \to ({F ↾}_{(−∞, X)}) : (−∞, X) ⟶ ℝ$
58		ioosscn	$⊢ (−∞, X) \subseteq ℂ$
59	58	a1i	$⊢ φ \to (−∞, X) \subseteq ℂ$
60		mnfxr	$⊢ −∞ \in ℝ^{*}$
61	60	a1i	$⊢ φ \to −∞ \in ℝ^{*}$
62	2	mnfltd	$⊢ φ \to −∞ < X$
63	49 61 2 62	lptioo2cn	$⊢ φ \to X \in limPt (TopOpen (ℂ_{fld})) ((−∞, X))$
64	57 59 63 21	limcrecl	$⊢ φ \to W \in ℝ$
65	1 2 54 64 13 14 15	fourierdlem55	$⊢ φ \to U : [- π, π] ⟶ ℝ$
66		ax-resscn	$⊢ ℝ \subseteq ℂ$
67	66	a1i	$⊢ φ \to ℝ \subseteq ℂ$
68	65 67	fssd	$⊢ φ \to U : [- π, π] ⟶ ℂ$
69	68	adantr	$⊢ (φ \land d \in (0, π)) \to U : [- π, π] ⟶ ℂ$
70	42	renegcli	$⊢ - π \in ℝ$
71	70	a1i	$⊢ (φ \land d \in (0, π)) \to - π \in ℝ$
72	70	a1i	$⊢ d \in (0, π) \to - π \in ℝ$
73		0red	$⊢ d \in (0, π) \to 0 \in ℝ$
74		negpilt0	$⊢ - π < 0$
75	74	a1i	$⊢ d \in (0, π) \to - π < 0$
76		0xr	$⊢ 0 \in ℝ^{*}$
77	42	rexri	$⊢ π \in ℝ^{*}$
78		ioogtlb	$⊢ (0 \in ℝ^{} \land π \in ℝ^{} \land d \in (0, π)) \to 0 < d$
79	76 77 78	mp3an12	$⊢ d \in (0, π) \to 0 < d$
80	72 73 40 75 79	lttrd	$⊢ d \in (0, π) \to - π < d$
81	72 40 80	ltled	$⊢ d \in (0, π) \to - π \leq d$
82	81	adantl	$⊢ (φ \land d \in (0, π)) \to - π \leq d$
83	43	leidd	$⊢ (φ \land d \in (0, π)) \to π \leq π$
84		iccss	$⊢ ((- π \in ℝ \land π \in ℝ) \land (- π \leq d \land π \leq π)) \to [d, π] \subseteq [- π, π]$
85	71 43 82 83 84	syl22anc	$⊢ (φ \land d \in (0, π)) \to [d, π] \subseteq [- π, π]$
86	69 85	fssresd	$⊢ (φ \land d \in (0, π)) \to ({U ↾}_{[d, π]}) : [d, π] ⟶ ℂ$
87	25	a1i	$⊢ (φ \land d \in (0, π)) \to O = {U ↾}_{[d, π]}$
88	87	feq1d	$⊢ (φ \land d \in (0, π)) \to (O : [d, π] ⟶ ℂ \leftrightarrow ({U ↾}_{[d, π]}) : [d, π] ⟶ ℂ)$
89	86 88	mpbird	$⊢ (φ \land d \in (0, π)) \to O : [d, π] ⟶ ℂ$
90	42	elexi	$⊢ π \in V$
91	90	prid2	$⊢ π \in \{d, π\}$
92		elun1	$⊢ π \in \{d, π\} \to π \in (\{d, π\} \cup (ran Q \cap (d, π)))$
93	91 92	ax-mp	$⊢ π \in (\{d, π\} \cup (ran Q \cap (d, π)))$
94	93 26	eleqtrri	$⊢ π \in T$
95	94	ne0ii	$⊢ T \neq \emptyset$
96	95	a1i	$⊢ φ \to T \neq \emptyset$
97		prfi	$⊢ \{d, π\} \in Fin$
98	97	a1i	$⊢ φ \to \{d, π\} \in Fin$
99		fzfi	$⊢ (0 \dots M) \in Fin$
100	29	rnmptfi	$⊢ (0 \dots M) \in Fin \to ran Q \in Fin$
101	99 100	ax-mp	$⊢ ran Q \in Fin$
102		infi	$⊢ ran Q \in Fin \to ran Q \cap (d, π) \in Fin$
103	101 102	mp1i	$⊢ φ \to ran Q \cap (d, π) \in Fin$
104		unfi	$⊢ (\{d, π\} \in Fin \land ran Q \cap (d, π) \in Fin) \to \{d, π\} \cup (ran Q \cap (d, π)) \in Fin$
105	98 103 104	syl2anc	$⊢ φ \to \{d, π\} \cup (ran Q \cap (d, π)) \in Fin$
106	26 105	eqeltrid	$⊢ φ \to T \in Fin$
107		hashnncl	$⊢ T \in Fin \to (\|T\| \in ℕ \leftrightarrow T \neq \emptyset)$
108	106 107	syl	$⊢ φ \to (\|T\| \in ℕ \leftrightarrow T \neq \emptyset)$
109	96 108	mpbird	$⊢ φ \to \|T\| \in ℕ$
110		nnm1nn0	$⊢ \|T\| \in ℕ \to \|T\| - 1 \in ℕ_{0}$
111	109 110	syl	$⊢ φ \to \|T\| - 1 \in ℕ_{0}$
112	27 111	eqeltrid	$⊢ φ \to N \in ℕ_{0}$
113	112	adantr	$⊢ (φ \land d \in (0, π)) \to N \in ℕ_{0}$
114		0red	$⊢ (φ \land d \in (0, π)) \to 0 \in ℝ$
115		1red	$⊢ (φ \land d \in (0, π)) \to 1 \in ℝ$
116	113	nn0red	$⊢ (φ \land d \in (0, π)) \to N \in ℝ$
117		0lt1	$⊢ 0 < 1$
118	117	a1i	$⊢ (φ \land d \in (0, π)) \to 0 < 1$
119		2re	$⊢ 2 \in ℝ$
120	119	a1i	$⊢ (φ \land d \in (0, π)) \to 2 \in ℝ$
121	109	nnred	$⊢ φ \to \|T\| \in ℝ$
122	121	adantr	$⊢ (φ \land d \in (0, π)) \to \|T\| \in ℝ$
123		iooltub	$⊢ (0 \in ℝ^{} \land π \in ℝ^{} \land d \in (0, π)) \to d < π$
124	76 77 123	mp3an12	$⊢ d \in (0, π) \to d < π$
125	40 124	ltned	$⊢ d \in (0, π) \to d \neq π$
126	125	adantl	$⊢ (φ \land d \in (0, π)) \to d \neq π$
127		hashprg	$⊢ (d \in ℝ \land π \in ℝ) \to (d \neq π \leftrightarrow \|\{d, π\}\| = 2)$
128	41 42 127	sylancl	$⊢ (φ \land d \in (0, π)) \to (d \neq π \leftrightarrow \|\{d, π\}\| = 2)$
129	126 128	mpbid	$⊢ (φ \land d \in (0, π)) \to \|\{d, π\}\| = 2$
130	129	eqcomd	$⊢ (φ \land d \in (0, π)) \to 2 = \|\{d, π\}\|$
131	106	adantr	$⊢ (φ \land d \in (0, π)) \to T \in Fin$
132		ssun1	$⊢ \{d, π\} \subseteq \{d, π\} \cup (ran Q \cap (d, π))$
133	132 26	sseqtrri	$⊢ \{d, π\} \subseteq T$
134		hashssle	$⊢ (T \in Fin \land \{d, π\} \subseteq T) \to \|\{d, π\}\| \leq \|T\|$
135	131 133 134	sylancl	$⊢ (φ \land d \in (0, π)) \to \|\{d, π\}\| \leq \|T\|$
136	130 135	eqbrtrd	$⊢ (φ \land d \in (0, π)) \to 2 \leq \|T\|$
137	120 122 115 136	lesub1dd	$⊢ (φ \land d \in (0, π)) \to 2 - 1 \leq \|T\| - 1$
138		1e2m1	$⊢ 1 = 2 - 1$
139	137 138 27	3brtr4g	$⊢ (φ \land d \in (0, π)) \to 1 \leq N$
140	114 115 116 118 139	ltletrd	$⊢ (φ \land d \in (0, π)) \to 0 < N$
141	140	gt0ne0d	$⊢ (φ \land d \in (0, π)) \to N \neq 0$
142		elnnne0	$⊢ N \in ℕ \leftrightarrow (N \in ℕ_{0} \land N \neq 0)$
143	113 141 142	sylanbrc	$⊢ (φ \land d \in (0, π)) \to N \in ℕ$
144	41	leidd	$⊢ (φ \land d \in (0, π)) \to d \leq d$
145	42	a1i	$⊢ d \in (0, π) \to π \in ℝ$
146	40 145 124	ltled	$⊢ d \in (0, π) \to d \leq π$
147	146	adantl	$⊢ (φ \land d \in (0, π)) \to d \leq π$
148	41 43 41 144 147	eliccd	$⊢ (φ \land d \in (0, π)) \to d \in [d, π]$
149	41 43 43 147 83	eliccd	$⊢ (φ \land d \in (0, π)) \to π \in [d, π]$
150	148 149	jca	$⊢ (φ \land d \in (0, π)) \to (d \in [d, π] \land π \in [d, π])$
151		vex	$⊢ d \in V$
152	151 90	prss	$⊢ (d \in [d, π] \land π \in [d, π]) \leftrightarrow \{d, π\} \subseteq [d, π]$
153	150 152	sylib	$⊢ (φ \land d \in (0, π)) \to \{d, π\} \subseteq [d, π]$
154		inss2	$⊢ ran Q \cap (d, π) \subseteq (d, π)$
155	154	a1i	$⊢ (φ \land d \in (0, π)) \to ran Q \cap (d, π) \subseteq (d, π)$
156		ioossicc	$⊢ (d, π) \subseteq [d, π]$
157	155 156	sstrdi	$⊢ (φ \land d \in (0, π)) \to ran Q \cap (d, π) \subseteq [d, π]$
158	153 157	unssd	$⊢ (φ \land d \in (0, π)) \to \{d, π\} \cup (ran Q \cap (d, π)) \subseteq [d, π]$
159	26 158	eqsstrid	$⊢ (φ \land d \in (0, π)) \to T \subseteq [d, π]$
160	151	prid1	$⊢ d \in \{d, π\}$
161		elun1	$⊢ d \in \{d, π\} \to d \in (\{d, π\} \cup (ran Q \cap (d, π)))$
162	160 161	ax-mp	$⊢ d \in (\{d, π\} \cup (ran Q \cap (d, π)))$
163	162 26	eleqtrri	$⊢ d \in T$
164	163	a1i	$⊢ (φ \land d \in (0, π)) \to d \in T$
165	94	a1i	$⊢ (φ \land d \in (0, π)) \to π \in T$
166	131 27 28 41 43 159 164 165	fourierdlem52	$⊢ (φ \land d \in (0, π)) \to ((J : (0 \dots N) ⟶ [d, π] \land J (0) = d) \land J (N) = π)$
167	166	simplld	$⊢ (φ \land d \in (0, π)) \to J : (0 \dots N) ⟶ [d, π]$
168	166	simplrd	$⊢ (φ \land d \in (0, π)) \to J (0) = d$
169	166	simprd	$⊢ (φ \land d \in (0, π)) \to J (N) = π$
170		elfzoelz	$⊢ k \in (0 ..^N) \to k \in ℤ$
171	170	zred	$⊢ k \in (0 ..^N) \to k \in ℝ$
172	171	adantl	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to k \in ℝ$
173	172	ltp1d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to k < k + 1$
174	40 145	jca	$⊢ d \in (0, π) \to (d \in ℝ \land π \in ℝ)$
175	151 90	prss	$⊢ (d \in ℝ \land π \in ℝ) \leftrightarrow \{d, π\} \subseteq ℝ$
176	174 175	sylib	$⊢ d \in (0, π) \to \{d, π\} \subseteq ℝ$
177	176	adantl	$⊢ (φ \land d \in (0, π)) \to \{d, π\} \subseteq ℝ$
178		ioossre	$⊢ (d, π) \subseteq ℝ$
179	154 178	sstri	$⊢ ran Q \cap (d, π) \subseteq ℝ$
180	179	a1i	$⊢ (φ \land d \in (0, π)) \to ran Q \cap (d, π) \subseteq ℝ$
181	177 180	unssd	$⊢ (φ \land d \in (0, π)) \to \{d, π\} \cup (ran Q \cap (d, π)) \subseteq ℝ$
182	26 181	eqsstrid	$⊢ (φ \land d \in (0, π)) \to T \subseteq ℝ$
183	131 182 28 27	fourierdlem36	$⊢ (φ \land d \in (0, π)) \to J {Isom}_{<, <} ((0 \dots N), T)$
184	183	adantr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J {Isom}_{<, <} ((0 \dots N), T)$
185		elfzofz	$⊢ k \in (0 ..^N) \to k \in (0 \dots N)$
186	185	adantl	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to k \in (0 \dots N)$
187		fzofzp1	$⊢ k \in (0 ..^N) \to k + 1 \in (0 \dots N)$
188	187	adantl	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to k + 1 \in (0 \dots N)$
189		isorel	$⊢ (J {Isom}_{<, <} ((0 \dots N), T) \land (k \in (0 \dots N) \land k + 1 \in (0 \dots N))) \to (k < k + 1 \leftrightarrow J (k) < J (k + 1))$
190	184 186 188 189	syl12anc	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (k < k + 1 \leftrightarrow J (k) < J (k + 1))$
191	173 190	mpbid	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J (k) < J (k + 1)$
192	65	adantr	$⊢ (φ \land d \in (0, π)) \to U : [- π, π] ⟶ ℝ$
193	192 85	feqresmpt	$⊢ (φ \land d \in (0, π)) \to {U ↾}_{[d, π]} = (s \in [d, π] ⟼ U (s))$
194	85	sselda	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to s \in [- π, π]$
195	1 2 54 64 13	fourierdlem9	$⊢ φ \to H : [- π, π] ⟶ ℝ$
196	195	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to H : [- π, π] ⟶ ℝ$
197	196 194	ffvelrnd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to H (s) \in ℝ$
198	14	fourierdlem43	$⊢ K : [- π, π] ⟶ ℝ$
199	198	a1i	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to K : [- π, π] ⟶ ℝ$
200	199 194	ffvelrnd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to K (s) \in ℝ$
201	197 200	remulcld	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to H (s) K (s) \in ℝ$
202	15	fvmpt2	$⊢ (s \in [- π, π] \land H (s) K (s) \in ℝ) \to U (s) = H (s) K (s)$
203	194 201 202	syl2anc	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to U (s) = H (s) K (s)$
204		0red	$⊢ (d \in (0, π) \land s \in [d, π]) \to 0 \in ℝ$
205	40	adantr	$⊢ (d \in (0, π) \land s \in [d, π]) \to d \in ℝ$
206	42	a1i	$⊢ (d \in (0, π) \land s \in [d, π]) \to π \in ℝ$
207		simpr	$⊢ (d \in (0, π) \land s \in [d, π]) \to s \in [d, π]$
208		eliccre	$⊢ (d \in ℝ \land π \in ℝ \land s \in [d, π]) \to s \in ℝ$
209	205 206 207 208	syl3anc	$⊢ (d \in (0, π) \land s \in [d, π]) \to s \in ℝ$
210	79	adantr	$⊢ (d \in (0, π) \land s \in [d, π]) \to 0 < d$
211	205	rexrd	$⊢ (d \in (0, π) \land s \in [d, π]) \to d \in ℝ^{*}$
212	77	a1i	$⊢ (d \in (0, π) \land s \in [d, π]) \to π \in ℝ^{*}$
213		iccgelb	$⊢ (d \in ℝ^{} \land π \in ℝ^{} \land s \in [d, π]) \to d \leq s$
214	211 212 207 213	syl3anc	$⊢ (d \in (0, π) \land s \in [d, π]) \to d \leq s$
215	204 205 209 210 214	ltletrd	$⊢ (d \in (0, π) \land s \in [d, π]) \to 0 < s$
216	215	gt0ne0d	$⊢ (d \in (0, π) \land s \in [d, π]) \to s \neq 0$
217	216	adantll	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to s \neq 0$
218	217	neneqd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to \neg s = 0$
219	218	iffalsed	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to if (s = 0, 0, \frac{F (X + s) - if (0 < s, Y, W)}{s}) = \frac{F (X + s) - if (0 < s, Y, W)}{s}$
220	215	adantll	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to 0 < s$
221	220	iftrued	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to if (0 < s, Y, W) = Y$
222	221	oveq2d	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to F (X + s) - if (0 < s, Y, W) = F (X + s) - Y$
223	222	oveq1d	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to \frac{F (X + s) - if (0 < s, Y, W)}{s} = \frac{F (X + s) - Y}{s}$
224	219 223	eqtrd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to if (s = 0, 0, \frac{F (X + s) - if (0 < s, Y, W)}{s}) = \frac{F (X + s) - Y}{s}$
225	1	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to F : ℝ ⟶ ℝ$
226	2	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to X \in ℝ$
227		iccssre	$⊢ (- π \in ℝ \land π \in ℝ) \to [- π, π] \subseteq ℝ$
228	70 42 227	mp2an	$⊢ [- π, π] \subseteq ℝ$
229	228 194	sselid	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to s \in ℝ$
230	226 229	readdcld	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to X + s \in ℝ$
231	225 230	ffvelrnd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to F (X + s) \in ℝ$
232	54	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to Y \in ℝ$
233	231 232	resubcld	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to F (X + s) - Y \in ℝ$
234	233 229 217	redivcld	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to \frac{F (X + s) - Y}{s} \in ℝ$
235	224 234	eqeltrd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to if (s = 0, 0, \frac{F (X + s) - if (0 < s, Y, W)}{s}) \in ℝ$
236	13	fvmpt2	$⊢ (s \in [- π, π] \land if (s = 0, 0, \frac{F (X + s) - if (0 < s, Y, W)}{s}) \in ℝ) \to H (s) = if (s = 0, 0, \frac{F (X + s) - if (0 < s, Y, W)}{s})$
237	194 235 236	syl2anc	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to H (s) = if (s = 0, 0, \frac{F (X + s) - if (0 < s, Y, W)}{s})$
238	237 219 223	3eqtrd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to H (s) = \frac{F (X + s) - Y}{s}$
239	206	renegcld	$⊢ (d \in (0, π) \land s \in [d, π]) \to - π \in ℝ$
240	74	a1i	$⊢ (d \in (0, π) \land s \in [d, π]) \to - π < 0$
241	239 204 209 240 215	lttrd	$⊢ (d \in (0, π) \land s \in [d, π]) \to - π < s$
242	239 209 241	ltled	$⊢ (d \in (0, π) \land s \in [d, π]) \to - π \leq s$
243		iccleub	$⊢ (d \in ℝ^{} \land π \in ℝ^{} \land s \in [d, π]) \to s \leq π$
244	211 212 207 243	syl3anc	$⊢ (d \in (0, π) \land s \in [d, π]) \to s \leq π$
245	239 206 209 242 244	eliccd	$⊢ (d \in (0, π) \land s \in [d, π]) \to s \in [- π, π]$
246	216	neneqd	$⊢ (d \in (0, π) \land s \in [d, π]) \to \neg s = 0$
247	246	iffalsed	$⊢ (d \in (0, π) \land s \in [d, π]) \to if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})}) = \frac{s}{2 \sin (\frac{s}{2})}$
248	119	a1i	$⊢ (d \in (0, π) \land s \in [d, π]) \to 2 \in ℝ$
249	209	rehalfcld	$⊢ (d \in (0, π) \land s \in [d, π]) \to \frac{s}{2} \in ℝ$
250	249	resincld	$⊢ (d \in (0, π) \land s \in [d, π]) \to \sin (\frac{s}{2}) \in ℝ$
251	248 250	remulcld	$⊢ (d \in (0, π) \land s \in [d, π]) \to 2 \sin (\frac{s}{2}) \in ℝ$
252		2cnd	$⊢ (d \in (0, π) \land s \in [d, π]) \to 2 \in ℂ$
253	209	recnd	$⊢ (d \in (0, π) \land s \in [d, π]) \to s \in ℂ$
254	253	halfcld	$⊢ (d \in (0, π) \land s \in [d, π]) \to \frac{s}{2} \in ℂ$
255	254	sincld	$⊢ (d \in (0, π) \land s \in [d, π]) \to \sin (\frac{s}{2}) \in ℂ$
256		2ne0	$⊢ 2 \neq 0$
257	256	a1i	$⊢ (d \in (0, π) \land s \in [d, π]) \to 2 \neq 0$
258		fourierdlem44	$⊢ (s \in [- π, π] \land s \neq 0) \to \sin (\frac{s}{2}) \neq 0$
259	245 216 258	syl2anc	$⊢ (d \in (0, π) \land s \in [d, π]) \to \sin (\frac{s}{2}) \neq 0$
260	252 255 257 259	mulne0d	$⊢ (d \in (0, π) \land s \in [d, π]) \to 2 \sin (\frac{s}{2}) \neq 0$
261	209 251 260	redivcld	$⊢ (d \in (0, π) \land s \in [d, π]) \to \frac{s}{2 \sin (\frac{s}{2})} \in ℝ$
262	247 261	eqeltrd	$⊢ (d \in (0, π) \land s \in [d, π]) \to if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})}) \in ℝ$
263	14	fvmpt2	$⊢ (s \in [- π, π] \land if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})}) \in ℝ) \to K (s) = if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})})$
264	245 262 263	syl2anc	$⊢ (d \in (0, π) \land s \in [d, π]) \to K (s) = if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})})$
265	264	adantll	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to K (s) = if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})})$
266	238 265	oveq12d	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to H (s) K (s) = (\frac{F (X + s) - Y}{s}) if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})})$
267	218	iffalsed	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})}) = \frac{s}{2 \sin (\frac{s}{2})}$
268	267	oveq2d	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to (\frac{F (X + s) - Y}{s}) if (s = 0, 1, \frac{s}{2 \sin (\frac{s}{2})}) = (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})$
269	203 266 268	3eqtrd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to U (s) = (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})$
270	269	mpteq2dva	$⊢ (φ \land d \in (0, π)) \to (s \in [d, π] ⟼ U (s)) = (s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})}))$
271	87 193 270	3eqtrd	$⊢ (φ \land d \in (0, π)) \to O = (s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})}))$
272	271	adantr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to O = (s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})}))$
273	272	reseq1d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {O ↾}_{(J (k), J (k + 1))} = {(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}$
274	1	adantr	$⊢ (φ \land d \in (0, π)) \to F : ℝ ⟶ ℝ$
275	2	adantr	$⊢ (φ \land d \in (0, π)) \to X \in ℝ$
276	4	adantr	$⊢ (φ \land d \in (0, π)) \to M \in ℕ$
277	5	adantr	$⊢ (φ \land d \in (0, π)) \to V \in P (M)$
278	7	adantlr	$⊢ ((φ \land d \in (0, π)) \land i \in (0 ..^M)) \to ({F ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) \underset{cn}{⟶} ℂ$
279	11	adantlr	$⊢ ((φ \land d \in (0, π)) \land i \in (0 ..^M)) \to R \in (({F ↾}_{(V (i), V (i + 1))}) {lim}_{ℂ} V (i))$
280	12	adantlr	$⊢ ((φ \land d \in (0, π)) \land i \in (0 ..^M)) \to L \in (({F ↾}_{(V (i), V (i + 1))}) {lim}_{ℂ} V (i + 1))$
281	124	adantl	$⊢ (φ \land d \in (0, π)) \to d < π$
282	73 40	ltnled	$⊢ d \in (0, π) \to (0 < d \leftrightarrow \neg d \leq 0)$
283	79 282	mpbid	$⊢ d \in (0, π) \to \neg d \leq 0$
284	283	intn3an2d	$⊢ d \in (0, π) \to \neg (0 \in ℝ \land d \leq 0 \land 0 \leq π)$
285		elicc2	$⊢ (d \in ℝ \land π \in ℝ) \to (0 \in [d, π] \leftrightarrow (0 \in ℝ \land d \leq 0 \land 0 \leq π))$
286	40 42 285	sylancl	$⊢ d \in (0, π) \to (0 \in [d, π] \leftrightarrow (0 \in ℝ \land d \leq 0 \land 0 \leq π))$
287	284 286	mtbird	$⊢ d \in (0, π) \to \neg 0 \in [d, π]$
288	287	adantl	$⊢ (φ \land d \in (0, π)) \to \neg 0 \in [d, π]$
289	54	adantr	$⊢ (φ \land d \in (0, π)) \to Y \in ℝ$
290		eqid	$⊢ (s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) = (s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})}))$
291		eqid	$⊢ (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}) = (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})})$
292		eqid	$⊢ (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}) = (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})})$
293		fveq2	$⊢ l = i \to Q (l) = Q (i)$
294		oveq1	$⊢ l = i \to l + 1 = i + 1$
295	294	fveq2d	$⊢ l = i \to Q (l + 1) = Q (i + 1)$
296	293 295	oveq12d	$⊢ l = i \to (Q (l), Q (l + 1)) = (Q (i), Q (i + 1))$
297	296	sseq2d	$⊢ l = i \to ((J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)) \leftrightarrow (J (k), J (k + 1)) \subseteq (Q (i), Q (i + 1)))$
298	297	cbvriotavw	$⊢ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) = (ι i \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (i), Q (i + 1)))$
299	274 275 3 276 277 278 279 280 41 43 281 85 288 289 290 29 26 27 28 291 292 298	fourierdlem86	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (((\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}) \in (({(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k + 1)) \land (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}) \in (({(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k))) \land ({(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}) : (J (k), J (k + 1)) \underset{cn}{⟶} ℂ)$
300	299	simprd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ({(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}) : (J (k), J (k + 1)) \underset{cn}{⟶} ℂ$
301	273 300	eqeltrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ({O ↾}_{(J (k), J (k + 1))}) : (J (k), J (k + 1)) \underset{cn}{⟶} ℂ$
302	299	simplld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}) \in (({(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k + 1))$
303	272	eqcomd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) = O$
304	303	reseq1d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))} = {O ↾}_{(J (k), J (k + 1))}$
305	304	oveq1d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ({(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k + 1) = ({O ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k + 1)$
306	302 305	eleqtrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}) \in (({O ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k + 1))$
307	299	simplrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}) \in (({(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k))$
308	304	oveq1d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ({(s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k) = ({O ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k)$
309	307 308	eleqtrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}) \in (({O ↾}_{(J (k), J (k + 1))}) {lim}_{ℂ} J (k))$
310		eqid	$⊢ ℝ D O = ℝ D O$
311	89	adantr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to O : [d, π] ⟶ ℂ$
312	41	ad2antrr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to d \in ℝ$
313	42	a1i	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to π \in ℝ$
314		elioore	$⊢ s \in (J (k), J (k + 1)) \to s \in ℝ$
315	314	adantl	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to s \in ℝ$
316	85 228	sstrdi	$⊢ (φ \land d \in (0, π)) \to [d, π] \subseteq ℝ$
317	316	adantr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to [d, π] \subseteq ℝ$
318	167	adantr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J : (0 \dots N) ⟶ [d, π]$
319	318 186	ffvelrnd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J (k) \in [d, π]$
320	317 319	sseldd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J (k) \in ℝ$
321	320	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to J (k) \in ℝ$
322	41	adantr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to d \in ℝ$
323	322	rexrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to d \in ℝ^{*}$
324	77	a1i	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to π \in ℝ^{*}$
325		iccgelb	$⊢ (d \in ℝ^{} \land π \in ℝ^{} \land J (k) \in [d, π]) \to d \leq J (k)$
326	323 324 319 325	syl3anc	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to d \leq J (k)$
327	326	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to d \leq J (k)$
328	321	rexrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to J (k) \in ℝ^{*}$
329	318 188	ffvelrnd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J (k + 1) \in [d, π]$
330	317 329	sseldd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J (k + 1) \in ℝ$
331	330	rexrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J (k + 1) \in ℝ^{*}$
332	331	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to J (k + 1) \in ℝ^{*}$
333		simpr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to s \in (J (k), J (k + 1))$
334		ioogtlb	$⊢ (J (k) \in ℝ^{} \land J (k + 1) \in ℝ^{} \land s \in (J (k), J (k + 1))) \to J (k) < s$
335	328 332 333 334	syl3anc	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to J (k) < s$
336	312 321 315 327 335	lelttrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to d < s$
337	312 315 336	ltled	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to d \leq s$
338	330	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to J (k + 1) \in ℝ$
339		iooltub	$⊢ (J (k) \in ℝ^{} \land J (k + 1) \in ℝ^{} \land s \in (J (k), J (k + 1))) \to s < J (k + 1)$
340	328 332 333 339	syl3anc	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to s < J (k + 1)$
341		iccleub	$⊢ (d \in ℝ^{} \land π \in ℝ^{} \land J (k + 1) \in [d, π]) \to J (k + 1) \leq π$
342	323 324 329 341	syl3anc	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J (k + 1) \leq π$
343	342	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to J (k + 1) \leq π$
344	315 338 313 340 343	ltletrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to s < π$
345	315 313 344	ltled	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to s \leq π$
346	312 313 315 337 345	eliccd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to s \in [d, π]$
347	346	ralrimiva	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \forall s \in (J (k), J (k + 1)) s \in [d, π]$
348		dfss3	$⊢ (J (k), J (k + 1)) \subseteq [d, π] \leftrightarrow \forall s \in (J (k), J (k + 1)) s \in [d, π]$
349	347 348	sylibr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (J (k), J (k + 1)) \subseteq [d, π]$
350	311 349	feqresmpt	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {O ↾}_{(J (k), J (k + 1))} = (s \in (J (k), J (k + 1)) ⟼ O (s))$
351		simplll	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to φ$
352		simpllr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to d \in (0, π)$
353	25	fveq1i	$⊢ O (s) = ({U ↾}_{[d, π]}) (s)$
354	353	a1i	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to O (s) = ({U ↾}_{[d, π]}) (s)$
355		fvres	$⊢ s \in [d, π] \to ({U ↾}_{[d, π]}) (s) = U (s)$
356	355	adantl	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to ({U ↾}_{[d, π]}) (s) = U (s)$
357	265 267	eqtrd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to K (s) = \frac{s}{2 \sin (\frac{s}{2})}$
358	238 357	oveq12d	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to H (s) K (s) = (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})$
359	233	recnd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to F (X + s) - Y \in ℂ$
360	253	adantll	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to s \in ℂ$
361		2cnd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to 2 \in ℂ$
362	360	halfcld	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to \frac{s}{2} \in ℂ$
363	362	sincld	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to \sin (\frac{s}{2}) \in ℂ$
364	361 363	mulcld	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to 2 \sin (\frac{s}{2}) \in ℂ$
365	260	adantll	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to 2 \sin (\frac{s}{2}) \neq 0$
366	359 360 364 217 365	dmdcan2d	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})}) = \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})}$
367	203 358 366	3eqtrd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to U (s) = \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})}$
368	354 356 367	3eqtrd	$⊢ ((φ \land d \in (0, π)) \land s \in [d, π]) \to O (s) = \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})}$
369	351 352 346 368	syl21anc	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to O (s) = \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})}$
370	351 352 346 366	syl21anc	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})}) = \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})}$
371	370	eqcomd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})} = (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})$
372		eqidd	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) = (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t})$
373		oveq2	$⊢ t = s \to X + t = X + s$
374	373	fveq2d	$⊢ t = s \to F (X + t) = F (X + s)$
375	374	oveq1d	$⊢ t = s \to F (X + t) - Y = F (X + s) - Y$
376		id	$⊢ t = s \to t = s$
377	375 376	oveq12d	$⊢ t = s \to \frac{F (X + t) - Y}{t} = \frac{F (X + s) - Y}{s}$
378	377	adantl	$⊢ (((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \land t = s) \to \frac{F (X + t) - Y}{t} = \frac{F (X + s) - Y}{s}$
379		simpr	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to s \in (J (k), J (k + 1))$
380		ovex	$⊢ \frac{F (X + s) - Y}{s} \in V$
381	380	a1i	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to \frac{F (X + s) - Y}{s} \in V$
382	372 378 379 381	fvmptd	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) = \frac{F (X + s) - Y}{s}$
383		eqidd	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) = (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})})$
384		oveq1	$⊢ t = s \to \frac{t}{2} = \frac{s}{2}$
385	384	fveq2d	$⊢ t = s \to \sin (\frac{t}{2}) = \sin (\frac{s}{2})$
386	385	oveq2d	$⊢ t = s \to 2 \sin (\frac{t}{2}) = 2 \sin (\frac{s}{2})$
387	376 386	oveq12d	$⊢ t = s \to \frac{t}{2 \sin (\frac{t}{2})} = \frac{s}{2 \sin (\frac{s}{2})}$
388	387	adantl	$⊢ (((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \land t = s) \to \frac{t}{2 \sin (\frac{t}{2})} = \frac{s}{2 \sin (\frac{s}{2})}$
389		ovex	$⊢ \frac{s}{2 \sin (\frac{s}{2})} \in V$
390	389	a1i	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to \frac{s}{2 \sin (\frac{s}{2})} \in V$
391	383 388 379 390	fvmptd	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s) = \frac{s}{2 \sin (\frac{s}{2})}$
392	382 391	oveq12d	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s) = (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})$
393	392	eqcomd	$⊢ ((φ \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})}) = (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s)$
394	393	adantllr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})}) = (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s)$
395	369 371 394	3eqtrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land s \in (J (k), J (k + 1))) \to O (s) = (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s)$
396	395	mpteq2dva	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (s \in (J (k), J (k + 1)) ⟼ O (s)) = (s \in (J (k), J (k + 1)) ⟼ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s))$
397	350 396	eqtr2d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (s \in (J (k), J (k + 1)) ⟼ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s)) = {O ↾}_{(J (k), J (k + 1))}$
398	397	oveq2d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \frac{d (s \in (J (k), J (k + 1))⟼ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s))}{d_{ℝ} s} = ℝ D ({O ↾}_{(J (k), J (k + 1))})$
399	66	a1i	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ℝ \subseteq ℂ$
400	349 317	sstrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (J (k), J (k + 1)) \subseteq ℝ$
401	49	tgioo2	$⊢ topGen (ran (.)) = TopOpen (ℂ_{fld}) ↾_{𝑡} ℝ$
402	49 401	dvres	$⊢ ((ℝ \subseteq ℂ \land O : [d, π] ⟶ ℂ) \land ([d, π] \subseteq ℝ \land (J (k), J (k + 1)) \subseteq ℝ)) \to ℝ D ({O ↾}_{(J (k), J (k + 1))}) = {O_{ℝ}^{'} ↾}_{int (topGen (ran (.))) ((J (k), J (k + 1)))}$
403	399 311 317 400 402	syl22anc	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ℝ D ({O ↾}_{(J (k), J (k + 1))}) = {O_{ℝ}^{'} ↾}_{int (topGen (ran (.))) ((J (k), J (k + 1)))}$
404		ioontr	$⊢ int (topGen (ran (.))) ((J (k), J (k + 1))) = (J (k), J (k + 1))$
405	404	a1i	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to int (topGen (ran (.))) ((J (k), J (k + 1))) = (J (k), J (k + 1))$
406	405	reseq2d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {O_{ℝ}^{'} ↾}_{int (topGen (ran (.))) ((J (k), J (k + 1)))} = {O_{ℝ}^{'} ↾}_{(J (k), J (k + 1))}$
407	398 403 406	3eqtrrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {O_{ℝ}^{'} ↾}_{(J (k), J (k + 1))} = \frac{d (s \in (J (k), J (k + 1))⟼ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s))}{d_{ℝ} s}$
408	1	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to F : ℝ ⟶ ℝ$
409	2	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to X \in ℝ$
410	4	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to M \in ℕ$
411	5	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V \in P (M)$
412	9	ad4ant14	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land i \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) \underset{cn}{⟶} ℝ$
413	85	adantr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to [d, π] \subseteq [- π, π]$
414	349 413	sstrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (J (k), J (k + 1)) \subseteq [- π, π]$
415	76	a1i	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to 0 \in ℝ^{*}$
416		0red	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to 0 \in ℝ$
417	79	ad2antlr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to 0 < d$
418	416 322 320 417 326	ltletrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to 0 < J (k)$
419	320 331 415 418	ltnelicc	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \neg 0 \in [J (k), J (k + 1)]$
420	54	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to Y \in ℝ$
421	42	a1i	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to π \in ℝ$
422	281	adantr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to d < π$
423		simpr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to k \in (0 ..^N)$
424		biid	$⊢ ((((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land i \in (0 ..^M)) \land (J (k), J (k + 1)) \subseteq (Q (i), Q (i + 1))) \land v \in (0 ..^M)) \land (J (k), J (k + 1)) \subseteq (Q (v), Q (v + 1))) \leftrightarrow ((((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land i \in (0 ..^M)) \land (J (k), J (k + 1)) \subseteq (Q (i), Q (i + 1))) \land v \in (0 ..^M)) \land (J (k), J (k + 1)) \subseteq (Q (v), Q (v + 1)))$
425	409 3 410 411 322 421 422 413 29 26 27 28 423 298 424	fourierdlem50	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) \in (0 ..^M) \land (J (k), J (k + 1)) \subseteq (Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1)))$
426	425	simpld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) \in (0 ..^M)$
427	425	simprd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (J (k), J (k + 1)) \subseteq (Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1))$
428	377	cbvmptv	$⊢ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) = (s \in (J (k), J (k + 1)) ⟼ \frac{F (X + s) - Y}{s})$
429	387	cbvmptv	$⊢ (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) = (s \in (J (k), J (k + 1)) ⟼ \frac{s}{2 \sin (\frac{s}{2})})$
430		eqid	$⊢ (s \in (J (k), J (k + 1)) ⟼ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s)) = (s \in (J (k), J (k + 1)) ⟼ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s))$
431	408 409 3 410 411 412 320 330 191 414 419 420 29 426 427 428 429 430	fourierdlem72	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \frac{d (s \in (J (k), J (k + 1))⟼ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{t}) (s) (t \in (J (k), J (k + 1)) ⟼ \frac{t}{2 \sin (\frac{t}{2})}) (s))}{d_{ℝ} s} : (J (k), J (k + 1)) \underset{cn}{⟶} ℂ$
432	407 431	eqeltrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ({O_{ℝ}^{'} ↾}_{(J (k), J (k + 1))}) : (J (k), J (k + 1)) \underset{cn}{⟶} ℂ$
433		eqid	$⊢ (s \in [d, π] ⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})}) = (s \in [d, π] ⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})$
434		eqid	$⊢ (X + J (k), X + J (k + 1)) = (X + J (k), X + J (k + 1))$
435	30 426	eqeltrid	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to C \in (0 ..^M)$
436		simpll	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to φ$
437	436 435	jca	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (φ \land C \in (0 ..^M))$
438		eleq1	$⊢ i = C \to (i \in (0 ..^M) \leftrightarrow C \in (0 ..^M))$
439	438	anbi2d	$⊢ i = C \to ((φ \land i \in (0 ..^M)) \leftrightarrow (φ \land C \in (0 ..^M)))$
440		fveq2	$⊢ i = C \to V (i) = V (C)$
441		oveq1	$⊢ i = C \to i + 1 = C + 1$
442	441	fveq2d	$⊢ i = C \to V (i + 1) = V (C + 1)$
443	440 442	oveq12d	$⊢ i = C \to (V (i), V (i + 1)) = (V (C), V (C + 1))$
444		raleq	$⊢ (V (i), V (i + 1)) = (V (C), V (C + 1)) \to (\forall t \in (V (i), V (i + 1)) \|F (t)\| \leq w \leftrightarrow \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w)$
445	443 444	syl	$⊢ i = C \to (\forall t \in (V (i), V (i + 1)) \|F (t)\| \leq w \leftrightarrow \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w)$
446	445	rexbidv	$⊢ i = C \to (\exists w \in ℝ \forall t \in (V (i), V (i + 1)) \|F (t)\| \leq w \leftrightarrow \exists w \in ℝ \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w)$
447	439 446	imbi12d	$⊢ i = C \to (((φ \land i \in (0 ..^M)) \to \exists w \in ℝ \forall t \in (V (i), V (i + 1)) \|F (t)\| \leq w) \leftrightarrow ((φ \land C \in (0 ..^M)) \to \exists w \in ℝ \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w))$
448	447 8	vtoclg	$⊢ C \in (0 ..^M) \to ((φ \land C \in (0 ..^M)) \to \exists w \in ℝ \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w)$
449	435 437 448	sylc	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \exists w \in ℝ \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w$
450		nfv	$⊢ Ⅎ t ((φ \land d \in (0, π)) \land k \in (0 ..^N))$
451		nfra1	$⊢ Ⅎ t \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w$
452	450 451	nfan	$⊢ Ⅎ t (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w)$
453		simplr	$⊢ ((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w) \land t \in (X + J (k), X + J (k + 1))) \to \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w$
454	70	a1i	$⊢ φ \to - π \in ℝ$
455	454 2	readdcld	$⊢ φ \to - π + X \in ℝ$
456	42	a1i	$⊢ φ \to π \in ℝ$
457	456 2	readdcld	$⊢ φ \to π + X \in ℝ$
458	455 457	iccssred	$⊢ φ \to [- π + X, π + X] \subseteq ℝ$
459		ressxr	$⊢ ℝ \subseteq ℝ^{*}$
460	458 459	sstrdi	$⊢ φ \to [- π + X, π + X] \subseteq ℝ^{*}$
461	460	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to [- π + X, π + X] \subseteq ℝ^{*}$
462	3 410 411	fourierdlem15	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V : (0 \dots M) ⟶ [- π + X, π + X]$
463		elfzofz	$⊢ C \in (0 ..^M) \to C \in (0 \dots M)$
464	435 463	syl	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to C \in (0 \dots M)$
465	462 464	ffvelrnd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C) \in [- π + X, π + X]$
466	461 465	sseldd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C) \in ℝ^{*}$
467	466	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to V (C) \in ℝ^{*}$
468		fzofzp1	$⊢ C \in (0 ..^M) \to C + 1 \in (0 \dots M)$
469	435 468	syl	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to C + 1 \in (0 \dots M)$
470	462 469	ffvelrnd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C + 1) \in [- π + X, π + X]$
471	461 470	sseldd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C + 1) \in ℝ^{*}$
472	471	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to V (C + 1) \in ℝ^{*}$
473		elioore	$⊢ t \in (X + J (k), X + J (k + 1)) \to t \in ℝ$
474	473	adantl	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to t \in ℝ$
475	70	a1i	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to - π \in ℝ$
476	475 421 409 3 410 411 464 29	fourierdlem13	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (Q (C) = V (C) - X \land V (C) = X + Q (C))$
477	476	simprd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C) = X + Q (C)$
478	477	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to V (C) = X + Q (C)$
479	458	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to [- π + X, π + X] \subseteq ℝ$
480	479 465	sseldd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C) \in ℝ$
481	480	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to V (C) \in ℝ$
482	478 481	eqeltrrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + Q (C) \in ℝ$
483	409 320	readdcld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to X + J (k) \in ℝ$
484	483	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + J (k) \in ℝ$
485	476	simpld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to Q (C) = V (C) - X$
486	480 409	resubcld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C) - X \in ℝ$
487	485 486	eqeltrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to Q (C) \in ℝ$
488	475 421 409 3 410 411 469 29	fourierdlem13	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (Q (C + 1) = V (C + 1) - X \land V (C + 1) = X + Q (C + 1))$
489	488	simpld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to Q (C + 1) = V (C + 1) - X$
490	479 470	sseldd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C + 1) \in ℝ$
491	490 409	resubcld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C + 1) - X \in ℝ$
492	489 491	eqeltrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to Q (C + 1) \in ℝ$
493	30	eqcomi	$⊢ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) = C$
494	493	fveq2i	$⊢ Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))) = Q (C)$
495	493	oveq1i	$⊢ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1 = C + 1$
496	495	fveq2i	$⊢ Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1) = Q (C + 1)$
497	494 496	oveq12i	$⊢ (Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1)) = (Q (C), Q (C + 1))$
498	427 497	sseqtrdi	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (J (k), J (k + 1)) \subseteq (Q (C), Q (C + 1))$
499	487 492 320 330 191 498	fourierdlem10	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (Q (C) \leq J (k) \land J (k + 1) \leq Q (C + 1))$
500	499	simpld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to Q (C) \leq J (k)$
501	487 320 409 500	leadd2dd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to X + Q (C) \leq X + J (k)$
502	501	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + Q (C) \leq X + J (k)$
503	484	rexrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + J (k) \in ℝ^{*}$
504	409 330	readdcld	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to X + J (k + 1) \in ℝ$
505	504	rexrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to X + J (k + 1) \in ℝ^{*}$
506	505	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + J (k + 1) \in ℝ^{*}$
507		simpr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to t \in (X + J (k), X + J (k + 1))$
508		ioogtlb	$⊢ (X + J (k) \in ℝ^{} \land X + J (k + 1) \in ℝ^{} \land t \in (X + J (k), X + J (k + 1))) \to X + J (k) < t$
509	503 506 507 508	syl3anc	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + J (k) < t$
510	482 484 474 502 509	lelttrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + Q (C) < t$
511	478 510	eqbrtrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to V (C) < t$
512	504	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + J (k + 1) \in ℝ$
513	488	simprd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to V (C + 1) = X + Q (C + 1)$
514	513 490	eqeltrrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to X + Q (C + 1) \in ℝ$
515	514	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + Q (C + 1) \in ℝ$
516		iooltub	$⊢ (X + J (k) \in ℝ^{} \land X + J (k + 1) \in ℝ^{} \land t \in (X + J (k), X + J (k + 1))) \to t < X + J (k + 1)$
517	503 506 507 516	syl3anc	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to t < X + J (k + 1)$
518	499	simprd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to J (k + 1) \leq Q (C + 1)$
519	330 492 409 518	leadd2dd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to X + J (k + 1) \leq X + Q (C + 1)$
520	519	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + J (k + 1) \leq X + Q (C + 1)$
521	474 512 515 517 520	ltletrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to t < X + Q (C + 1)$
522	513	eqcomd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to X + Q (C + 1) = V (C + 1)$
523	522	adantr	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to X + Q (C + 1) = V (C + 1)$
524	521 523	breqtrd	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to t < V (C + 1)$
525	467 472 474 511 524	eliood	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to t \in (V (C), V (C + 1))$
526	525	adantlr	$⊢ ((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w) \land t \in (X + J (k), X + J (k + 1))) \to t \in (V (C), V (C + 1))$
527		rspa	$⊢ (\forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w \land t \in (V (C), V (C + 1))) \to \|F (t)\| \leq w$
528	453 526 527	syl2anc	$⊢ ((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w) \land t \in (X + J (k), X + J (k + 1))) \to \|F (t)\| \leq w$
529	528	ex	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w) \to (t \in (X + J (k), X + J (k + 1)) \to \|F (t)\| \leq w)$
530	452 529	ralrimi	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w) \to \forall t \in (X + J (k), X + J (k + 1)) \|F (t)\| \leq w$
531	530	ex	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (\forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w \to \forall t \in (X + J (k), X + J (k + 1)) \|F (t)\| \leq w)$
532	531	reximdv	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (\exists w \in ℝ \forall t \in (V (C), V (C + 1)) \|F (t)\| \leq w \to \exists w \in ℝ \forall t \in (X + J (k), X + J (k + 1)) \|F (t)\| \leq w)$
533	449 532	mpd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \exists w \in ℝ \forall t \in (X + J (k), X + J (k + 1)) \|F (t)\| \leq w$
534	443	raleqdv	$⊢ i = C \to (\forall t \in (V (i), V (i + 1)) \|F_{ℝ}^{'} (t)\| \leq z \leftrightarrow \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z)$
535	534	rexbidv	$⊢ i = C \to (\exists z \in ℝ \forall t \in (V (i), V (i + 1)) \|F_{ℝ}^{'} (t)\| \leq z \leftrightarrow \exists z \in ℝ \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z)$
536	439 535	imbi12d	$⊢ i = C \to (((φ \land i \in (0 ..^M)) \to \exists z \in ℝ \forall t \in (V (i), V (i + 1)) \|F_{ℝ}^{'} (t)\| \leq z) \leftrightarrow ((φ \land C \in (0 ..^M)) \to \exists z \in ℝ \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z))$
537	536 10	vtoclg	$⊢ C \in (0 ..^M) \to ((φ \land C \in (0 ..^M)) \to \exists z \in ℝ \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z)$
538	435 437 537	sylc	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \exists z \in ℝ \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z$
539		nfra1	$⊢ Ⅎ t \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z$
540	450 539	nfan	$⊢ Ⅎ t (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z)$
541	1 67	fssd	$⊢ φ \to F : ℝ ⟶ ℂ$
542		ssid	$⊢ ℝ \subseteq ℝ$
543	542	a1i	$⊢ φ \to ℝ \subseteq ℝ$
544		ioossre	$⊢ (X + J (k), X + J (k + 1)) \subseteq ℝ$
545	544	a1i	$⊢ φ \to (X + J (k), X + J (k + 1)) \subseteq ℝ$
546	49 401	dvres	$⊢ ((ℝ \subseteq ℂ \land F : ℝ ⟶ ℂ) \land (ℝ \subseteq ℝ \land (X + J (k), X + J (k + 1)) \subseteq ℝ)) \to ℝ D ({F ↾}_{(X + J (k), X + J (k + 1))}) = {F_{ℝ}^{'} ↾}_{int (topGen (ran (.))) ((X + J (k), X + J (k + 1)))}$
547	67 541 543 545 546	syl22anc	$⊢ φ \to ℝ D ({F ↾}_{(X + J (k), X + J (k + 1))}) = {F_{ℝ}^{'} ↾}_{int (topGen (ran (.))) ((X + J (k), X + J (k + 1)))}$
548		ioontr	$⊢ int (topGen (ran (.))) ((X + J (k), X + J (k + 1))) = (X + J (k), X + J (k + 1))$
549	548	reseq2i	$⊢ {F_{ℝ}^{'} ↾}_{int (topGen (ran (.))) ((X + J (k), X + J (k + 1)))} = {F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}$
550	547 549	eqtrdi	$⊢ φ \to ℝ D ({F ↾}_{(X + J (k), X + J (k + 1))}) = {F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}$
551	550	fveq1d	$⊢ φ \to {({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t) = ({F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}) (t)$
552		fvres	$⊢ t \in (X + J (k), X + J (k + 1)) \to ({F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}) (t) = F_{ℝ}^{'} (t)$
553	551 552	sylan9eq	$⊢ (φ \land t \in (X + J (k), X + J (k + 1))) \to {({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t) = F_{ℝ}^{'} (t)$
554	553	ad4ant14	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to {({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t) = F_{ℝ}^{'} (t)$
555	554	fveq2d	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land t \in (X + J (k), X + J (k + 1))) \to \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| = \|F_{ℝ}^{'} (t)\|$
556	555	adantlr	$⊢ ((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z) \land t \in (X + J (k), X + J (k + 1))) \to \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| = \|F_{ℝ}^{'} (t)\|$
557		simplr	$⊢ ((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z) \land t \in (X + J (k), X + J (k + 1))) \to \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z$
558	525	adantlr	$⊢ ((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z) \land t \in (X + J (k), X + J (k + 1))) \to t \in (V (C), V (C + 1))$
559		rspa	$⊢ (\forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z \land t \in (V (C), V (C + 1))) \to \|F_{ℝ}^{'} (t)\| \leq z$
560	557 558 559	syl2anc	$⊢ ((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z) \land t \in (X + J (k), X + J (k + 1))) \to \|F_{ℝ}^{'} (t)\| \leq z$
561	556 560	eqbrtrd	$⊢ ((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z) \land t \in (X + J (k), X + J (k + 1))) \to \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z$
562	561	ex	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z) \to (t \in (X + J (k), X + J (k + 1)) \to \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z)$
563	540 562	ralrimi	$⊢ (((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z) \to \forall t \in (X + J (k), X + J (k + 1)) \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z$
564	563	ex	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (\forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z \to \forall t \in (X + J (k), X + J (k + 1)) \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z)$
565	564	reximdv	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (\exists z \in ℝ \forall t \in (V (C), V (C + 1)) \|F_{ℝ}^{'} (t)\| \leq z \to \exists z \in ℝ \forall t \in (X + J (k), X + J (k + 1)) \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z)$
566	538 565	mpd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \exists z \in ℝ \forall t \in (X + J (k), X + J (k + 1)) \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z$
567	323 324 318 423	fourierdlem8	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to [J (k), J (k + 1)] \subseteq [d, π]$
568	143	ad2antrr	$⊢ (((φ \land d \in (0, π)) \land r \in [d, π]) \land \neg r \in ran J) \to N \in ℕ$
569	167 316	fssd	$⊢ (φ \land d \in (0, π)) \to J : (0 \dots N) ⟶ ℝ$
570	569	ad2antrr	$⊢ (((φ \land d \in (0, π)) \land r \in [d, π]) \land \neg r \in ran J) \to J : (0 \dots N) ⟶ ℝ$
571		simpr	$⊢ ((φ \land d \in (0, π)) \land r \in [d, π]) \to r \in [d, π]$
572	168	eqcomd	$⊢ (φ \land d \in (0, π)) \to d = J (0)$
573	169	eqcomd	$⊢ (φ \land d \in (0, π)) \to π = J (N)$
574	572 573	oveq12d	$⊢ (φ \land d \in (0, π)) \to [d, π] = [J (0), J (N)]$
575	574	adantr	$⊢ ((φ \land d \in (0, π)) \land r \in [d, π]) \to [d, π] = [J (0), J (N)]$
576	571 575	eleqtrd	$⊢ ((φ \land d \in (0, π)) \land r \in [d, π]) \to r \in [J (0), J (N)]$
577	576	adantr	$⊢ (((φ \land d \in (0, π)) \land r \in [d, π]) \land \neg r \in ran J) \to r \in [J (0), J (N)]$
578		simpr	$⊢ (((φ \land d \in (0, π)) \land r \in [d, π]) \land \neg r \in ran J) \to \neg r \in ran J$
579		fveq2	$⊢ j = k \to J (j) = J (k)$
580	579	breq1d	$⊢ j = k \to (J (j) < r \leftrightarrow J (k) < r)$
581	580	cbvrabv	$⊢ \{j \in (0 ..^N) \| J (j) < r\} = \{k \in (0 ..^N) \| J (k) < r\}$
582	581	supeq1i	$⊢ sup (\{j \in (0 ..^N) \| J (j) < r\} ℝ <) = sup (\{k \in (0 ..^N) \| J (k) < r\} ℝ <)$
583	568 570 577 578 582	fourierdlem25	$⊢ (((φ \land d \in (0, π)) \land r \in [d, π]) \land \neg r \in ran J) \to \exists m \in (0 ..^N) r \in (J (m), J (m + 1))$
584	541	ad2antrr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to F : ℝ ⟶ ℂ$
585	542	a1i	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ℝ \subseteq ℝ$
586	544	a1i	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (X + J (k), X + J (k + 1)) \subseteq ℝ$
587	399 584 585 586 546	syl22anc	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ℝ D ({F ↾}_{(X + J (k), X + J (k + 1))}) = {F_{ℝ}^{'} ↾}_{int (topGen (ran (.))) ((X + J (k), X + J (k + 1)))}$
588	525	ralrimiva	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to \forall t \in (X + J (k), X + J (k + 1)) t \in (V (C), V (C + 1))$
589		dfss3	$⊢ (X + J (k), X + J (k + 1)) \subseteq (V (C), V (C + 1)) \leftrightarrow \forall t \in (X + J (k), X + J (k + 1)) t \in (V (C), V (C + 1))$
590	588 589	sylibr	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to (X + J (k), X + J (k + 1)) \subseteq (V (C), V (C + 1))$
591		resabs2	$⊢ (X + J (k), X + J (k + 1)) \subseteq (V (C), V (C + 1)) \to {({F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}) ↾}_{(V (C), V (C + 1))} = {F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}$
592	590 591	syl	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {({F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}) ↾}_{(V (C), V (C + 1))} = {F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}$
593	549 587 592	3eqtr4a	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ℝ D ({F ↾}_{(X + J (k), X + J (k + 1))}) = {({F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}) ↾}_{(V (C), V (C + 1))}$
594	590	resabs1d	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) ↾}_{(X + J (k), X + J (k + 1))} = {F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))}$
595	594	eqcomd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {F_{ℝ}^{'} ↾}_{(X + J (k), X + J (k + 1))} = {({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) ↾}_{(X + J (k), X + J (k + 1))}$
596	593 592 595	3eqtrrd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) ↾}_{(X + J (k), X + J (k + 1))} = ℝ D ({F ↾}_{(X + J (k), X + J (k + 1))})$
597	443	reseq2d	$⊢ i = C \to {F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))} = {F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}$
598	597 443	feq12d	$⊢ i = C \to (({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ \leftrightarrow ({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) : (V (C), V (C + 1)) ⟶ ℝ)$
599	439 598	imbi12d	$⊢ i = C \to (((φ \land i \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ) \leftrightarrow ((φ \land C \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) : (V (C), V (C + 1)) ⟶ ℝ))$
600		cncff	$⊢ ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) \underset{cn}{⟶} ℝ \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ$
601	9 600	syl	$⊢ (φ \land i \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ$
602	599 601	vtoclg	$⊢ C \in (0 ..^M) \to ((φ \land C \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) : (V (C), V (C + 1)) ⟶ ℝ)$
603	602	anabsi7	$⊢ (φ \land C \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) : (V (C), V (C + 1)) ⟶ ℝ$
604	437 603	syl	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) : (V (C), V (C + 1)) ⟶ ℝ$
605	604 590	fssresd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to ({({F_{ℝ}^{'} ↾}_{(V (C), V (C + 1))}) ↾}_{(X + J (k), X + J (k + 1))}) : (X + J (k), X + J (k + 1)) ⟶ ℝ$
606	596 605	feq1dd	$⊢ ((φ \land d \in (0, π)) \land k \in (0 ..^N)) \to {({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} : (X + J (k), X + J (k + 1)) ⟶ ℝ$
607	375 386	oveq12d	$⊢ t = s \to \frac{F (X + t) - Y}{2 \sin (\frac{t}{2})} = \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})}$
608	607	cbvmptv	$⊢ (t \in (J (k), J (k + 1)) ⟼ \frac{F (X + t) - Y}{2 \sin (\frac{t}{2})}) = (s \in (J (k), J (k + 1)) ⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})$
609		fveq2	$⊢ r = t \to F (r) = F (t)$
610	609	fveq2d	$⊢ r = t \to \|F (r)\| = \|F (t)\|$
611	610	breq1d	$⊢ r = t \to (\|F (r)\| \leq w \leftrightarrow \|F (t)\| \leq w)$
612	611	cbvralvw	$⊢ \forall r \in (X + J (k), X + J (k + 1)) \|F (r)\| \leq w \leftrightarrow \forall t \in (X + J (k), X + J (k + 1)) \|F (t)\| \leq w$
613	612	anbi2i	$⊢ (((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land w \in ℝ) \land z \in ℝ) \land \forall r \in (X + J (k), X + J (k + 1)) \|F (r)\| \leq w) \leftrightarrow (((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land w \in ℝ) \land z \in ℝ) \land \forall t \in (X + J (k), X + J (k + 1)) \|F (t)\| \leq w)$
614		fveq2	$⊢ r = t \to {({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (r) = {({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)$
615	614	fveq2d	$⊢ r = t \to \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (r)\| = \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\|$
616	615	breq1d	$⊢ r = t \to (\|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (r)\| \leq z \leftrightarrow \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z)$
617	616	cbvralvw	$⊢ \forall r \in (X + J (k), X + J (k + 1)) \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (r)\| \leq z \leftrightarrow \forall t \in (X + J (k), X + J (k + 1)) \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z$
618	613 617	anbi12i	$⊢ ((((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land w \in ℝ) \land z \in ℝ) \land \forall r \in (X + J (k), X + J (k + 1)) \|F (r)\| \leq w) \land \forall r \in (X + J (k), X + J (k + 1)) \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (r)\| \leq z) \leftrightarrow ((((((φ \land d \in (0, π)) \land k \in (0 ..^N)) \land w \in ℝ) \land z \in ℝ) \land \forall t \in (X + J (k), X + J (k + 1)) \|F (t)\| \leq w) \land \forall t \in (X + J (k), X + J (k + 1)) \|{({F ↾}_{(X + J (k), X + J (k + 1))})}_{ℝ}^{'} (t)\| \leq z)$
619	274 275 41 43 85 288 289 433 434 533 566 167 191 567 583 606 608 618	fourierdlem80	$⊢ (φ \land d \in (0, π)) \to \exists b \in ℝ \forall s \in dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} \|\frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} (s)\| \leq b$
620	366	mpteq2dva	$⊢ (φ \land d \in (0, π)) \to (s \in [d, π] ⟼ (\frac{F (X + s) - Y}{s}) (\frac{s}{2 \sin (\frac{s}{2})})) = (s \in [d, π] ⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})$
621	271 620	eqtrd	$⊢ (φ \land d \in (0, π)) \to O = (s \in [d, π] ⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})$
622	621	oveq2d	$⊢ (φ \land d \in (0, π)) \to ℝ D O = \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s}$
623	622	dmeqd	$⊢ (φ \land d \in (0, π)) \to dom O_{ℝ}^{'} = dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s}$
624		nfcv	$⊢ \underline{Ⅎ} s dom O_{ℝ}^{'}$
625		nfcv	$⊢ \underline{Ⅎ} s ℝ$
626		nfcv	$⊢ \underline{Ⅎ} s D$
627		nfmpt1	$⊢ \underline{Ⅎ} s (s \in [d, π] ⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})$
628	625 626 627	nfov	$⊢ \underline{Ⅎ} s \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s}$
629	628	nfdm	$⊢ \underline{Ⅎ} s dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s}$
630	624 629	raleqf	$⊢ dom O_{ℝ}^{'} = dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} \to (\forall s \in dom O_{ℝ}^{'} \|O_{ℝ}^{'} (s)\| \leq b \leftrightarrow \forall s \in dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} \|O_{ℝ}^{'} (s)\| \leq b)$
631	623 630	syl	$⊢ (φ \land d \in (0, π)) \to (\forall s \in dom O_{ℝ}^{'} \|O_{ℝ}^{'} (s)\| \leq b \leftrightarrow \forall s \in dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} \|O_{ℝ}^{'} (s)\| \leq b)$
632	622	fveq1d	$⊢ (φ \land d \in (0, π)) \to O_{ℝ}^{'} (s) = \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} (s)$
633	632	fveq2d	$⊢ (φ \land d \in (0, π)) \to \|O_{ℝ}^{'} (s)\| = \|\frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} (s)\|$
634	633	breq1d	$⊢ (φ \land d \in (0, π)) \to (\|O_{ℝ}^{'} (s)\| \leq b \leftrightarrow \|\frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} (s)\| \leq b)$
635	634	ralbidv	$⊢ (φ \land d \in (0, π)) \to (\forall s \in dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} \|O_{ℝ}^{'} (s)\| \leq b \leftrightarrow \forall s \in dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} \|\frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} (s)\| \leq b)$
636	631 635	bitrd	$⊢ (φ \land d \in (0, π)) \to (\forall s \in dom O_{ℝ}^{'} \|O_{ℝ}^{'} (s)\| \leq b \leftrightarrow \forall s \in dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} \|\frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} (s)\| \leq b)$
637	636	rexbidv	$⊢ (φ \land d \in (0, π)) \to (\exists b \in ℝ \forall s \in dom O_{ℝ}^{'} \|O_{ℝ}^{'} (s)\| \leq b \leftrightarrow \exists b \in ℝ \forall s \in dom \frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} \|\frac{d (s \in [d, π]⟼ \frac{F (X + s) - Y}{2 \sin (\frac{s}{2})})}{d_{ℝ} s} (s)\| \leq b)$
638	619 637	mpbird	$⊢ (φ \land d \in (0, π)) \to \exists b \in ℝ \forall s \in dom O_{ℝ}^{'} \|O_{ℝ}^{'} (s)\| \leq b$
639		eqid	$⊢ (l \in ℝ^{+} ⟼ \int_{(d, π)} O (s) \sin l s d s) = (l \in ℝ^{+} ⟼ \int_{(d, π)} O (s) \sin l s d s)$
640		eqeq1	$⊢ t = s \to (t = J (k) \leftrightarrow s = J (k))$
641		fveq2	$⊢ h = l \to Q (h) = Q (l)$
642		oveq1	$⊢ h = l \to h + 1 = l + 1$
643	642	fveq2d	$⊢ h = l \to Q (h + 1) = Q (l + 1)$
644	641 643	oveq12d	$⊢ h = l \to (Q (h), Q (h + 1)) = (Q (l), Q (l + 1))$
645	644	sseq2d	$⊢ h = l \to ((J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)) \leftrightarrow (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))$
646	645	cbvriotavw	$⊢ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) = (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))$
647	646	fveq2i	$⊢ Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))))$
648	647	eqeq2i	$⊢ J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))) \leftrightarrow J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))))$
649	648	a1i	$⊢ ⊤ \to (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))) \leftrightarrow J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))))$
650		csbeq1	$⊢ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) = (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) \to ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R = ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R$
651	646 650	mp1i	$⊢ ⊤ \to ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R = ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R$
652	649 651	ifbieq1d	$⊢ ⊤ \to if (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R, F (X + J (k))) = if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k)))$
653	652	mptru	$⊢ if (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R, F (X + J (k))) = if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k)))$
654	653	oveq1i	$⊢ if (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R, F (X + J (k))) - Y = if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y$
655	654	oveq1i	$⊢ \frac{if (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)} = \frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}$
656	655	oveq1i	$⊢ (\frac{if (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}) = (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})})$
657	656	a1i	$⊢ t = s \to (\frac{if (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}) = (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})})$
658		eqeq1	$⊢ t = s \to (t = J (k + 1) \leftrightarrow s = J (k + 1))$
659	646	oveq1i	$⊢ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1 = (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1$
660	659	fveq2i	$⊢ Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1)$
661	660	eqeq2i	$⊢ J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1) \leftrightarrow J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1)$
662	661	a1i	$⊢ ⊤ \to (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1) \leftrightarrow J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1))$
663		csbeq1	$⊢ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) = (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) \to ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L = ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L$
664	646 663	mp1i	$⊢ ⊤ \to ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L = ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L$
665	662 664	ifbieq1d	$⊢ ⊤ \to if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) = if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1)))$
666	665	mptru	$⊢ if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) = if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1)))$
667	666	oveq1i	$⊢ if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) - Y = if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y$
668	667	oveq1i	$⊢ \frac{if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)} = \frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}$
669	668	oveq1i	$⊢ (\frac{if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}) = (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})})$
670	669	a1i	$⊢ t = s \to (\frac{if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}) = (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})})$
671		fveq2	$⊢ t = s \to O (t) = O (s)$
672	658 670 671	ifbieq12d	$⊢ t = s \to if (t = J (k + 1), (\frac{if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}), O (t)) = if (s = J (k + 1), (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}), O (s))$
673	640 657 672	ifbieq12d	$⊢ t = s \to if (t = J (k), (\frac{if (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}), if (t = J (k + 1), (\frac{if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}), O (t))) = if (s = J (k), (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}), if (s = J (k + 1), (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}), O (s)))$
674	673	cbvmptv	$⊢ (t \in [J (k), J (k + 1)] ⟼ if (t = J (k), (\frac{if (J (k) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1)))), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}), if (t = J (k + 1), (\frac{if (J (k + 1) = Q ((ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) + 1), ⦋ (ι h \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (h), Q (h + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}), O (t)))) = (s \in [J (k), J (k + 1)] ⟼ if (s = J (k), (\frac{if (J (k) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1)))), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ R, F (X + J (k))) - Y}{J (k)}) (\frac{J (k)}{2 \sin (\frac{J (k)}{2})}), if (s = J (k + 1), (\frac{if (J (k + 1) = Q ((ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) + 1), ⦋ (ι l \in (0 ..^M) \| (J (k), J (k + 1)) \subseteq (Q (l), Q (l + 1))) / i ⦌ L, F (X + J (k + 1))) - Y}{J (k + 1)}) (\frac{J (k + 1)}{2 \sin (\frac{J (k + 1)}{2})}), O (s))))$
675	41 43 89 143 167 168 169 191 301 306 309 310 432 638 639 674	fourierdlem73	$⊢ (φ \land d \in (0, π)) \to \forall e \in ℝ^{+} \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < e$
676		breq2	$⊢ e = a \to (\|\int_{(d, π)} O (s) \sin l s d s\| < e \leftrightarrow \|\int_{(d, π)} O (s) \sin l s d s\| < a)$
677	676	rexralbidv	$⊢ e = a \to (\exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < e \leftrightarrow \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < a)$
678	677	cbvralvw	$⊢ \forall e \in ℝ^{+} \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < e \leftrightarrow \forall a \in ℝ^{+} \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < a$
679	675 678	sylib	$⊢ (φ \land d \in (0, π)) \to \forall a \in ℝ^{+} \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < a$
680	679	adantlr	$⊢ ((φ \land e \in ℝ^{+}) \land d \in (0, π)) \to \forall a \in ℝ^{+} \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < a$
681		rphalfcl	$⊢ e \in ℝ^{+} \to \frac{e}{2} \in ℝ^{+}$
682	681	ad2antlr	$⊢ ((φ \land e \in ℝ^{+}) \land d \in (0, π)) \to \frac{e}{2} \in ℝ^{+}$
683		breq2	$⊢ a = \frac{e}{2} \to (\|\int_{(d, π)} O (s) \sin l s d s\| < a \leftrightarrow \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2})$
684	683	rexralbidv	$⊢ a = \frac{e}{2} \to (\exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < a \leftrightarrow \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2})$
685	684	rspccva	$⊢ (\forall a \in ℝ^{+} \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < a \land \frac{e}{2} \in ℝ^{+}) \to \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2}$
686	680 682 685	syl2anc	$⊢ ((φ \land e \in ℝ^{+}) \land d \in (0, π)) \to \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2}$
687	156	a1i	$⊢ (φ \land d \in (0, π)) \to (d, π) \subseteq [d, π]$
688	687	sselda	$⊢ ((φ \land d \in (0, π)) \land s \in (d, π)) \to s \in [d, π]$
689	688 355	syl	$⊢ ((φ \land d \in (0, π)) \land s \in (d, π)) \to ({U ↾}_{[d, π]}) (s) = U (s)$
690	353 689	eqtr2id	$⊢ ((φ \land d \in (0, π)) \land s \in (d, π)) \to U (s) = O (s)$
691	690	oveq1d	$⊢ ((φ \land d \in (0, π)) \land s \in (d, π)) \to U (s) \sin l s = O (s) \sin l s$
692	691	itgeq2dv	$⊢ (φ \land d \in (0, π)) \to \int_{(d, π)} U (s) \sin l s d s = \int_{(d, π)} O (s) \sin l s d s$
693	692	adantr	$⊢ ((φ \land d \in (0, π)) \land \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2}) \to \int_{(d, π)} U (s) \sin l s d s = \int_{(d, π)} O (s) \sin l s d s$
694	693	fveq2d	$⊢ ((φ \land d \in (0, π)) \land \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2}) \to \|\int_{(d, π)} U (s) \sin l s d s\| = \|\int_{(d, π)} O (s) \sin l s d s\|$
695		simpr	$⊢ ((φ \land d \in (0, π)) \land \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2}) \to \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2}$
696	694 695	eqbrtrd	$⊢ ((φ \land d \in (0, π)) \land \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2}) \to \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}$
697	696	ex	$⊢ (φ \land d \in (0, π)) \to (\|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2} \to \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2})$
698	697	adantlr	$⊢ ((φ \land e \in ℝ^{+}) \land d \in (0, π)) \to (\|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2} \to \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2})$
699	698	ralimdv	$⊢ ((φ \land e \in ℝ^{+}) \land d \in (0, π)) \to (\forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2} \to \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2})$
700	699	reximdv	$⊢ ((φ \land e \in ℝ^{+}) \land d \in (0, π)) \to (\exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} O (s) \sin l s d s\| < \frac{e}{2} \to \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2})$
701	686 700	mpd	$⊢ ((φ \land e \in ℝ^{+}) \land d \in (0, π)) \to \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}$
702	701	adantr	$⊢ (((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to \exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}$
703		nfv	$⊢ Ⅎ k ((φ \land e \in ℝ^{+}) \land d \in (0, π))$
704		nfra1	$⊢ Ⅎ k \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
705	703 704	nfan	$⊢ Ⅎ k (((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
706		nfv	$⊢ Ⅎ k j \in ℕ$
707	705 706	nfan	$⊢ Ⅎ k ((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ)$
708		nfv	$⊢ Ⅎ k \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}$
709	707 708	nfan	$⊢ Ⅎ k (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2})$
710		simpll	$⊢ ((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land j \in ℕ) \land k \in ℤ_{\geq j}) \to ((φ \land e \in ℝ^{+}) \land d \in (0, π))$
711		eluznn	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to k \in ℕ$
712	711	adantll	$⊢ ((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land j \in ℕ) \land k \in ℤ_{\geq j}) \to k \in ℕ$
713	710 712	jca	$⊢ ((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land j \in ℕ) \land k \in ℤ_{\geq j}) \to (((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ)$
714	713	adantllr	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land k \in ℤ_{\geq j}) \to (((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ)$
715		simpllr	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land k \in ℤ_{\geq j}) \to \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
716	711	adantll	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land k \in ℤ_{\geq j}) \to k \in ℕ$
717		rspa	$⊢ (\forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2} \land k \in ℕ) \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
718	715 716 717	syl2anc	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land k \in ℤ_{\geq j}) \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
719	714 718	jca	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land k \in ℤ_{\geq j}) \to ((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
720	719	adantlr	$⊢ ((((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \land k \in ℤ_{\geq j}) \to ((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
721		nnre	$⊢ j \in ℕ \to j \in ℝ$
722	721	rexrd	$⊢ j \in ℕ \to j \in ℝ^{*}$
723	722	adantr	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to j \in ℝ^{*}$
724	50	a1i	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to +∞ \in ℝ^{*}$
725		eluzelre	$⊢ k \in ℤ_{\geq j} \to k \in ℝ$
726		halfre	$⊢ \frac{1}{2} \in ℝ$
727	726	a1i	$⊢ k \in ℤ_{\geq j} \to \frac{1}{2} \in ℝ$
728	725 727	readdcld	$⊢ k \in ℤ_{\geq j} \to k + (\frac{1}{2}) \in ℝ$
729	728	adantl	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to k + (\frac{1}{2}) \in ℝ$
730	721	adantr	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to j \in ℝ$
731	725	adantl	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to k \in ℝ$
732		eluzle	$⊢ k \in ℤ_{\geq j} \to j \leq k$
733	732	adantl	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to j \leq k$
734		halfgt0	$⊢ 0 < \frac{1}{2}$
735	734	a1i	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to 0 < \frac{1}{2}$
736	726	a1i	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to \frac{1}{2} \in ℝ$
737	736 731	ltaddposd	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to (0 < \frac{1}{2} \leftrightarrow k < k + (\frac{1}{2}))$
738	735 737	mpbid	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to k < k + (\frac{1}{2})$
739	730 731 729 733 738	lelttrd	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to j < k + (\frac{1}{2})$
740	729	ltpnfd	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to k + (\frac{1}{2}) < +∞$
741	723 724 729 739 740	eliood	$⊢ (j \in ℕ \land k \in ℤ_{\geq j}) \to k + (\frac{1}{2}) \in (j, +∞)$
742	741	adantlr	$⊢ ((j \in ℕ \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \land k \in ℤ_{\geq j}) \to k + (\frac{1}{2}) \in (j, +∞)$
743		simplr	$⊢ ((j \in ℕ \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \land k \in ℤ_{\geq j}) \to \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}$
744		oveq1	$⊢ l = k + (\frac{1}{2}) \to l s = (k + (\frac{1}{2})) s$
745	744	fveq2d	$⊢ l = k + (\frac{1}{2}) \to \sin l s = \sin (k + (\frac{1}{2})) s$
746	745	oveq2d	$⊢ l = k + (\frac{1}{2}) \to U (s) \sin l s = U (s) \sin (k + (\frac{1}{2})) s$
747	746	adantr	$⊢ (l = k + (\frac{1}{2}) \land s \in (d, π)) \to U (s) \sin l s = U (s) \sin (k + (\frac{1}{2})) s$
748	747	itgeq2dv	$⊢ l = k + (\frac{1}{2}) \to \int_{(d, π)} U (s) \sin l s d s = \int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s$
749	748	fveq2d	$⊢ l = k + (\frac{1}{2}) \to \|\int_{(d, π)} U (s) \sin l s d s\| = \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\|$
750	749	breq1d	$⊢ l = k + (\frac{1}{2}) \to (\|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2} \leftrightarrow \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
751	750	rspcv	$⊢ k + (\frac{1}{2}) \in (j, +∞) \to (\forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2} \to \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
752	742 743 751	sylc	$⊢ ((j \in ℕ \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \land k \in ℤ_{\geq j}) \to \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
753	752	adantlll	$⊢ ((((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \land k \in ℤ_{\geq j}) \to \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
754	720 753 31	sylanbrc	$⊢ ((((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \land k \in ℤ_{\geq j}) \to χ$
755		0red	$⊢ χ \to 0 \in ℝ$
756	42	a1i	$⊢ χ \to π \in ℝ$
757		ioossicc	$⊢ (0, π) \subseteq [0, π]$
758	31	biimpi	$⊢ χ \to (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
759		simp-4r	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to d \in (0, π)$
760	758 759	syl	$⊢ χ \to d \in (0, π)$
761	757 760	sselid	$⊢ χ \to d \in [0, π]$
762		simp-5l	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to φ$
763	758 762	syl	$⊢ χ \to φ$
764	65	adantr	$⊢ (φ \land s \in (0, π)) \to U : [- π, π] ⟶ ℝ$
765	70	rexri	$⊢ - π \in ℝ^{*}$
766		0re	$⊢ 0 \in ℝ$
767	70 766 74	ltleii	$⊢ - π \leq 0$
768		iooss1	$⊢ (- π \in ℝ^{*} \land - π \leq 0) \to (0, π) \subseteq (- π, π)$
769	765 767 768	mp2an	$⊢ (0, π) \subseteq (- π, π)$
770		ioossicc	$⊢ (- π, π) \subseteq [- π, π]$
771	769 770	sstri	$⊢ (0, π) \subseteq [- π, π]$
772	771	sseli	$⊢ s \in (0, π) \to s \in [- π, π]$
773	772	adantl	$⊢ (φ \land s \in (0, π)) \to s \in [- π, π]$
774	764 773	ffvelrnd	$⊢ (φ \land s \in (0, π)) \to U (s) \in ℝ$
775	763 774	sylan	$⊢ (χ \land s \in (0, π)) \to U (s) \in ℝ$
776		simpllr	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to k \in ℕ$
777	758 776	syl	$⊢ χ \to k \in ℕ$
778	777	nnred	$⊢ χ \to k \in ℝ$
779	726	a1i	$⊢ χ \to \frac{1}{2} \in ℝ$
780	778 779	readdcld	$⊢ χ \to k + (\frac{1}{2}) \in ℝ$
781	780	adantr	$⊢ (χ \land s \in (0, π)) \to k + (\frac{1}{2}) \in ℝ$
782		elioore	$⊢ s \in (0, π) \to s \in ℝ$
783	782	adantl	$⊢ (χ \land s \in (0, π)) \to s \in ℝ$
784	781 783	remulcld	$⊢ (χ \land s \in (0, π)) \to (k + (\frac{1}{2})) s \in ℝ$
785	784	resincld	$⊢ (χ \land s \in (0, π)) \to \sin (k + (\frac{1}{2})) s \in ℝ$
786	775 785	remulcld	$⊢ (χ \land s \in (0, π)) \to U (s) \sin (k + (\frac{1}{2})) s \in ℝ$
787	786	recnd	$⊢ (χ \land s \in (0, π)) \to U (s) \sin (k + (\frac{1}{2})) s \in ℂ$
788	76	a1i	$⊢ χ \to 0 \in ℝ^{*}$
789	77	a1i	$⊢ χ \to π \in ℝ^{*}$
790	755	leidd	$⊢ χ \to 0 \leq 0$
791		ioossre	$⊢ (0, π) \subseteq ℝ$
792	791 760	sselid	$⊢ χ \to d \in ℝ$
793	788 789 760 123	syl3anc	$⊢ χ \to d < π$
794	792 756 793	ltled	$⊢ χ \to d \leq π$
795		ioossioo	$⊢ ((0 \in ℝ^{} \land π \in ℝ^{}) \land (0 \leq 0 \land d \leq π)) \to (0, d) \subseteq (0, π)$
796	788 789 790 794 795	syl22anc	$⊢ χ \to (0, d) \subseteq (0, π)$
797		ioombl	$⊢ (0, d) \in dom vol$
798	797	a1i	$⊢ χ \to (0, d) \in dom vol$
799		eleq1	$⊢ n = k \to (n \in ℕ \leftrightarrow k \in ℕ)$
800	799	anbi2d	$⊢ n = k \to ((φ \land n \in ℕ) \leftrightarrow (φ \land k \in ℕ))$
801		simpl	$⊢ (n = k \land s \in (0, π)) \to n = k$
802	801	oveq1d	$⊢ (n = k \land s \in (0, π)) \to n + (\frac{1}{2}) = k + (\frac{1}{2})$
803	802	oveq1d	$⊢ (n = k \land s \in (0, π)) \to (n + (\frac{1}{2})) s = (k + (\frac{1}{2})) s$
804	803	fveq2d	$⊢ (n = k \land s \in (0, π)) \to \sin (n + (\frac{1}{2})) s = \sin (k + (\frac{1}{2})) s$
805	804	oveq2d	$⊢ (n = k \land s \in (0, π)) \to U (s) \sin (n + (\frac{1}{2})) s = U (s) \sin (k + (\frac{1}{2})) s$
806	805	mpteq2dva	$⊢ n = k \to (s \in (0, π) ⟼ U (s) \sin (n + (\frac{1}{2})) s) = (s \in (0, π) ⟼ U (s) \sin (k + (\frac{1}{2})) s)$
807	806	eleq1d	$⊢ n = k \to ((s \in (0, π) ⟼ U (s) \sin (n + (\frac{1}{2})) s) \in 𝐿^{1} \leftrightarrow (s \in (0, π) ⟼ U (s) \sin (k + (\frac{1}{2})) s) \in 𝐿^{1})$
808	800 807	imbi12d	$⊢ n = k \to (((φ \land n \in ℕ) \to (s \in (0, π) ⟼ U (s) \sin (n + (\frac{1}{2})) s) \in 𝐿^{1}) \leftrightarrow ((φ \land k \in ℕ) \to (s \in (0, π) ⟼ U (s) \sin (k + (\frac{1}{2})) s) \in 𝐿^{1}))$
809	771	a1i	$⊢ (φ \land n \in ℕ) \to (0, π) \subseteq [- π, π]$
810		ioombl	$⊢ (0, π) \in dom vol$
811	810	a1i	$⊢ (φ \land n \in ℕ) \to (0, π) \in dom vol$
812	65	ffvelrnda	$⊢ (φ \land s \in [- π, π]) \to U (s) \in ℝ$
813	812	adantlr	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to U (s) \in ℝ$
814		nnre	$⊢ n \in ℕ \to n \in ℝ$
815		readdcl	$⊢ (n \in ℝ \land \frac{1}{2} \in ℝ) \to n + (\frac{1}{2}) \in ℝ$
816	814 726 815	sylancl	$⊢ n \in ℕ \to n + (\frac{1}{2}) \in ℝ$
817	816	adantr	$⊢ (n \in ℕ \land s \in [- π, π]) \to n + (\frac{1}{2}) \in ℝ$
818		simpr	$⊢ (n \in ℕ \land s \in [- π, π]) \to s \in [- π, π]$
819	228 818	sselid	$⊢ (n \in ℕ \land s \in [- π, π]) \to s \in ℝ$
820	817 819	remulcld	$⊢ (n \in ℕ \land s \in [- π, π]) \to (n + (\frac{1}{2})) s \in ℝ$
821	820	resincld	$⊢ (n \in ℕ \land s \in [- π, π]) \to \sin (n + (\frac{1}{2})) s \in ℝ$
822	821	adantll	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to \sin (n + (\frac{1}{2})) s \in ℝ$
823	813 822	remulcld	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to U (s) \sin (n + (\frac{1}{2})) s \in ℝ$
824	16	fvmpt2	$⊢ (s \in [- π, π] \land \sin (n + (\frac{1}{2})) s \in ℝ) \to S (s) = \sin (n + (\frac{1}{2})) s$
825	818 821 824	syl2anc	$⊢ (n \in ℕ \land s \in [- π, π]) \to S (s) = \sin (n + (\frac{1}{2})) s$
826	825	adantll	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to S (s) = \sin (n + (\frac{1}{2})) s$
827	826	oveq2d	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to U (s) S (s) = U (s) \sin (n + (\frac{1}{2})) s$
828	827	mpteq2dva	$⊢ (φ \land n \in ℕ) \to (s \in [- π, π] ⟼ U (s) S (s)) = (s \in [- π, π] ⟼ U (s) \sin (n + (\frac{1}{2})) s)$
829	17 828	eqtr2id	$⊢ (φ \land n \in ℕ) \to (s \in [- π, π] ⟼ U (s) \sin (n + (\frac{1}{2})) s) = G$
830	1	adantr	$⊢ (φ \land n \in ℕ) \to F : ℝ ⟶ ℝ$
831	6	adantr	$⊢ (φ \land n \in ℕ) \to X \in ran V$
832	20	adantr	$⊢ (φ \land n \in ℕ) \to Y \in (({F ↾}_{(X, +∞)}) {lim}_{ℂ} X)$
833	21	adantr	$⊢ (φ \land n \in ℕ) \to W \in (({F ↾}_{(−∞, X)}) {lim}_{ℂ} X)$
834	814	adantl	$⊢ (φ \land n \in ℕ) \to n \in ℝ$
835	4	adantr	$⊢ (φ \land n \in ℕ) \to M \in ℕ$
836	5	adantr	$⊢ (φ \land n \in ℕ) \to V \in P (M)$
837	7	adantlr	$⊢ ((φ \land n \in ℕ) \land i \in (0 ..^M)) \to ({F ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) \underset{cn}{⟶} ℂ$
838	11	adantlr	$⊢ ((φ \land n \in ℕ) \land i \in (0 ..^M)) \to R \in (({F ↾}_{(V (i), V (i + 1))}) {lim}_{ℂ} V (i))$
839	12	adantlr	$⊢ ((φ \land n \in ℕ) \land i \in (0 ..^M)) \to L \in (({F ↾}_{(V (i), V (i + 1))}) {lim}_{ℂ} V (i + 1))$
840		eqid	$⊢ (m \in ℕ ⟼ \{p \in (ℝ^{(0 \dots m)}) \| ((p (0) = - π \land p (m) = π) \land \forall i \in (0 ..^m) p (i) < p (i + 1))\}) = (m \in ℕ ⟼ \{p \in (ℝ^{(0 \dots m)}) \| ((p (0) = - π \land p (m) = π) \land \forall i \in (0 ..^m) p (i) < p (i + 1))\})$
841		eqid	$⊢ ℝ D F = ℝ D F$
842	601	adantlr	$⊢ ((φ \land n \in ℕ) \land i \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ$
843	22	adantr	$⊢ (φ \land n \in ℕ) \to A \in (({F_{ℝ}^{'} ↾}_{(−∞, X)}) {lim}_{ℂ} X)$
844	23	adantr	$⊢ (φ \land n \in ℕ) \to B \in (({F_{ℝ}^{'} ↾}_{(X, +∞)}) {lim}_{ℂ} X)$
845	3 830 831 832 833 13 14 15 834 16 17 835 836 837 838 839 29 840 841 842 843 844	fourierdlem88	$⊢ (φ \land n \in ℕ) \to G \in 𝐿^{1}$
846	829 845	eqeltrd	$⊢ (φ \land n \in ℕ) \to (s \in [- π, π] ⟼ U (s) \sin (n + (\frac{1}{2})) s) \in 𝐿^{1}$
847	809 811 823 846	iblss	$⊢ (φ \land n \in ℕ) \to (s \in (0, π) ⟼ U (s) \sin (n + (\frac{1}{2})) s) \in 𝐿^{1}$
848	808 847	chvarvv	$⊢ (φ \land k \in ℕ) \to (s \in (0, π) ⟼ U (s) \sin (k + (\frac{1}{2})) s) \in 𝐿^{1}$
849	763 777 848	syl2anc	$⊢ χ \to (s \in (0, π) ⟼ U (s) \sin (k + (\frac{1}{2})) s) \in 𝐿^{1}$
850	796 798 786 849	iblss	$⊢ χ \to (s \in (0, d) ⟼ U (s) \sin (k + (\frac{1}{2})) s) \in 𝐿^{1}$
851	788 789 760 78	syl3anc	$⊢ χ \to 0 < d$
852	755 792 851	ltled	$⊢ χ \to 0 \leq d$
853	756	leidd	$⊢ χ \to π \leq π$
854		ioossioo	$⊢ ((0 \in ℝ^{} \land π \in ℝ^{}) \land (0 \leq d \land π \leq π)) \to (d, π) \subseteq (0, π)$
855	788 789 852 853 854	syl22anc	$⊢ χ \to (d, π) \subseteq (0, π)$
856		ioombl	$⊢ (d, π) \in dom vol$
857	856	a1i	$⊢ χ \to (d, π) \in dom vol$
858	855 857 786 849	iblss	$⊢ χ \to (s \in (d, π) ⟼ U (s) \sin (k + (\frac{1}{2})) s) \in 𝐿^{1}$
859	755 756 761 787 850 858	itgsplitioo	$⊢ χ \to \int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s = \int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s + \int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s$
860	859	fveq2d	$⊢ χ \to \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| = \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s + \int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\|$
861	796	sselda	$⊢ (χ \land s \in (0, d)) \to s \in (0, π)$
862	861 786	syldan	$⊢ (χ \land s \in (0, d)) \to U (s) \sin (k + (\frac{1}{2})) s \in ℝ$
863	862 850	itgcl	$⊢ χ \to \int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s \in ℂ$
864	855	sselda	$⊢ (χ \land s \in (d, π)) \to s \in (0, π)$
865	864 786	syldan	$⊢ (χ \land s \in (d, π)) \to U (s) \sin (k + (\frac{1}{2})) s \in ℝ$
866	865 858	itgcl	$⊢ χ \to \int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s \in ℂ$
867	863 866	addcld	$⊢ χ \to \int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s + \int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s \in ℂ$
868	867	abscld	$⊢ χ \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s + \int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| \in ℝ$
869	863	abscld	$⊢ χ \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| \in ℝ$
870	866	abscld	$⊢ χ \to \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| \in ℝ$
871	869 870	readdcld	$⊢ χ \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| + \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| \in ℝ$
872		simp-5r	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land k \in ℕ) \land \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to e \in ℝ^{+}$
873	758 872	syl	$⊢ χ \to e \in ℝ^{+}$
874	873	rpred	$⊢ χ \to e \in ℝ$
875	863 866	abstrid	$⊢ χ \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s + \int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| \leq \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| + \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\|$
876	758	simplrd	$⊢ χ \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
877	758	simprd	$⊢ χ \to \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
878	869 870 874 876 877	lt2halvesd	$⊢ χ \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| + \|\int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e$
879	868 871 874 875 878	lelttrd	$⊢ χ \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s + \int_{(d, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e$
880	860 879	eqbrtrd	$⊢ χ \to \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e$
881	754 880	syl	$⊢ ((((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \land k \in ℤ_{\geq j}) \to \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e$
882	881	ex	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \to (k \in ℤ_{\geq j} \to \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e)$
883	709 882	ralrimi	$⊢ (((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \land \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2}) \to \forall k \in ℤ_{\geq j} \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e$
884	883	ex	$⊢ ((((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \land j \in ℕ) \to (\forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2} \to \forall k \in ℤ_{\geq j} \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e)$
885	884	reximdva	$⊢ (((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to (\exists j \in ℕ \forall l \in (j, +∞) \|\int_{(d, π)} U (s) \sin l s d s\| < \frac{e}{2} \to \exists j \in ℕ \forall k \in ℤ_{\geq j} \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e)$
886	702 885	mpd	$⊢ (((φ \land e \in ℝ^{+}) \land d \in (0, π)) \land \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to \exists j \in ℕ \forall k \in ℤ_{\geq j} \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e$
887		pipos	$⊢ 0 < π$
888	70 766 42	lttri	$⊢ (- π < 0 \land 0 < π) \to - π < π$
889	74 887 888	mp2an	$⊢ - π < π$
890	70 42 889	ltleii	$⊢ - π \leq π$
891	890	a1i	$⊢ φ \to - π \leq π$
892	3	fourierdlem2	$⊢ M \in ℕ \to (V \in P (M) \leftrightarrow (V \in (ℝ^{(0 \dots M)}) \land ((V (0) = - π + X \land V (M) = π + X) \land \forall i \in (0 ..^M) V (i) < V (i + 1))))$
893	4 892	syl	$⊢ φ \to (V \in P (M) \leftrightarrow (V \in (ℝ^{(0 \dots M)}) \land ((V (0) = - π + X \land V (M) = π + X) \land \forall i \in (0 ..^M) V (i) < V (i + 1))))$
894	5 893	mpbid	$⊢ φ \to (V \in (ℝ^{(0 \dots M)}) \land ((V (0) = - π + X \land V (M) = π + X) \land \forall i \in (0 ..^M) V (i) < V (i + 1)))$
895	894	simpld	$⊢ φ \to V \in (ℝ^{(0 \dots M)})$
896		elmapi	$⊢ V \in (ℝ^{(0 \dots M)}) \to V : (0 \dots M) ⟶ ℝ$
897	895 896	syl	$⊢ φ \to V : (0 \dots M) ⟶ ℝ$
898	897	ffvelrnda	$⊢ (φ \land i \in (0 \dots M)) \to V (i) \in ℝ$
899	2	adantr	$⊢ (φ \land i \in (0 \dots M)) \to X \in ℝ$
900	898 899	resubcld	$⊢ (φ \land i \in (0 \dots M)) \to V (i) - X \in ℝ$
901	900 29	fmptd	$⊢ φ \to Q : (0 \dots M) ⟶ ℝ$
902	29	a1i	$⊢ φ \to Q = (i \in (0 \dots M) ⟼ V (i) - X)$
903		fveq2	$⊢ i = 0 \to V (i) = V (0)$
904	903	oveq1d	$⊢ i = 0 \to V (i) - X = V (0) - X$
905	904	adantl	$⊢ (φ \land i = 0) \to V (i) - X = V (0) - X$
906	4	nnnn0d	$⊢ φ \to M \in ℕ_{0}$
907		nn0uz	$⊢ ℕ_{0} = ℤ_{\geq 0}$
908	906 907	eleqtrdi	$⊢ φ \to M \in ℤ_{\geq 0}$
909		eluzfz1	$⊢ M \in ℤ_{\geq 0} \to 0 \in (0 \dots M)$
910	908 909	syl	$⊢ φ \to 0 \in (0 \dots M)$
911	897 910	ffvelrnd	$⊢ φ \to V (0) \in ℝ$
912	911 2	resubcld	$⊢ φ \to V (0) - X \in ℝ$
913	902 905 910 912	fvmptd	$⊢ φ \to Q (0) = V (0) - X$
914	894	simprd	$⊢ φ \to ((V (0) = - π + X \land V (M) = π + X) \land \forall i \in (0 ..^M) V (i) < V (i + 1))$
915	914	simplld	$⊢ φ \to V (0) = - π + X$
916	915	oveq1d	$⊢ φ \to V (0) - X = (- π) + X - X$
917	454	recnd	$⊢ φ \to - π \in ℂ$
918	2	recnd	$⊢ φ \to X \in ℂ$
919	917 918	pncand	$⊢ φ \to (- π) + X - X = - π$
920	913 916 919	3eqtrd	$⊢ φ \to Q (0) = - π$
921	454 456 2 3 840 4 5 29	fourierdlem14	$⊢ φ \to Q \in (m \in ℕ ⟼ \{p \in (ℝ^{(0 \dots m)}) \| ((p (0) = - π \land p (m) = π) \land \forall i \in (0 ..^m) p (i) < p (i + 1))\}) (M)$
922	840	fourierdlem2	$⊢ M \in ℕ \to (Q \in (m \in ℕ ⟼ \{p \in (ℝ^{(0 \dots m)}) \| ((p (0) = - π \land p (m) = π) \land \forall i \in (0 ..^m) p (i) < p (i + 1))\}) (M) \leftrightarrow (Q \in (ℝ^{(0 \dots M)}) \land ((Q (0) = - π \land Q (M) = π) \land \forall i \in (0 ..^M) Q (i) < Q (i + 1))))$
923	4 922	syl	$⊢ φ \to (Q \in (m \in ℕ ⟼ \{p \in (ℝ^{(0 \dots m)}) \| ((p (0) = - π \land p (m) = π) \land \forall i \in (0 ..^m) p (i) < p (i + 1))\}) (M) \leftrightarrow (Q \in (ℝ^{(0 \dots M)}) \land ((Q (0) = - π \land Q (M) = π) \land \forall i \in (0 ..^M) Q (i) < Q (i + 1))))$
924	921 923	mpbid	$⊢ φ \to (Q \in (ℝ^{(0 \dots M)}) \land ((Q (0) = - π \land Q (M) = π) \land \forall i \in (0 ..^M) Q (i) < Q (i + 1)))$
925	924	simprd	$⊢ φ \to ((Q (0) = - π \land Q (M) = π) \land \forall i \in (0 ..^M) Q (i) < Q (i + 1))$
926	925	simplrd	$⊢ φ \to Q (M) = π$
927	925	simprd	$⊢ φ \to \forall i \in (0 ..^M) Q (i) < Q (i + 1)$
928	927	r19.21bi	$⊢ (φ \land i \in (0 ..^M)) \to Q (i) < Q (i + 1)$
929	1	adantr	$⊢ (φ \land i \in (0 ..^M)) \to F : ℝ ⟶ ℝ$
930	840 4 921	fourierdlem15	$⊢ φ \to Q : (0 \dots M) ⟶ [- π, π]$
931	930	adantr	$⊢ (φ \land i \in (0 ..^M)) \to Q : (0 \dots M) ⟶ [- π, π]$
932		elfzofz	$⊢ i \in (0 ..^M) \to i \in (0 \dots M)$
933	932	adantl	$⊢ (φ \land i \in (0 ..^M)) \to i \in (0 \dots M)$
934	931 933	ffvelrnd	$⊢ (φ \land i \in (0 ..^M)) \to Q (i) \in [- π, π]$
935		fzofzp1	$⊢ i \in (0 ..^M) \to i + 1 \in (0 \dots M)$
936	935	adantl	$⊢ (φ \land i \in (0 ..^M)) \to i + 1 \in (0 \dots M)$
937	931 936	ffvelrnd	$⊢ (φ \land i \in (0 ..^M)) \to Q (i + 1) \in [- π, π]$
938	2	adantr	$⊢ (φ \land i \in (0 ..^M)) \to X \in ℝ$
939		ffn	$⊢ V : (0 \dots M) ⟶ ℝ \to V Fn (0 \dots M)$
940	895 896 939	3syl	$⊢ φ \to V Fn (0 \dots M)$
941		fvelrnb	$⊢ V Fn (0 \dots M) \to (X \in ran V \leftrightarrow \exists i \in (0 \dots M) V (i) = X)$
942	940 941	syl	$⊢ φ \to (X \in ran V \leftrightarrow \exists i \in (0 \dots M) V (i) = X)$
943	6 942	mpbid	$⊢ φ \to \exists i \in (0 \dots M) V (i) = X$
944		oveq1	$⊢ V (i) = X \to V (i) - X = X - X$
945	944	adantl	$⊢ ((φ \land i \in (0 \dots M)) \land V (i) = X) \to V (i) - X = X - X$
946	918	subidd	$⊢ φ \to X - X = 0$
947	946	ad2antrr	$⊢ ((φ \land i \in (0 \dots M)) \land V (i) = X) \to X - X = 0$
948	945 947	eqtr2d	$⊢ ((φ \land i \in (0 \dots M)) \land V (i) = X) \to 0 = V (i) - X$
949	948	ex	$⊢ (φ \land i \in (0 \dots M)) \to (V (i) = X \to 0 = V (i) - X)$
950	949	reximdva	$⊢ φ \to (\exists i \in (0 \dots M) V (i) = X \to \exists i \in (0 \dots M) 0 = V (i) - X)$
951	943 950	mpd	$⊢ φ \to \exists i \in (0 \dots M) 0 = V (i) - X$
952	29	elrnmpt	$⊢ 0 \in ℝ \to (0 \in ran Q \leftrightarrow \exists i \in (0 \dots M) 0 = V (i) - X)$
953	766 952	ax-mp	$⊢ 0 \in ran Q \leftrightarrow \exists i \in (0 \dots M) 0 = V (i) - X$
954	951 953	sylibr	$⊢ φ \to 0 \in ran Q$
955	840 4 921 954	fourierdlem12	$⊢ (φ \land i \in (0 ..^M)) \to \neg 0 \in (Q (i), Q (i + 1))$
956	897	adantr	$⊢ (φ \land i \in (0 ..^M)) \to V : (0 \dots M) ⟶ ℝ$
957	956 933	ffvelrnd	$⊢ (φ \land i \in (0 ..^M)) \to V (i) \in ℝ$
958	957 938	resubcld	$⊢ (φ \land i \in (0 ..^M)) \to V (i) - X \in ℝ$
959	29	fvmpt2	$⊢ (i \in (0 \dots M) \land V (i) - X \in ℝ) \to Q (i) = V (i) - X$
960	933 958 959	syl2anc	$⊢ (φ \land i \in (0 ..^M)) \to Q (i) = V (i) - X$
961	960	oveq1d	$⊢ (φ \land i \in (0 ..^M)) \to Q (i) + X = V (i) - X + X$
962	957	recnd	$⊢ (φ \land i \in (0 ..^M)) \to V (i) \in ℂ$
963	918	adantr	$⊢ (φ \land i \in (0 ..^M)) \to X \in ℂ$
964	962 963	npcand	$⊢ (φ \land i \in (0 ..^M)) \to V (i) - X + X = V (i)$
965	961 964	eqtrd	$⊢ (φ \land i \in (0 ..^M)) \to Q (i) + X = V (i)$
966		fveq2	$⊢ j = i \to V (j) = V (i)$
967	966	oveq1d	$⊢ j = i \to V (j) - X = V (i) - X$
968	967	cbvmptv	$⊢ (j \in (0 \dots M) ⟼ V (j) - X) = (i \in (0 \dots M) ⟼ V (i) - X)$
969	29 968	eqtr4i	$⊢ Q = (j \in (0 \dots M) ⟼ V (j) - X)$
970	969	a1i	$⊢ (φ \land i \in (0 ..^M)) \to Q = (j \in (0 \dots M) ⟼ V (j) - X)$
971		fveq2	$⊢ j = i + 1 \to V (j) = V (i + 1)$
972	971	oveq1d	$⊢ j = i + 1 \to V (j) - X = V (i + 1) - X$
973	972	adantl	$⊢ ((φ \land i \in (0 ..^M)) \land j = i + 1) \to V (j) - X = V (i + 1) - X$
974	956 936	ffvelrnd	$⊢ (φ \land i \in (0 ..^M)) \to V (i + 1) \in ℝ$
975	974 938	resubcld	$⊢ (φ \land i \in (0 ..^M)) \to V (i + 1) - X \in ℝ$
976	970 973 936 975	fvmptd	$⊢ (φ \land i \in (0 ..^M)) \to Q (i + 1) = V (i + 1) - X$
977	976	oveq1d	$⊢ (φ \land i \in (0 ..^M)) \to Q (i + 1) + X = V (i + 1) - X + X$
978	974	recnd	$⊢ (φ \land i \in (0 ..^M)) \to V (i + 1) \in ℂ$
979	978 963	npcand	$⊢ (φ \land i \in (0 ..^M)) \to V (i + 1) - X + X = V (i + 1)$
980	977 979	eqtrd	$⊢ (φ \land i \in (0 ..^M)) \to Q (i + 1) + X = V (i + 1)$
981	965 980	oveq12d	$⊢ (φ \land i \in (0 ..^M)) \to (Q (i) + X, Q (i + 1) + X) = (V (i), V (i + 1))$
982	981	reseq2d	$⊢ (φ \land i \in (0 ..^M)) \to {F ↾}_{(Q (i) + X, Q (i + 1) + X)} = {F ↾}_{(V (i), V (i + 1))}$
983	981	oveq1d	$⊢ (φ \land i \in (0 ..^M)) \to (Q (i) + X, Q (i + 1) + X) \underset{cn}{⟶} ℂ = (V (i), V (i + 1)) \underset{cn}{⟶} ℂ$
984	7 982 983	3eltr4d	$⊢ (φ \land i \in (0 ..^M)) \to ({F ↾}_{(Q (i) + X, Q (i + 1) + X)}) : (Q (i) + X, Q (i + 1) + X) \underset{cn}{⟶} ℂ$
985	54	adantr	$⊢ (φ \land i \in (0 ..^M)) \to Y \in ℝ$
986	64	adantr	$⊢ (φ \land i \in (0 ..^M)) \to W \in ℝ$
987	929 934 937 938 955 984 985 986 13	fourierdlem40	$⊢ (φ \land i \in (0 ..^M)) \to ({H ↾}_{(Q (i), Q (i + 1))}) : (Q (i), Q (i + 1)) \underset{cn}{⟶} ℂ$
988		id	$⊢ ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ$
989	66	a1i	$⊢ ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ \to ℝ \subseteq ℂ$
990	988 989	fssd	$⊢ ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℝ \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℂ$
991	9 600 990	3syl	$⊢ (φ \land i \in (0 ..^M)) \to ({F_{ℝ}^{'} ↾}_{(V (i), V (i + 1))}) : (V (i), V (i + 1)) ⟶ ℂ$
992		eqid	$⊢ if (V (i) = X, B, \frac{R - if (V (i) < X, W, Y)}{Q (i)}) = if (V (i) = X, B, \frac{R - if (V (i) < X, W, Y)}{Q (i)})$
993	2 3 1 6 20 64 13 4 5 11 29 840 841 991 23 992	fourierdlem75	$⊢ (φ \land i \in (0 ..^M)) \to if (V (i) = X, B, \frac{R - if (V (i) < X, W, Y)}{Q (i)}) \in (({H ↾}_{(Q (i), Q (i + 1))}) {lim}_{ℂ} Q (i))$
994		eqid	$⊢ if (V (i + 1) = X, A, \frac{L - if (V (i + 1) < X, W, Y)}{Q (i + 1)}) = if (V (i + 1) = X, A, \frac{L - if (V (i + 1) < X, W, Y)}{Q (i + 1)})$
995	2 3 1 6 54 21 13 4 5 12 29 840 841 601 22 994	fourierdlem74	$⊢ (φ \land i \in (0 ..^M)) \to if (V (i + 1) = X, A, \frac{L - if (V (i + 1) < X, W, Y)}{Q (i + 1)}) \in (({H ↾}_{(Q (i), Q (i + 1))}) {lim}_{ℂ} Q (i + 1))$
996		fveq2	$⊢ j = i \to Q (j) = Q (i)$
997		oveq1	$⊢ j = i \to j + 1 = i + 1$
998	997	fveq2d	$⊢ j = i \to Q (j + 1) = Q (i + 1)$
999	996 998	oveq12d	$⊢ j = i \to (Q (j), Q (j + 1)) = (Q (i), Q (i + 1))$
1000	999	cbvmptv	$⊢ (j \in (0 ..^M) ⟼ (Q (j), Q (j + 1))) = (i \in (0 ..^M) ⟼ (Q (i), Q (i + 1)))$
1001	454 456 891 195 4 901 920 926 928 987 993 995 1000	fourierdlem70	$⊢ φ \to \exists x \in ℝ \forall s \in [- π, π] \|H (s)\| \leq x$
1002		eqid	$⊢ \frac{\frac{e}{3}}{y} = \frac{\frac{e}{3}}{y}$
1003		fveq2	$⊢ t = s \to G (t) = G (s)$
1004	1003	fveq2d	$⊢ t = s \to \|G (t)\| = \|G (s)\|$
1005	1004	breq1d	$⊢ t = s \to (\|G (t)\| \leq y \leftrightarrow \|G (s)\| \leq y)$
1006	1005	cbvralvw	$⊢ \forall t \in [- π, π] \|G (t)\| \leq y \leftrightarrow \forall s \in [- π, π] \|G (s)\| \leq y$
1007	1006	ralbii	$⊢ \forall n \in ℕ \forall t \in [- π, π] \|G (t)\| \leq y \leftrightarrow \forall n \in ℕ \forall s \in [- π, π] \|G (s)\| \leq y$
1008	1007	3anbi3i	$⊢ ((φ \land e \in ℝ^{+}) \land y \in ℝ^{+} \land \forall n \in ℕ \forall t \in [- π, π] \|G (t)\| \leq y) \leftrightarrow ((φ \land e \in ℝ^{+}) \land y \in ℝ^{+} \land \forall n \in ℕ \forall s \in [- π, π] \|G (s)\| \leq y)$
1009	1008	anbi1i	$⊢ (((φ \land e \in ℝ^{+}) \land y \in ℝ^{+} \land \forall n \in ℕ \forall t \in [- π, π] \|G (t)\| \leq y) \land u \in dom vol) \leftrightarrow (((φ \land e \in ℝ^{+}) \land y \in ℝ^{+} \land \forall n \in ℕ \forall s \in [- π, π] \|G (s)\| \leq y) \land u \in dom vol)$
1010	1009	anbi1i	$⊢ ((((φ \land e \in ℝ^{+}) \land y \in ℝ^{+} \land \forall n \in ℕ \forall t \in [- π, π] \|G (t)\| \leq y) \land u \in dom vol) \land (u \subseteq [- π, π] \land vol (u) \leq \frac{\frac{e}{3}}{y})) \leftrightarrow ((((φ \land e \in ℝ^{+}) \land y \in ℝ^{+} \land \forall n \in ℕ \forall s \in [- π, π] \|G (s)\| \leq y) \land u \in dom vol) \land (u \subseteq [- π, π] \land vol (u) \leq \frac{\frac{e}{3}}{y}))$
1011	1010	anbi1i	$⊢ (((((φ \land e \in ℝ^{+}) \land y \in ℝ^{+} \land \forall n \in ℕ \forall t \in [- π, π] \|G (t)\| \leq y) \land u \in dom vol) \land (u \subseteq [- π, π] \land vol (u) \leq \frac{\frac{e}{3}}{y})) \land n \in ℕ) \leftrightarrow (((((φ \land e \in ℝ^{+}) \land y \in ℝ^{+} \land \forall n \in ℕ \forall s \in [- π, π] \|G (s)\| \leq y) \land u \in dom vol) \land (u \subseteq [- π, π] \land vol (u) \leq \frac{\frac{e}{3}}{y})) \land n \in ℕ)$
1012	1 2 54 64 13 14 15 16 17 1001 845 1002 1011	fourierdlem87	$⊢ (φ \land e \in ℝ^{+}) \to \exists c \in ℝ^{+} \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
1013		iftrue	$⊢ c \leq \frac{π}{2} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) = c$
1014	1013	adantl	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) = c$
1015	76	a1i	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to 0 \in ℝ^{*}$
1016	77	a1i	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to π \in ℝ^{*}$
1017		rpre	$⊢ c \in ℝ^{+} \to c \in ℝ$
1018	1017	adantr	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to c \in ℝ$
1019		rpgt0	$⊢ c \in ℝ^{+} \to 0 < c$
1020	1019	adantr	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to 0 < c$
1021	42	rehalfcli	$⊢ \frac{π}{2} \in ℝ$
1022	1021	a1i	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to \frac{π}{2} \in ℝ$
1023	42	a1i	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to π \in ℝ$
1024		simpr	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to c \leq \frac{π}{2}$
1025		halfpos	$⊢ π \in ℝ \to (0 < π \leftrightarrow \frac{π}{2} < π)$
1026	42 1025	ax-mp	$⊢ 0 < π \leftrightarrow \frac{π}{2} < π$
1027	887 1026	mpbi	$⊢ \frac{π}{2} < π$
1028	1027	a1i	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to \frac{π}{2} < π$
1029	1018 1022 1023 1024 1028	lelttrd	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to c < π$
1030	1015 1016 1018 1020 1029	eliood	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to c \in (0, π)$
1031	1014 1030	eqeltrd	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in (0, π)$
1032		iffalse	$⊢ \neg c \leq \frac{π}{2} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) = \frac{π}{2}$
1033		2pos	$⊢ 0 < 2$
1034	42 119 887 1033	divgt0ii	$⊢ 0 < \frac{π}{2}$
1035		elioo2	$⊢ (0 \in ℝ^{} \land π \in ℝ^{}) \to (\frac{π}{2} \in (0, π) \leftrightarrow (\frac{π}{2} \in ℝ \land 0 < \frac{π}{2} \land \frac{π}{2} < π))$
1036	76 77 1035	mp2an	$⊢ \frac{π}{2} \in (0, π) \leftrightarrow (\frac{π}{2} \in ℝ \land 0 < \frac{π}{2} \land \frac{π}{2} < π)$
1037	1021 1034 1027 1036	mpbir3an	$⊢ \frac{π}{2} \in (0, π)$
1038	1037	a1i	$⊢ \neg c \leq \frac{π}{2} \to \frac{π}{2} \in (0, π)$
1039	1032 1038	eqeltrd	$⊢ \neg c \leq \frac{π}{2} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in (0, π)$
1040	1039	adantl	$⊢ (c \in ℝ^{+} \land \neg c \leq \frac{π}{2}) \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in (0, π)$
1041	1031 1040	pm2.61dan	$⊢ c \in ℝ^{+} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in (0, π)$
1042	1041	3ad2ant2	$⊢ ((φ \land e \in ℝ^{+}) \land c \in ℝ^{+} \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in (0, π)$
1043		ioombl	$⊢ (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \in dom vol$
1044	1043	a1i	$⊢ (c \in ℝ^{+} \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \in dom vol$
1045		simpr	$⊢ (c \in ℝ^{+} \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
1046	1044 1045	jca	$⊢ (c \in ℝ^{+} \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \in dom vol \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}))$
1047		ioossicc	$⊢ (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \subseteq [0, if (c \leq \frac{π}{2}, c, \frac{π}{2})]$
1048	70	a1i	$⊢ c \in ℝ^{+} \to - π \in ℝ$
1049	42	a1i	$⊢ c \in ℝ^{+} \to π \in ℝ$
1050	767	a1i	$⊢ c \in ℝ^{+} \to - π \leq 0$
1051	791 1041	sselid	$⊢ c \in ℝ^{+} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in ℝ$
1052	1021	a1i	$⊢ c \in ℝ^{+} \to \frac{π}{2} \in ℝ$
1053		min2	$⊢ (c \in ℝ \land \frac{π}{2} \in ℝ) \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \leq \frac{π}{2}$
1054	1017 1021 1053	sylancl	$⊢ c \in ℝ^{+} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \leq \frac{π}{2}$
1055	1027	a1i	$⊢ c \in ℝ^{+} \to \frac{π}{2} < π$
1056	1051 1052 1049 1054 1055	lelttrd	$⊢ c \in ℝ^{+} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) < π$
1057	1051 1049 1056	ltled	$⊢ c \in ℝ^{+} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \leq π$
1058		iccss	$⊢ ((- π \in ℝ \land π \in ℝ) \land (- π \leq 0 \land if (c \leq \frac{π}{2}, c, \frac{π}{2}) \leq π)) \to [0, if (c \leq \frac{π}{2}, c, \frac{π}{2})] \subseteq [- π, π]$
1059	1048 1049 1050 1057 1058	syl22anc	$⊢ c \in ℝ^{+} \to [0, if (c \leq \frac{π}{2}, c, \frac{π}{2})] \subseteq [- π, π]$
1060	1047 1059	sstrid	$⊢ c \in ℝ^{+} \to (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \subseteq [- π, π]$
1061		0red	$⊢ c \in ℝ^{+} \to 0 \in ℝ$
1062	1020 1014	breqtrrd	$⊢ (c \in ℝ^{+} \land c \leq \frac{π}{2}) \to 0 < if (c \leq \frac{π}{2}, c, \frac{π}{2})$
1063	1034 1032	breqtrrid	$⊢ \neg c \leq \frac{π}{2} \to 0 < if (c \leq \frac{π}{2}, c, \frac{π}{2})$
1064	1063	adantl	$⊢ (c \in ℝ^{+} \land \neg c \leq \frac{π}{2}) \to 0 < if (c \leq \frac{π}{2}, c, \frac{π}{2})$
1065	1062 1064	pm2.61dan	$⊢ c \in ℝ^{+} \to 0 < if (c \leq \frac{π}{2}, c, \frac{π}{2})$
1066	1061 1051 1065	ltled	$⊢ c \in ℝ^{+} \to 0 \leq if (c \leq \frac{π}{2}, c, \frac{π}{2})$
1067		volioo	$⊢ (0 \in ℝ \land if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in ℝ \land 0 \leq if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) = if (c \leq \frac{π}{2}, c, \frac{π}{2}) - 0$
1068	1061 1051 1066 1067	syl3anc	$⊢ c \in ℝ^{+} \to vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) = if (c \leq \frac{π}{2}, c, \frac{π}{2}) - 0$
1069	1051	recnd	$⊢ c \in ℝ^{+} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in ℂ$
1070	1069	subid1d	$⊢ c \in ℝ^{+} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) - 0 = if (c \leq \frac{π}{2}, c, \frac{π}{2})$
1071	1068 1070	eqtrd	$⊢ c \in ℝ^{+} \to vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) = if (c \leq \frac{π}{2}, c, \frac{π}{2})$
1072		min1	$⊢ (c \in ℝ \land \frac{π}{2} \in ℝ) \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \leq c$
1073	1017 1021 1072	sylancl	$⊢ c \in ℝ^{+} \to if (c \leq \frac{π}{2}, c, \frac{π}{2}) \leq c$
1074	1071 1073	eqbrtrd	$⊢ c \in ℝ^{+} \to vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) \leq c$
1075	1060 1074	jca	$⊢ c \in ℝ^{+} \to ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \subseteq [- π, π] \land vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) \leq c)$
1076	1075	adantr	$⊢ (c \in ℝ^{+} \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \subseteq [- π, π] \land vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) \leq c)$
1077		sseq1	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to (u \subseteq [- π, π] \leftrightarrow (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \subseteq [- π, π])$
1078		fveq2	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to vol (u) = vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2})))$
1079	1078	breq1d	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to (vol (u) \leq c \leftrightarrow vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) \leq c)$
1080	1077 1079	anbi12d	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to ((u \subseteq [- π, π] \land vol (u) \leq c) \leftrightarrow ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \subseteq [- π, π] \land vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) \leq c))$
1081		itgeq1	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to \int_{u} U (s) \sin (k + (\frac{1}{2})) s d s = \int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s$
1082	1081	fveq2d	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| = \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\|$
1083	1082	breq1d	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to (\|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2} \leftrightarrow \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
1084	1083	ralbidv	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to (\forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2} \leftrightarrow \forall k \in ℕ \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
1085	1080 1084	imbi12d	$⊢ u = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \to (((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \leftrightarrow (((0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \subseteq [- π, π] \land vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) \leq c) \to \forall k \in ℕ \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}))$
1086	1085	rspcva	$⊢ ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \in dom vol \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to (((0, if (c \leq \frac{π}{2}, c, \frac{π}{2})) \subseteq [- π, π] \land vol ((0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))) \leq c) \to \forall k \in ℕ \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
1087	1046 1076 1086	sylc	$⊢ (c \in ℝ^{+} \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to \forall k \in ℕ \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
1088	1087	3adant1	$⊢ ((φ \land e \in ℝ^{+}) \land c \in ℝ^{+} \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to \forall k \in ℕ \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
1089		oveq2	$⊢ d = if (c \leq \frac{π}{2}, c, \frac{π}{2}) \to (0, d) = (0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))$
1090	1089	itgeq1d	$⊢ d = if (c \leq \frac{π}{2}, c, \frac{π}{2}) \to \int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s = \int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s$
1091	1090	fveq2d	$⊢ d = if (c \leq \frac{π}{2}, c, \frac{π}{2}) \to \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| = \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\|$
1092	1091	breq1d	$⊢ d = if (c \leq \frac{π}{2}, c, \frac{π}{2}) \to (\|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2} \leftrightarrow \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
1093	1092	ralbidv	$⊢ d = if (c \leq \frac{π}{2}, c, \frac{π}{2}) \to (\forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2} \leftrightarrow \forall k \in ℕ \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
1094	1093	rspcev	$⊢ (if (c \leq \frac{π}{2}, c, \frac{π}{2}) \in (0, π) \land \forall k \in ℕ \|\int_{(0, if (c \leq \frac{π}{2}, c, \frac{π}{2}))} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to \exists d \in (0, π) \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
1095	1042 1088 1094	syl2anc	$⊢ ((φ \land e \in ℝ^{+}) \land c \in ℝ^{+} \land \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})) \to \exists d \in (0, π) \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
1096	1095	rexlimdv3a	$⊢ (φ \land e \in ℝ^{+}) \to (\exists c \in ℝ^{+} \forall u \in dom vol ((u \subseteq [- π, π] \land vol (u) \leq c) \to \forall k \in ℕ \|\int_{u} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}) \to \exists d \in (0, π) \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2})$
1097	1012 1096	mpd	$⊢ (φ \land e \in ℝ^{+}) \to \exists d \in (0, π) \forall k \in ℕ \|\int_{(0, d)} U (s) \sin (k + (\frac{1}{2})) s d s\| < \frac{e}{2}$
1098	886 1097	r19.29a	$⊢ (φ \land e \in ℝ^{+}) \to \exists j \in ℕ \forall k \in ℤ_{\geq j} \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e$
1099	1098	ralrimiva	$⊢ φ \to \forall e \in ℝ^{+} \exists j \in ℕ \forall k \in ℤ_{\geq j} \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e$
1100		nnex	$⊢ ℕ \in V$
1101	1100	mptex	$⊢ (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) \in V$
1102	1101	a1i	$⊢ φ \to (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) \in V$
1103		eqidd	$⊢ (φ \land k \in ℕ) \to (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) = (n \in ℕ ⟼ \int_{(0, π)} G (s) d s)$
1104	772	adantl	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to s \in [- π, π]$
1105	774	ad4ant14	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to U (s) \in ℝ$
1106	772	adantl	$⊢ ((k \in ℕ \land n = k) \land s \in (0, π)) \to s \in [- π, π]$
1107		simpr	$⊢ (k \in ℕ \land n = k) \to n = k$
1108		simpl	$⊢ (k \in ℕ \land n = k) \to k \in ℕ$
1109	1107 1108	eqeltrd	$⊢ (k \in ℕ \land n = k) \to n \in ℕ$
1110	1109	nnred	$⊢ (k \in ℕ \land n = k) \to n \in ℝ$
1111	726	a1i	$⊢ (k \in ℕ \land n = k) \to \frac{1}{2} \in ℝ$
1112	1110 1111	readdcld	$⊢ (k \in ℕ \land n = k) \to n + (\frac{1}{2}) \in ℝ$
1113	1112	adantr	$⊢ ((k \in ℕ \land n = k) \land s \in (0, π)) \to n + (\frac{1}{2}) \in ℝ$
1114	228 1106	sselid	$⊢ ((k \in ℕ \land n = k) \land s \in (0, π)) \to s \in ℝ$
1115	1113 1114	remulcld	$⊢ ((k \in ℕ \land n = k) \land s \in (0, π)) \to (n + (\frac{1}{2})) s \in ℝ$
1116	1115	resincld	$⊢ ((k \in ℕ \land n = k) \land s \in (0, π)) \to \sin (n + (\frac{1}{2})) s \in ℝ$
1117	1106 1116 824	syl2anc	$⊢ ((k \in ℕ \land n = k) \land s \in (0, π)) \to S (s) = \sin (n + (\frac{1}{2})) s$
1118	1117	adantlll	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to S (s) = \sin (n + (\frac{1}{2})) s$
1119	1110	adantll	$⊢ ((φ \land k \in ℕ) \land n = k) \to n \in ℝ$
1120	1119	adantr	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to n \in ℝ$
1121		1red	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to 1 \in ℝ$
1122	1121	rehalfcld	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to \frac{1}{2} \in ℝ$
1123	1120 1122	readdcld	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to n + (\frac{1}{2}) \in ℝ$
1124	228 1104	sselid	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to s \in ℝ$
1125	1123 1124	remulcld	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to (n + (\frac{1}{2})) s \in ℝ$
1126	1125	resincld	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to \sin (n + (\frac{1}{2})) s \in ℝ$
1127	1118 1126	eqeltrd	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to S (s) \in ℝ$
1128	1105 1127	remulcld	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to U (s) S (s) \in ℝ$
1129	17	fvmpt2	$⊢ (s \in [- π, π] \land U (s) S (s) \in ℝ) \to G (s) = U (s) S (s)$
1130	1104 1128 1129	syl2anc	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to G (s) = U (s) S (s)$
1131		oveq1	$⊢ n = k \to n + (\frac{1}{2}) = k + (\frac{1}{2})$
1132	1131	oveq1d	$⊢ n = k \to (n + (\frac{1}{2})) s = (k + (\frac{1}{2})) s$
1133	1132	fveq2d	$⊢ n = k \to \sin (n + (\frac{1}{2})) s = \sin (k + (\frac{1}{2})) s$
1134	1133	ad2antlr	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to \sin (n + (\frac{1}{2})) s = \sin (k + (\frac{1}{2})) s$
1135	1118 1134	eqtrd	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to S (s) = \sin (k + (\frac{1}{2})) s$
1136	1135	oveq2d	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to U (s) S (s) = U (s) \sin (k + (\frac{1}{2})) s$
1137	1130 1136	eqtrd	$⊢ (((φ \land k \in ℕ) \land n = k) \land s \in (0, π)) \to G (s) = U (s) \sin (k + (\frac{1}{2})) s$
1138	1137	itgeq2dv	$⊢ ((φ \land k \in ℕ) \land n = k) \to \int_{(0, π)} G (s) d s = \int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s$
1139		simpr	$⊢ (φ \land k \in ℕ) \to k \in ℕ$
1140	805	itgeq2dv	$⊢ n = k \to \int_{(0, π)} U (s) \sin (n + (\frac{1}{2})) s d s = \int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s$
1141	1140	eleq1d	$⊢ n = k \to (\int_{(0, π)} U (s) \sin (n + (\frac{1}{2})) s d s \in ℂ \leftrightarrow \int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s \in ℂ)$
1142	800 1141	imbi12d	$⊢ n = k \to (((φ \land n \in ℕ) \to \int_{(0, π)} U (s) \sin (n + (\frac{1}{2})) s d s \in ℂ) \leftrightarrow ((φ \land k \in ℕ) \to \int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s \in ℂ))$
1143	774	adantlr	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to U (s) \in ℝ$
1144		simpr	$⊢ (φ \land n \in ℕ) \to n \in ℕ$
1145	1144 772 821	syl2an	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to \sin (n + (\frac{1}{2})) s \in ℝ$
1146	1143 1145	remulcld	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to U (s) \sin (n + (\frac{1}{2})) s \in ℝ$
1147	1146 847	itgcl	$⊢ (φ \land n \in ℕ) \to \int_{(0, π)} U (s) \sin (n + (\frac{1}{2})) s d s \in ℂ$
1148	1142 1147	chvarvv	$⊢ (φ \land k \in ℕ) \to \int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s \in ℂ$
1149	1103 1138 1139 1148	fvmptd	$⊢ (φ \land k \in ℕ) \to (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) (k) = \int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s$
1150	39 33 1102 1149 1148	clim0c	$⊢ φ \to ((n \in ℕ ⟼ \int_{(0, π)} G (s) d s) ⇝ 0 \leftrightarrow \forall e \in ℝ^{+} \exists j \in ℕ \forall k \in ℤ_{\geq j} \|\int_{(0, π)} U (s) \sin (k + (\frac{1}{2})) s d s\| < e)$
1151	1099 1150	mpbird	$⊢ φ \to (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) ⇝ 0$
1152	1100	mptex	$⊢ (n \in ℕ ⟼ \frac{\int_{(0, π)} G (s) d s}{π}) \in V$
1153	19 1152	eqeltri	$⊢ E \in V$
1154	1153	a1i	$⊢ φ \to E \in V$
1155	1100	mptex	$⊢ (n \in ℕ ⟼ π) \in V$
1156	1155	a1i	$⊢ φ \to (n \in ℕ ⟼ π) \in V$
1157	42	recni	$⊢ π \in ℂ$
1158	1157	a1i	$⊢ φ \to π \in ℂ$
1159		eqidd	$⊢ m \in ℕ \to (n \in ℕ ⟼ π) = (n \in ℕ ⟼ π)$
1160		eqidd	$⊢ (m \in ℕ \land n = m) \to π = π$
1161		id	$⊢ m \in ℕ \to m \in ℕ$
1162	42	a1i	$⊢ m \in ℕ \to π \in ℝ$
1163	1159 1160 1161 1162	fvmptd	$⊢ m \in ℕ \to (n \in ℕ ⟼ π) (m) = π$
1164	1163	adantl	$⊢ (φ \land m \in ℕ) \to (n \in ℕ ⟼ π) (m) = π$
1165	39 33 1156 1158 1164	climconst	$⊢ φ \to (n \in ℕ ⟼ π) ⇝ π$
1166	766 887	gtneii	$⊢ π \neq 0$
1167	1166	a1i	$⊢ φ \to π \neq 0$
1168	2	adantr	$⊢ (φ \land n \in ℕ) \to X \in ℝ$
1169	54	adantr	$⊢ (φ \land n \in ℕ) \to Y \in ℝ$
1170	64	adantr	$⊢ (φ \land n \in ℕ) \to W \in ℝ$
1171	830 1168 1169 1170 13 14 15 834 16 17	fourierdlem67	$⊢ (φ \land n \in ℕ) \to G : [- π, π] ⟶ ℝ$
1172	1171	adantr	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to G : [- π, π] ⟶ ℝ$
1173	809	sselda	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to s \in [- π, π]$
1174	1172 1173	ffvelrnd	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to G (s) \in ℝ$
1175	1171	ffvelrnda	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to G (s) \in ℝ$
1176	1171	feqmptd	$⊢ (φ \land n \in ℕ) \to G = (s \in [- π, π] ⟼ G (s))$
1177	1176 845	eqeltrrd	$⊢ (φ \land n \in ℕ) \to (s \in [- π, π] ⟼ G (s)) \in 𝐿^{1}$
1178	809 811 1175 1177	iblss	$⊢ (φ \land n \in ℕ) \to (s \in (0, π) ⟼ G (s)) \in 𝐿^{1}$
1179	1174 1178	itgcl	$⊢ (φ \land n \in ℕ) \to \int_{(0, π)} G (s) d s \in ℂ$
1180		eqid	$⊢ (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) = (n \in ℕ ⟼ \int_{(0, π)} G (s) d s)$
1181	1180	fvmpt2	$⊢ (n \in ℕ \land \int_{(0, π)} G (s) d s \in ℂ) \to (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) (n) = \int_{(0, π)} G (s) d s$
1182	1144 1179 1181	syl2anc	$⊢ (φ \land n \in ℕ) \to (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) (n) = \int_{(0, π)} G (s) d s$
1183	1182 1179	eqeltrd	$⊢ (φ \land n \in ℕ) \to (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) (n) \in ℂ$
1184		eqid	$⊢ (n \in ℕ ⟼ π) = (n \in ℕ ⟼ π)$
1185	1184	fvmpt2	$⊢ (n \in ℕ \land π \in ℝ) \to (n \in ℕ ⟼ π) (n) = π$
1186	42 1185	mpan2	$⊢ n \in ℕ \to (n \in ℕ ⟼ π) (n) = π$
1187	1157	a1i	$⊢ n \in ℕ \to π \in ℂ$
1188	1166	a1i	$⊢ n \in ℕ \to π \neq 0$
1189		eldifsn	$⊢ π \in (ℂ ∖ \{0\}) \leftrightarrow (π \in ℂ \land π \neq 0)$
1190	1187 1188 1189	sylanbrc	$⊢ n \in ℕ \to π \in (ℂ ∖ \{0\})$
1191	1186 1190	eqeltrd	$⊢ n \in ℕ \to (n \in ℕ ⟼ π) (n) \in (ℂ ∖ \{0\})$
1192	1191	adantl	$⊢ (φ \land n \in ℕ) \to (n \in ℕ ⟼ π) (n) \in (ℂ ∖ \{0\})$
1193	1157	a1i	$⊢ (φ \land n \in ℕ) \to π \in ℂ$
1194	1166	a1i	$⊢ (φ \land n \in ℕ) \to π \neq 0$
1195	1179 1193 1194	divcld	$⊢ (φ \land n \in ℕ) \to \frac{\int_{(0, π)} G (s) d s}{π} \in ℂ$
1196	19	fvmpt2	$⊢ (n \in ℕ \land \frac{\int_{(0, π)} G (s) d s}{π} \in ℂ) \to E (n) = \frac{\int_{(0, π)} G (s) d s}{π}$
1197	1144 1195 1196	syl2anc	$⊢ (φ \land n \in ℕ) \to E (n) = \frac{\int_{(0, π)} G (s) d s}{π}$
1198	1182	eqcomd	$⊢ (φ \land n \in ℕ) \to \int_{(0, π)} G (s) d s = (n \in ℕ ⟼ \int_{(0, π)} G (s) d s) (n)$
1199	1186	eqcomd	$⊢ n \in ℕ \to π = (n \in ℕ ⟼ π) (n)$
1200	1199	adantl	$⊢ (φ \land n \in ℕ) \to π = (n \in ℕ ⟼ π) (n)$
1201	1198 1200	oveq12d	$⊢ (φ \land n \in ℕ) \to \frac{\int_{(0, π)} G (s) d s}{π} = \frac{(n \in ℕ ⟼ \int_{(0, π)} G (s) d s) (n)}{(n \in ℕ ⟼ π) (n)}$
1202	1197 1201	eqtrd	$⊢ (φ \land n \in ℕ) \to E (n) = \frac{(n \in ℕ ⟼ \int_{(0, π)} G (s) d s) (n)}{(n \in ℕ ⟼ π) (n)}$
1203	34 35 36 38 39 33 1151 1154 1165 1167 1183 1192 1202	climdivf	$⊢ φ \to E ⇝ (\frac{0}{π})$
1204	1157 1166	div0i	$⊢ \frac{0}{π} = 0$
1205	1204	a1i	$⊢ φ \to \frac{0}{π} = 0$
1206	1203 1205	breqtrd	$⊢ φ \to E ⇝ 0$
1207	1100	mptex	$⊢ (m \in ℕ ⟼ \int_{(0, π)} F (X + s) D (m) (s) d s) \in V$
1208	18 1207	eqeltri	$⊢ Z \in V$
1209	1208	a1i	$⊢ φ \to Z \in V$
1210	1100	mptex	$⊢ (m \in ℕ ⟼ \frac{Y}{2}) \in V$
1211	1210	a1i	$⊢ φ \to (m \in ℕ ⟼ \frac{Y}{2}) \in V$
1212		limccl	$⊢ ({F ↾}_{(X, +∞)}) {lim}_{ℂ} X \subseteq ℂ$
1213	1212 20	sselid	$⊢ φ \to Y \in ℂ$
1214	1213	halfcld	$⊢ φ \to \frac{Y}{2} \in ℂ$
1215		eqidd	$⊢ (φ \land n \in ℤ_{\geq 1}) \to (m \in ℕ ⟼ \frac{Y}{2}) = (m \in ℕ ⟼ \frac{Y}{2})$
1216		eqidd	$⊢ ((φ \land n \in ℤ_{\geq 1}) \land m = n) \to \frac{Y}{2} = \frac{Y}{2}$
1217	39	eqcomi	$⊢ ℤ_{\geq 1} = ℕ$
1218	1217	eleq2i	$⊢ n \in ℤ_{\geq 1} \leftrightarrow n \in ℕ$
1219	1218	biimpi	$⊢ n \in ℤ_{\geq 1} \to n \in ℕ$
1220	1219	adantl	$⊢ (φ \land n \in ℤ_{\geq 1}) \to n \in ℕ$
1221	1214	adantr	$⊢ (φ \land n \in ℤ_{\geq 1}) \to \frac{Y}{2} \in ℂ$
1222	1215 1216 1220 1221	fvmptd	$⊢ (φ \land n \in ℤ_{\geq 1}) \to (m \in ℕ ⟼ \frac{Y}{2}) (n) = \frac{Y}{2}$
1223	32 33 1211 1214 1222	climconst	$⊢ φ \to (m \in ℕ ⟼ \frac{Y}{2}) ⇝ (\frac{Y}{2})$
1224	1195 19	fmptd	$⊢ φ \to E : ℕ ⟶ ℂ$
1225	1224	adantr	$⊢ (φ \land n \in ℤ_{\geq 1}) \to E : ℕ ⟶ ℂ$
1226	1225 1220	ffvelrnd	$⊢ (φ \land n \in ℤ_{\geq 1}) \to E (n) \in ℂ$
1227	1222 1221	eqeltrd	$⊢ (φ \land n \in ℤ_{\geq 1}) \to (m \in ℕ ⟼ \frac{Y}{2}) (n) \in ℂ$
1228	1222	oveq2d	$⊢ (φ \land n \in ℤ_{\geq 1}) \to E (n) + (m \in ℕ ⟼ \frac{Y}{2}) (n) = E (n) + (\frac{Y}{2})$
1229	810	a1i	$⊢ φ \to (0, π) \in dom vol$
1230		0red	$⊢ s \in (0, π) \to 0 \in ℝ$
1231	1230	rexrd	$⊢ s \in (0, π) \to 0 \in ℝ^{*}$
1232	77	a1i	$⊢ s \in (0, π) \to π \in ℝ^{*}$
1233		id	$⊢ s \in (0, π) \to s \in (0, π)$
1234		ioogtlb	$⊢ (0 \in ℝ^{} \land π \in ℝ^{} \land s \in (0, π)) \to 0 < s$
1235	1231 1232 1233 1234	syl3anc	$⊢ s \in (0, π) \to 0 < s$
1236	1235	gt0ne0d	$⊢ s \in (0, π) \to s \neq 0$
1237	1236	neneqd	$⊢ s \in (0, π) \to \neg s = 0$
1238		velsn	$⊢ s \in \{0\} \leftrightarrow s = 0$
1239	1237 1238	sylnibr	$⊢ s \in (0, π) \to \neg s \in \{0\}$
1240	772 1239	eldifd	$⊢ s \in (0, π) \to s \in ([- π, π] ∖ \{0\})$
1241	1240	ssriv	$⊢ (0, π) \subseteq [- π, π] ∖ \{0\}$
1242	1241	a1i	$⊢ φ \to (0, π) \subseteq [- π, π] ∖ \{0\}$
1243	1235	adantl	$⊢ (φ \land s \in (0, π)) \to 0 < s$
1244	1243	iftrued	$⊢ (φ \land s \in (0, π)) \to if (0 < s, Y, W) = Y$
1245		eqid	$⊢ D (n) = D (n)$
1246		0red	$⊢ (φ \land n \in ℕ) \to 0 \in ℝ$
1247	42	a1i	$⊢ (φ \land n \in ℕ) \to π \in ℝ$
1248	766 42 887	ltleii	$⊢ 0 \leq π$
1249	1248	a1i	$⊢ (φ \land n \in ℕ) \to 0 \leq π$
1250		eqid	$⊢ (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) = (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π})$
1251	24 1144 1245 1246 1247 1249 1250	dirkeritg	$⊢ (φ \land n \in ℕ) \to \int_{(0, π)} D (n) (s) d s = (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (π) - (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (0)$
1252		ubicc2	$⊢ (0 \in ℝ^{} \land π \in ℝ^{} \land 0 \leq π) \to π \in [0, π]$
1253	76 77 1248 1252	mp3an	$⊢ π \in [0, π]$
1254		oveq1	$⊢ s = π \to \frac{s}{2} = \frac{π}{2}$
1255		oveq2	$⊢ s = π \to k s = k π$
1256	1255	fveq2d	$⊢ s = π \to \sin k s = \sin k π$
1257	1256	oveq1d	$⊢ s = π \to \frac{\sin k s}{k} = \frac{\sin k π}{k}$
1258		elfzelz	$⊢ k \in (1 \dots n) \to k \in ℤ$
1259	1258	zcnd	$⊢ k \in (1 \dots n) \to k \in ℂ$
1260	1157	a1i	$⊢ k \in (1 \dots n) \to π \in ℂ$
1261	1166	a1i	$⊢ k \in (1 \dots n) \to π \neq 0$
1262	1259 1260 1261	divcan4d	$⊢ k \in (1 \dots n) \to \frac{k π}{π} = k$
1263	1262 1258	eqeltrd	$⊢ k \in (1 \dots n) \to \frac{k π}{π} \in ℤ$
1264	1259 1260	mulcld	$⊢ k \in (1 \dots n) \to k π \in ℂ$
1265		sineq0	$⊢ k π \in ℂ \to (\sin k π = 0 \leftrightarrow \frac{k π}{π} \in ℤ)$
1266	1264 1265	syl	$⊢ k \in (1 \dots n) \to (\sin k π = 0 \leftrightarrow \frac{k π}{π} \in ℤ)$
1267	1263 1266	mpbird	$⊢ k \in (1 \dots n) \to \sin k π = 0$
1268	1267	oveq1d	$⊢ k \in (1 \dots n) \to \frac{\sin k π}{k} = \frac{0}{k}$
1269		0red	$⊢ k \in (1 \dots n) \to 0 \in ℝ$
1270		1red	$⊢ k \in (1 \dots n) \to 1 \in ℝ$
1271	1258	zred	$⊢ k \in (1 \dots n) \to k \in ℝ$
1272	117	a1i	$⊢ k \in (1 \dots n) \to 0 < 1$
1273		elfzle1	$⊢ k \in (1 \dots n) \to 1 \leq k$
1274	1269 1270 1271 1272 1273	ltletrd	$⊢ k \in (1 \dots n) \to 0 < k$
1275	1274	gt0ne0d	$⊢ k \in (1 \dots n) \to k \neq 0$
1276	1259 1275	div0d	$⊢ k \in (1 \dots n) \to \frac{0}{k} = 0$
1277	1268 1276	eqtrd	$⊢ k \in (1 \dots n) \to \frac{\sin k π}{k} = 0$
1278	1257 1277	sylan9eq	$⊢ (s = π \land k \in (1 \dots n)) \to \frac{\sin k s}{k} = 0$
1279	1278	sumeq2dv	$⊢ s = π \to \sum_{k = 1}^{n} (\frac{\sin k s}{k}) = \sum_{k = 1}^{n} 0$
1280		fzfi	$⊢ (1 \dots n) \in Fin$
1281	1280	olci	$⊢ ((1 \dots n) \subseteq ℤ_{\geq \underset{˙}{∥}} \lor (1 \dots n) \in Fin)$
1282		sumz	$⊢ ((1 \dots n) \subseteq ℤ_{\geq \underset{˙}{∥}} \lor (1 \dots n) \in Fin) \to \sum_{k = 1}^{n} 0 = 0$
1283	1281 1282	ax-mp	$⊢ \sum_{k = 1}^{n} 0 = 0$
1284	1279 1283	eqtrdi	$⊢ s = π \to \sum_{k = 1}^{n} (\frac{\sin k s}{k}) = 0$
1285	1254 1284	oveq12d	$⊢ s = π \to (\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k}) = (\frac{π}{2}) + 0$
1286	1285	oveq1d	$⊢ s = π \to \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π} = \frac{(\frac{π}{2}) + 0}{π}$
1287		ovex	$⊢ \frac{(\frac{π}{2}) + 0}{π} \in V$
1288	1286 1250 1287	fvmpt	$⊢ π \in [0, π] \to (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (π) = \frac{(\frac{π}{2}) + 0}{π}$
1289	1253 1288	ax-mp	$⊢ (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (π) = \frac{(\frac{π}{2}) + 0}{π}$
1290		lbicc2	$⊢ (0 \in ℝ^{} \land π \in ℝ^{} \land 0 \leq π) \to 0 \in [0, π]$
1291	76 77 1248 1290	mp3an	$⊢ 0 \in [0, π]$
1292		oveq1	$⊢ s = 0 \to \frac{s}{2} = \frac{0}{2}$
1293		2cn	$⊢ 2 \in ℂ$
1294	1293 256	div0i	$⊢ \frac{0}{2} = 0$
1295	1292 1294	eqtrdi	$⊢ s = 0 \to \frac{s}{2} = 0$
1296		oveq2	$⊢ s = 0 \to k s = k \cdot 0$
1297	1259	mul01d	$⊢ k \in (1 \dots n) \to k \cdot 0 = 0$
1298	1296 1297	sylan9eq	$⊢ (s = 0 \land k \in (1 \dots n)) \to k s = 0$
1299	1298	fveq2d	$⊢ (s = 0 \land k \in (1 \dots n)) \to \sin k s = \sin 0$
1300		sin0	$⊢ \sin 0 = 0$
1301	1299 1300	eqtrdi	$⊢ (s = 0 \land k \in (1 \dots n)) \to \sin k s = 0$
1302	1301	oveq1d	$⊢ (s = 0 \land k \in (1 \dots n)) \to \frac{\sin k s}{k} = \frac{0}{k}$
1303	1276	adantl	$⊢ (s = 0 \land k \in (1 \dots n)) \to \frac{0}{k} = 0$
1304	1302 1303	eqtrd	$⊢ (s = 0 \land k \in (1 \dots n)) \to \frac{\sin k s}{k} = 0$
1305	1304	sumeq2dv	$⊢ s = 0 \to \sum_{k = 1}^{n} (\frac{\sin k s}{k}) = \sum_{k = 1}^{n} 0$
1306	1305 1283	eqtrdi	$⊢ s = 0 \to \sum_{k = 1}^{n} (\frac{\sin k s}{k}) = 0$
1307	1295 1306	oveq12d	$⊢ s = 0 \to (\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k}) = 0 + 0$
1308		00id	$⊢ 0 + 0 = 0$
1309	1307 1308	eqtrdi	$⊢ s = 0 \to (\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k}) = 0$
1310	1309	oveq1d	$⊢ s = 0 \to \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π} = \frac{0}{π}$
1311	1310 1204	eqtrdi	$⊢ s = 0 \to \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π} = 0$
1312		c0ex	$⊢ 0 \in V$
1313	1311 1250 1312	fvmpt	$⊢ 0 \in [0, π] \to (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (0) = 0$
1314	1291 1313	ax-mp	$⊢ (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (0) = 0$
1315	1289 1314	oveq12i	$⊢ (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (π) - (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (0) = (\frac{(\frac{π}{2}) + 0}{π}) - 0$
1316	1315	a1i	$⊢ (φ \land n \in ℕ) \to (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (π) - (s \in [0, π] ⟼ \frac{(\frac{s}{2}) + \sum_{k = 1}^{n} (\frac{\sin k s}{k})}{π}) (0) = (\frac{(\frac{π}{2}) + 0}{π}) - 0$
1317	1021	recni	$⊢ \frac{π}{2} \in ℂ$
1318	1317	addid1i	$⊢ (\frac{π}{2}) + 0 = \frac{π}{2}$
1319	1318	oveq1i	$⊢ \frac{(\frac{π}{2}) + 0}{π} = \frac{\frac{π}{2}}{π}$
1320	1157 1293 1157 256 1166	divdiv32i	$⊢ \frac{\frac{π}{2}}{π} = \frac{\frac{π}{π}}{2}$
1321	1157 1166	dividi	$⊢ \frac{π}{π} = 1$
1322	1321	oveq1i	$⊢ \frac{\frac{π}{π}}{2} = \frac{1}{2}$
1323	1319 1320 1322	3eqtri	$⊢ \frac{(\frac{π}{2}) + 0}{π} = \frac{1}{2}$
1324	1323	oveq1i	$⊢ (\frac{(\frac{π}{2}) + 0}{π}) - 0 = (\frac{1}{2}) - 0$
1325		halfcn	$⊢ \frac{1}{2} \in ℂ$
1326	1325	subid1i	$⊢ (\frac{1}{2}) - 0 = \frac{1}{2}$
1327	1324 1326	eqtri	$⊢ (\frac{(\frac{π}{2}) + 0}{π}) - 0 = \frac{1}{2}$
1328	1327	a1i	$⊢ (φ \land n \in ℕ) \to (\frac{(\frac{π}{2}) + 0}{π}) - 0 = \frac{1}{2}$
1329	1251 1316 1328	3eqtrd	$⊢ (φ \land n \in ℕ) \to \int_{(0, π)} D (n) (s) d s = \frac{1}{2}$
1330	1 2 3 4 5 6 7 11 12 13 14 15 16 17 841 601 22 23 20 21 1229 1242 19 24 54 1244 1329	fourierdlem95	$⊢ (φ \land n \in ℕ) \to E (n) + (\frac{Y}{2}) = \int_{(0, π)} F (X + s) D (n) (s) d s$
1331	1220 1330	syldan	$⊢ (φ \land n \in ℤ_{\geq 1}) \to E (n) + (\frac{Y}{2}) = \int_{(0, π)} F (X + s) D (n) (s) d s$
1332	18	a1i	$⊢ (φ \land n \in ℕ) \to Z = (m \in ℕ ⟼ \int_{(0, π)} F (X + s) D (m) (s) d s)$
1333		fveq2	$⊢ m = n \to D (m) = D (n)$
1334	1333	fveq1d	$⊢ m = n \to D (m) (s) = D (n) (s)$
1335	1334	oveq2d	$⊢ m = n \to F (X + s) D (m) (s) = F (X + s) D (n) (s)$
1336	1335	adantr	$⊢ (m = n \land s \in (0, π)) \to F (X + s) D (m) (s) = F (X + s) D (n) (s)$
1337	1336	itgeq2dv	$⊢ m = n \to \int_{(0, π)} F (X + s) D (m) (s) d s = \int_{(0, π)} F (X + s) D (n) (s) d s$
1338	1337	adantl	$⊢ ((φ \land n \in ℕ) \land m = n) \to \int_{(0, π)} F (X + s) D (m) (s) d s = \int_{(0, π)} F (X + s) D (n) (s) d s$
1339	1	adantr	$⊢ (φ \land s \in (0, π)) \to F : ℝ ⟶ ℝ$
1340	2	adantr	$⊢ (φ \land s \in (0, π)) \to X \in ℝ$
1341	782	adantl	$⊢ (φ \land s \in (0, π)) \to s \in ℝ$
1342	1340 1341	readdcld	$⊢ (φ \land s \in (0, π)) \to X + s \in ℝ$
1343	1339 1342	ffvelrnd	$⊢ (φ \land s \in (0, π)) \to F (X + s) \in ℝ$
1344	1343	adantlr	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to F (X + s) \in ℝ$
1345	24	dirkerf	$⊢ n \in ℕ \to D (n) : ℝ ⟶ ℝ$
1346	1345	ad2antlr	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to D (n) : ℝ ⟶ ℝ$
1347	782	adantl	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to s \in ℝ$
1348	1346 1347	ffvelrnd	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to D (n) (s) \in ℝ$
1349	1344 1348	remulcld	$⊢ ((φ \land n \in ℕ) \land s \in (0, π)) \to F (X + s) D (n) (s) \in ℝ$
1350	1	adantr	$⊢ (φ \land s \in [- π, π]) \to F : ℝ ⟶ ℝ$
1351	2	adantr	$⊢ (φ \land s \in [- π, π]) \to X \in ℝ$
1352	228	sseli	$⊢ s \in [- π, π] \to s \in ℝ$
1353	1352	adantl	$⊢ (φ \land s \in [- π, π]) \to s \in ℝ$
1354	1351 1353	readdcld	$⊢ (φ \land s \in [- π, π]) \to X + s \in ℝ$
1355	1350 1354	ffvelrnd	$⊢ (φ \land s \in [- π, π]) \to F (X + s) \in ℝ$
1356	1355	adantlr	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to F (X + s) \in ℝ$
1357	1345	ad2antlr	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to D (n) : ℝ ⟶ ℝ$
1358	1352	adantl	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to s \in ℝ$
1359	1357 1358	ffvelrnd	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to D (n) (s) \in ℝ$
1360	1356 1359	remulcld	$⊢ ((φ \land n \in ℕ) \land s \in [- π, π]) \to F (X + s) D (n) (s) \in ℝ$
1361	70	a1i	$⊢ (φ \land n \in ℕ) \to - π \in ℝ$
1362	24	dirkercncf	$⊢ n \in ℕ \to D (n) : ℝ \underset{cn}{⟶} ℝ$
1363	1362	adantl	$⊢ (φ \land n \in ℕ) \to D (n) : ℝ \underset{cn}{⟶} ℝ$
1364		eqid	$⊢ (s \in [- π, π] ⟼ F (X + s) D (n) (s)) = (s \in [- π, π] ⟼ F (X + s) D (n) (s))$
1365	1361 1247 830 1168 3 835 836 837 838 839 29 840 1363 1364	fourierdlem84	$⊢ (φ \land n \in ℕ) \to (s \in [- π, π] ⟼ F (X + s) D (n) (s)) \in 𝐿^{1}$
1366	809 811 1360 1365	iblss	$⊢ (φ \land n \in ℕ) \to (s \in (0, π) ⟼ F (X + s) D (n) (s)) \in 𝐿^{1}$
1367	1349 1366	itgrecl	$⊢ (φ \land n \in ℕ) \to \int_{(0, π)} F (X + s) D (n) (s) d s \in ℝ$
1368	1332 1338 1144 1367	fvmptd	$⊢ (φ \land n \in ℕ) \to Z (n) = \int_{(0, π)} F (X + s) D (n) (s) d s$
1369	1368	eqcomd	$⊢ (φ \land n \in ℕ) \to \int_{(0, π)} F (X + s) D (n) (s) d s = Z (n)$
1370	1220 1369	syldan	$⊢ (φ \land n \in ℤ_{\geq 1}) \to \int_{(0, π)} F (X + s) D (n) (s) d s = Z (n)$
1371	1228 1331 1370	3eqtrrd	$⊢ (φ \land n \in ℤ_{\geq 1}) \to Z (n) = E (n) + (m \in ℕ ⟼ \frac{Y}{2}) (n)$
1372	32 33 1206 1209 1223 1226 1227 1371	climadd	$⊢ φ \to Z ⇝ (0 + (\frac{Y}{2}))$
1373	1214	addid2d	$⊢ φ \to 0 + (\frac{Y}{2}) = \frac{Y}{2}$
1374	1372 1373	breqtrd	$⊢ φ \to Z ⇝ (\frac{Y}{2})$

Metamath Proof Explorer

Theorem fourierdlem104

Proof