fourierswlem - Mathbox for Glauco Siliprandi

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Theorem fourierswlem 46680

Description: The Fourier series for the square wave 𝐹 converges to 𝑌, a simpler expression for this special case. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierswlem.t	⊢ 𝑇 = (2 · π)
fourierswlem.f	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if((𝑥 mod 𝑇) < π, 1, -1))
fourierswlem.x	⊢ 𝑋 ∈ ℝ
fourierswlem.y	⊢ 𝑌 = if((𝑋 mod π) = 0, 0, (𝐹‘𝑋))

**Assertion**
Ref	Expression
fourierswlem	⊢ 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2)

Distinct variable groups: 𝑥,𝑇 𝑥,𝑋 Allowed substitution hints: 𝐹(𝑥) 𝑌(𝑥)

Proof of Theorem fourierswlem Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → 2 ∥ (𝑋 / π))
2		2z 12557	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
3	2	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → 2 ∈ ℤ)
4		fourierswlem.x	. . . . . . . . . . . . 13 ⊢ 𝑋 ∈ ℝ
5		pirp 26450	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ⁺
6		mod0 13833	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℝ ∧ π ∈ ℝ⁺) → ((𝑋 mod π) = 0 ↔ (𝑋 / π) ∈ ℤ))
7	4, 5, 6	mp2an 698	. . . . . . . . . . . 12 ⊢ ((𝑋 mod π) = 0 ↔ (𝑋 / π) ∈ ℤ)
8	7	birani 504	. . . . . . . . . . 11 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 / π) ∈ ℤ)
9		divides 16221	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ (𝑋 / π) ∈ ℤ) → (2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π)))
10	3, 8, 9	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π)))
11	1, 10	mpbid 233	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π))
12		2cnd 12257	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → 2 ∈ ℂ)
13		picn 26447	. . . . . . . . . . . . . . . . . . . 20 ⊢ π ∈ ℂ
14	13	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → π ∈ ℂ)
15		zcn 12527	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
16	12, 14, 15	mulassd 11166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((2 · π) · 𝑘) = (2 · (π · 𝑘)))
17	14, 15	mulcld 11163	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (π · 𝑘) ∈ ℂ)
18	12, 17	mulcomd 11164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → (2 · (π · 𝑘)) = ((π · 𝑘) · 2))
19	16, 18	eqtrd 2775	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((2 · π) · 𝑘) = ((π · 𝑘) · 2))
20	19	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((2 · π) · 𝑘) = ((π · 𝑘) · 2))
21	14, 15, 12	mulassd 11166	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((π · 𝑘) · 2) = (π · (𝑘 · 2)))
22	21	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((π · 𝑘) · 2) = (π · (𝑘 · 2)))
23		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 · 2) = (𝑋 / π) → (𝑘 · 2) = (𝑋 / π))
24	23	eqcomd 2746	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 · 2) = (𝑋 / π) → (𝑋 / π) = (𝑘 · 2))
25	24	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / π) = (𝑘 · 2))
26	4	recni 11157	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑋 ∈ ℂ
27	26	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑋 ∈ ℂ)
28	13	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → π ∈ ℂ)
29	15	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑘 ∈ ℂ)
30		2cnd 12257	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 2 ∈ ℂ)
31	29, 30	mulcld 11163	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑘 · 2) ∈ ℂ)
32		pire 26446	. . . . . . . . . . . . . . . . . . . 20 ⊢ π ∈ ℝ
33		pipos 26448	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < π
34	32, 33	gt0ne0ii 11684	. . . . . . . . . . . . . . . . . . 19 ⊢ π ≠ 0
35	34	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → π ≠ 0)
36	27, 28, 31, 35	divmuld 11951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((𝑋 / π) = (𝑘 · 2) ↔ (π · (𝑘 · 2)) = 𝑋))
37	25, 36	mpbid 233	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (π · (𝑘 · 2)) = 𝑋)
38	20, 22, 37	3eqtrrd 2780	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑋 = ((2 · π) · 𝑘))
39		fourierswlem.t	. . . . . . . . . . . . . . . 16 ⊢ 𝑇 = (2 · π)
40	39	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑇 = (2 · π))
41	38, 40	oveq12d 7381	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) = (((2 · π) · 𝑘) / (2 · π)))
42	12, 14	mulcld 11163	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (2 · π) ∈ ℂ)
43		2ne0 12283	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
44	43	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → 2 ≠ 0)
45	34	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → π ≠ 0)
46	12, 14, 44, 45	mulne0d 11800	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (2 · π) ≠ 0)
47	15, 42, 46	divcan3d 11934	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (((2 · π) · 𝑘) / (2 · π)) = 𝑘)
48	47	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (((2 · π) · 𝑘) / (2 · π)) = 𝑘)
49	41, 48	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) = 𝑘)
50		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑘 ∈ ℤ)
51	49, 50	eqeltrd 2840	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) ∈ ℤ)
52	51	ex 413	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ → ((𝑘 · 2) = (𝑋 / π) → (𝑋 / 𝑇) ∈ ℤ))
53	52	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑘 ∈ ℤ → ((𝑘 · 2) = (𝑋 / π) → (𝑋 / 𝑇) ∈ ℤ)))
54	53	rexlimdv 3139	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π) → (𝑋 / 𝑇) ∈ ℤ))
55	11, 54	mpd 15	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 / 𝑇) ∈ ℤ)
56		2re 12253	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
57	56, 32	remulcli 11159	. . . . . . . . . . 11 ⊢ (2 · π) ∈ ℝ
58	39, 57	eqeltri 2836	. . . . . . . . . 10 ⊢ 𝑇 ∈ ℝ
59		2pos 12282	. . . . . . . . . . . 12 ⊢ 0 < 2
60	56, 32, 59, 33	mulgt0ii 11277	. . . . . . . . . . 11 ⊢ 0 < (2 · π)
61	60, 39	breqtrri 5106	. . . . . . . . . 10 ⊢ 0 < 𝑇
62	58, 61	elrpii 12943	. . . . . . . . 9 ⊢ 𝑇 ∈ ℝ⁺
63		mod0 13833	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → ((𝑋 mod 𝑇) = 0 ↔ (𝑋 / 𝑇) ∈ ℤ))
64	4, 62, 63	mp2an 698	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 ↔ (𝑋 / 𝑇) ∈ ℤ)
65	55, 64	sylibr 235	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 mod 𝑇) = 0)
66	65	orcd 879	. . . . . 6 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
67		odd2np1 16308	. . . . . . . . . 10 ⊢ ((𝑋 / π) ∈ ℤ → (¬ 2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π)))
68	7, 67	sylbi 218	. . . . . . . . 9 ⊢ ((𝑋 mod π) = 0 → (¬ 2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π)))
69	68	biimpa 477	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π))
70	12, 15	mulcld 11163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (2 · 𝑘) ∈ ℂ)
71	70	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (2 · 𝑘) ∈ ℂ)
72		1cnd 11137	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 1 ∈ ℂ)
73	13	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π ∈ ℂ)
74	71, 72, 73	adddird 11168	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) + 1) · π) = (((2 · 𝑘) · π) + (1 · π)))
75	12, 15	mulcomd 11164	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (2 · 𝑘) = (𝑘 · 2))
76	75	oveq1d 7378	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) · π) = ((𝑘 · 2) · π))
77	15, 12, 14	mulassd 11166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((𝑘 · 2) · π) = (𝑘 · (2 · π)))
78	39	eqcomi 2749	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · π) = 𝑇
79	78	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (2 · π) = 𝑇)
80	79	oveq2d 7379	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → (𝑘 · (2 · π)) = (𝑘 · 𝑇))
81	76, 77, 80	3eqtrd 2779	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) · π) = (𝑘 · 𝑇))
82	13	mullidi 11148	. . . . . . . . . . . . . . . . . 18 ⊢ (1 · π) = π
83	82	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (1 · π) = π)
84	81, 83	oveq12d 7381	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (((2 · 𝑘) · π) + (1 · π)) = ((𝑘 · 𝑇) + π))
85	84	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) · π) + (1 · π)) = ((𝑘 · 𝑇) + π))
86	39, 42	eqeltrid 2844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → 𝑇 ∈ ℂ)
87	15, 86	mulcld 11163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (𝑘 · 𝑇) ∈ ℂ)
88	87, 14	addcomd 11346	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → ((𝑘 · 𝑇) + π) = (π + (𝑘 · 𝑇)))
89	88	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((𝑘 · 𝑇) + π) = (π + (𝑘 · 𝑇)))
90	74, 85, 89	3eqtrrd 2780	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π + (𝑘 · 𝑇)) = (((2 · 𝑘) + 1) · π))
91		peano2cn 11316	. . . . . . . . . . . . . . . . 17 ⊢ ((2 · 𝑘) ∈ ℂ → ((2 · 𝑘) + 1) ∈ ℂ)
92	70, 91	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) + 1) ∈ ℂ)
93	92, 14	mulcomd 11164	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (((2 · 𝑘) + 1) · π) = (π · ((2 · 𝑘) + 1)))
94	93	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) + 1) · π) = (π · ((2 · 𝑘) + 1)))
95		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (((2 · 𝑘) + 1) = (𝑋 / π) → ((2 · 𝑘) + 1) = (𝑋 / π))
96	95	eqcomd 2746	. . . . . . . . . . . . . . . 16 ⊢ (((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 / π) = ((2 · 𝑘) + 1))
97	96	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (𝑋 / π) = ((2 · 𝑘) + 1))
98	26	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 𝑋 ∈ ℂ)
99	92	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((2 · 𝑘) + 1) ∈ ℂ)
100	34	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π ≠ 0)
101	98, 73, 99, 100	divmuld 11951	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((𝑋 / π) = ((2 · 𝑘) + 1) ↔ (π · ((2 · 𝑘) + 1)) = 𝑋))
102	97, 101	mpbid 233	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π · ((2 · 𝑘) + 1)) = 𝑋)
103	90, 94, 102	3eqtrrd 2780	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 𝑋 = (π + (𝑘 · 𝑇)))
104	103	oveq1d 7378	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (𝑋 mod 𝑇) = ((π + (𝑘 · 𝑇)) mod 𝑇))
105		modcyc 13863	. . . . . . . . . . . . . 14 ⊢ ((π ∈ ℝ ∧ 𝑇 ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → ((π + (𝑘 · 𝑇)) mod 𝑇) = (π mod 𝑇))
106	32, 62, 105	mp3an12 1459	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤ → ((π + (𝑘 · 𝑇)) mod 𝑇) = (π mod 𝑇))
107	106	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((π + (𝑘 · 𝑇)) mod 𝑇) = (π mod 𝑇))
108	32	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π ∈ ℝ)
109	62	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 𝑇 ∈ ℝ⁺)
110		0re 11144	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
111	110, 32, 33	ltleii 11267	. . . . . . . . . . . . . 14 ⊢ 0 ≤ π
112	111	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 0 ≤ π)
113		2timesgt 45743	. . . . . . . . . . . . . . . 16 ⊢ (π ∈ ℝ⁺ → π < (2 · π))
114	5, 113	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ π < (2 · π)
115	114, 39	breqtrri 5106	. . . . . . . . . . . . . 14 ⊢ π < 𝑇
116	115	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π < 𝑇)
117		modid 13853	. . . . . . . . . . . . 13 ⊢ (((π ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) ∧ (0 ≤ π ∧ π < 𝑇)) → (π mod 𝑇) = π)
118	108, 109, 112, 116, 117	syl22anc 844	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π mod 𝑇) = π)
119	104, 107, 118	3eqtrd 2779	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (𝑋 mod 𝑇) = π)
120	119	ex 413	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → (((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 mod 𝑇) = π))
121	120	a1i 11	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → (𝑘 ∈ ℤ → (((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 mod 𝑇) = π)))
122	121	rexlimdv 3139	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → (∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 mod 𝑇) = π))
123	69, 122	mpd 15	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → (𝑋 mod 𝑇) = π)
124	123	olcd 880	. . . . . 6 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
125	66, 124	pm2.61dan 818	. . . . 5 ⊢ ((𝑋 mod π) = 0 → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
126		0xr 11190	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
127	32	rexri 11201	. . . . . . . 8 ⊢ π ∈ ℝ^*
128		iocgtlb 45954	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (0(,]π)) → 0 < (𝑋 mod 𝑇))
129	126, 127, 128	mp3an12 1459	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → 0 < (𝑋 mod 𝑇))
130	129	gt0ne0d 11712	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → (𝑋 mod 𝑇) ≠ 0)
131	130	neneqd 2940	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → ¬ (𝑋 mod 𝑇) = 0)
132		pm2.53 857	. . . . . 6 ⊢ (((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π) → (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) = π))
133	132	imp 407	. . . . 5 ⊢ ((((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) = π)
134	125, 131, 133	syl2anr 603	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) = π)
135	126	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → 0 ∈ ℝ^*)
136	127	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → π ∈ ℝ^*)
137		modcl 13830	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) ∈ ℝ)
138	4, 62, 137	mp2an 698	. . . . . . . . . . . . . 14 ⊢ (𝑋 mod 𝑇) ∈ ℝ
139	138	rexri 11201	. . . . . . . . . . . . 13 ⊢ (𝑋 mod 𝑇) ∈ ℝ^*
140	139	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ∈ ℝ^*)
141		id 22	. . . . . . . . . . . . 13 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) = π)
142	33, 141	breqtrrid 5117	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → 0 < (𝑋 mod 𝑇))
143	32	eqlei2 11255	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ≤ π)
144	135, 136, 140, 142, 143	eliocd 45959	. . . . . . . . . . 11 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ∈ (0(,]π))
145	144	iftrued 4469	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = π → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
146	145	adantl 482	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
147		oveq1 7370	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → (𝑥 mod 𝑇) = (𝑋 mod 𝑇))
148	147	breq1d 5089	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → ((𝑥 mod 𝑇) < π ↔ (𝑋 mod 𝑇) < π))
149	148	ifbid 4485	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → if((𝑥 mod 𝑇) < π, 1, -1) = if((𝑋 mod 𝑇) < π, 1, -1))
150		fourierswlem.f	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if((𝑥 mod 𝑇) < π, 1, -1))
151		1ex 11138	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
152		negex 11389	. . . . . . . . . . . . . 14 ⊢ -1 ∈ V
153	151, 152	ifex 4512	. . . . . . . . . . . . 13 ⊢ if((𝑋 mod 𝑇) < π, 1, -1) ∈ V
154	149, 150, 153	fvmpt 6942	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℝ → (𝐹‘𝑋) = if((𝑋 mod 𝑇) < π, 1, -1))
155	4, 154	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹‘𝑋) = if((𝑋 mod 𝑇) < π, 1, -1)
156	138	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑋 mod 𝑇) < π → (𝑋 mod 𝑇) ∈ ℝ)
157		id 22	. . . . . . . . . . . . . 14 ⊢ ((𝑋 mod 𝑇) < π → (𝑋 mod 𝑇) < π)
158	156, 157	ltned 11280	. . . . . . . . . . . . 13 ⊢ ((𝑋 mod 𝑇) < π → (𝑋 mod 𝑇) ≠ π)
159	158	necon2bi 2965	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → ¬ (𝑋 mod 𝑇) < π)
160	159	iffalsed 4472	. . . . . . . . . . 11 ⊢ ((𝑋 mod 𝑇) = π → if((𝑋 mod 𝑇) < π, 1, -1) = -1)
161	155, 160	eqtrid 2787	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = π → (𝐹‘𝑋) = -1)
162	161	adantl 482	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (𝐹‘𝑋) = -1)
163	146, 162	oveq12d 7381	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (1 + -1))
164		1pneg1e0 12293	. . . . . . . 8 ⊢ (1 + -1) = 0
165	163, 164	eqtrdi 2791	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = 0)
166	165	oveq1d 7378	. . . . . 6 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = (0 / 2))
167	166	adantll 720	. . . . 5 ⊢ ((((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) ∧ (𝑋 mod 𝑇) = π) → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = (0 / 2))
168		2cn 12254	. . . . . . 7 ⊢ 2 ∈ ℂ
169	168, 43	div0i 11887	. . . . . 6 ⊢ (0 / 2) = 0
170	169	a1i 11	. . . . 5 ⊢ ((((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) ∧ (𝑋 mod 𝑇) = π) → (0 / 2) = 0)
171		fourierswlem.y	. . . . . . 7 ⊢ 𝑌 = if((𝑋 mod π) = 0, 0, (𝐹‘𝑋))
172		iftrue 4467	. . . . . . 7 ⊢ ((𝑋 mod π) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = 0)
173	171, 172	eqtr2id 2788	. . . . . 6 ⊢ ((𝑋 mod π) = 0 → 0 = 𝑌)
174	173	ad2antlr 733	. . . . 5 ⊢ ((((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) ∧ (𝑋 mod 𝑇) = π) → 0 = 𝑌)
175	167, 170, 174	3eqtrrd 2780	. . . 4 ⊢ ((((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) ∧ (𝑋 mod 𝑇) = π) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
176	134, 175	mpdan 693	. . 3 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
177		iftrue 4467	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
178	177	adantr 481	. . . . . 6 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
179	138	a1i 11	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) ∈ ℝ)
180	32	a1i 11	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → π ∈ ℝ)
181		iocleub 45955	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (0(,]π)) → (𝑋 mod 𝑇) ≤ π)
182	126, 127, 181	mp3an12 1459	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → (𝑋 mod 𝑇) ≤ π)
183	182	adantr 481	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) ≤ π)
184		ax-1cn 11094	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
185	184, 13	mulcomi 11151	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 · π) = (π · 1)
186	82, 185	eqtr3i 2765	. . . . . . . . . . . . . . . . . 18 ⊢ π = (π · 1)
187	186	oveq1i 7373	. . . . . . . . . . . . . . . . 17 ⊢ (π + (π · (2 · (⌊‘(𝑋 / 𝑇))))) = ((π · 1) + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
188	168, 13	mulcomi 11151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2 · π) = (π · 2)
189	39, 188	eqtri 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑇 = (π · 2)
190	189	oveq1i 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) = ((π · 2) · (⌊‘(𝑋 / 𝑇)))
191	110, 61	gtneii 11256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑇 ≠ 0
192	4, 58, 191	redivcli 11920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 / 𝑇) ∈ ℝ
193		flcl 13752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 / 𝑇) ∈ ℝ → (⌊‘(𝑋 / 𝑇)) ∈ ℤ)
194	192, 193	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⌊‘(𝑋 / 𝑇)) ∈ ℤ
195		zcn 12527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⌊‘(𝑋 / 𝑇)) ∈ ℤ → (⌊‘(𝑋 / 𝑇)) ∈ ℂ)
196	194, 195	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⌊‘(𝑋 / 𝑇)) ∈ ℂ
197	13, 168, 196	mulassi 11154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((π · 2) · (⌊‘(𝑋 / 𝑇))) = (π · (2 · (⌊‘(𝑋 / 𝑇))))
198	190, 197	eqtri 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) = (π · (2 · (⌊‘(𝑋 / 𝑇))))
199	198	oveq2i 7374	. . . . . . . . . . . . . . . . 17 ⊢ (π + (𝑇 · (⌊‘(𝑋 / 𝑇)))) = (π + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
200	168, 196	mulcli 11150	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ
201	13, 184, 200	adddii 11155	. . . . . . . . . . . . . . . . 17 ⊢ (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) = ((π · 1) + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
202	187, 199, 201	3eqtr4ri 2774	. . . . . . . . . . . . . . . 16 ⊢ (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) = (π + (𝑇 · (⌊‘(𝑋 / 𝑇))))
203	202	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (π = (𝑋 mod 𝑇) → (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) = (π + (𝑇 · (⌊‘(𝑋 / 𝑇)))))
204		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (π = (𝑋 mod 𝑇) → π = (𝑋 mod 𝑇))
205		modval 13828	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))))
206	4, 62, 205	mp2an 698	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 mod 𝑇) = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇))))
207	204, 206	eqtrdi 2791	. . . . . . . . . . . . . . . 16 ⊢ (π = (𝑋 mod 𝑇) → π = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))))
208	207	oveq1d 7378	. . . . . . . . . . . . . . 15 ⊢ (π = (𝑋 mod 𝑇) → (π + (𝑇 · (⌊‘(𝑋 / 𝑇)))) = ((𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))) + (𝑇 · (⌊‘(𝑋 / 𝑇)))))
209	26	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (π = (𝑋 mod 𝑇) → 𝑋 ∈ ℂ)
210	58	recni 11157	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 ∈ ℂ
211	210, 196	mulcli 11150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ
212	211	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (π = (𝑋 mod 𝑇) → (𝑇 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ)
213	209, 212	npcand 11507	. . . . . . . . . . . . . . 15 ⊢ (π = (𝑋 mod 𝑇) → ((𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))) + (𝑇 · (⌊‘(𝑋 / 𝑇)))) = 𝑋)
214	203, 208, 213	3eqtrrd 2780	. . . . . . . . . . . . . 14 ⊢ (π = (𝑋 mod 𝑇) → 𝑋 = (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))))
215	214	oveq1d 7378	. . . . . . . . . . . . 13 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) = ((π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) / π))
216	184, 200	addcli 11149	. . . . . . . . . . . . . 14 ⊢ (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℂ
217	216, 13, 34	divcan3i 11899	. . . . . . . . . . . . 13 ⊢ ((π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) / π) = (1 + (2 · (⌊‘(𝑋 / 𝑇))))
218	215, 217	eqtrdi 2791	. . . . . . . . . . . 12 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) = (1 + (2 · (⌊‘(𝑋 / 𝑇)))))
219		1z 12555	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
220		zmulcl 12574	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℤ ∧ (⌊‘(𝑋 / 𝑇)) ∈ ℤ) → (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ)
221	2, 194, 220	mp2an 698	. . . . . . . . . . . . . 14 ⊢ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ
222		zaddcl 12565	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ) → (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ)
223	219, 221, 222	mp2an 698	. . . . . . . . . . . . 13 ⊢ (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ
224	223	a1i 11	. . . . . . . . . . . 12 ⊢ (π = (𝑋 mod 𝑇) → (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ)
225	218, 224	eqeltrd 2840	. . . . . . . . . . 11 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) ∈ ℤ)
226	225, 7	sylibr 235	. . . . . . . . . 10 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 mod π) = 0)
227	226	necon3bi 2961	. . . . . . . . 9 ⊢ (¬ (𝑋 mod π) = 0 → π ≠ (𝑋 mod 𝑇))
228	227	adantl 482	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → π ≠ (𝑋 mod 𝑇))
229	179, 180, 183, 228	leneltd 11298	. . . . . . 7 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) < π)
230		iftrue 4467	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) < π → if((𝑋 mod 𝑇) < π, 1, -1) = 1)
231	155, 230	eqtrid 2787	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) < π → (𝐹‘𝑋) = 1)
232	229, 231	syl 17	. . . . . 6 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝐹‘𝑋) = 1)
233	178, 232	oveq12d 7381	. . . . 5 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (1 + 1))
234	233	oveq1d 7378	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = ((1 + 1) / 2))
235		1p1e2 12299	. . . . . . 7 ⊢ (1 + 1) = 2
236	235	oveq1i 7373	. . . . . 6 ⊢ ((1 + 1) / 2) = (2 / 2)
237		2div2e1 12315	. . . . . 6 ⊢ (2 / 2) = 1
238	236, 237	eqtr2i 2764	. . . . 5 ⊢ 1 = ((1 + 1) / 2)
239	232, 238	eqtr2di 2792	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → ((1 + 1) / 2) = (𝐹‘𝑋))
240		iffalse 4470	. . . . . 6 ⊢ (¬ (𝑋 mod π) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = (𝐹‘𝑋))
241	171, 240	eqtr2id 2788	. . . . 5 ⊢ (¬ (𝑋 mod π) = 0 → (𝐹‘𝑋) = 𝑌)
242	241	adantl 482	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝐹‘𝑋) = 𝑌)
243	234, 239, 242	3eqtrrd 2780	. . 3 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
244	176, 243	pm2.61dan 818	. 2 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
245	130	necon2bi 2965	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → ¬ (𝑋 mod 𝑇) ∈ (0(,]π))
246	245	iffalsed 4472	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = -1)
247		id 22	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) = 0)
248	247, 33	eqbrtrdi 5118	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) < π)
249	248	iftrued 4469	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod 𝑇) < π, 1, -1) = 1)
250	155, 249	eqtrid 2787	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → (𝐹‘𝑋) = 1)
251	246, 250	oveq12d 7381	. . . . . 6 ⊢ ((𝑋 mod 𝑇) = 0 → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (-1 + 1))
252	251	oveq1d 7378	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = ((-1 + 1) / 2))
253		neg1cn 12142	. . . . . . . . 9 ⊢ -1 ∈ ℂ
254	184, 253, 164	addcomli 11336	. . . . . . . 8 ⊢ (-1 + 1) = 0
255	254	oveq1i 7373	. . . . . . 7 ⊢ ((-1 + 1) / 2) = (0 / 2)
256	255, 169	eqtri 2763	. . . . . 6 ⊢ ((-1 + 1) / 2) = 0
257	256	a1i 11	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → ((-1 + 1) / 2) = 0)
258	39	oveq2i 7374	. . . . . . . . . . . . 13 ⊢ (𝑋 / 𝑇) = (𝑋 / (2 · π))
259		2cnne0 12384	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
260	13, 34	pm3.2i 471	. . . . . . . . . . . . . 14 ⊢ (π ∈ ℂ ∧ π ≠ 0)
261		divdiv1 11864	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (π ∈ ℂ ∧ π ≠ 0)) → ((𝑋 / 2) / π) = (𝑋 / (2 · π)))
262	26, 259, 260, 261	mp3an 1469	. . . . . . . . . . . . 13 ⊢ ((𝑋 / 2) / π) = (𝑋 / (2 · π))
263	26, 168, 13, 43, 34	divdiv32i 11908	. . . . . . . . . . . . 13 ⊢ ((𝑋 / 2) / π) = ((𝑋 / π) / 2)
264	258, 262, 263	3eqtr2i 2769	. . . . . . . . . . . 12 ⊢ (𝑋 / 𝑇) = ((𝑋 / π) / 2)
265	264	oveq2i 7374	. . . . . . . . . . 11 ⊢ (2 · (𝑋 / 𝑇)) = (2 · ((𝑋 / π) / 2))
266	26, 13, 34	divcli 11895	. . . . . . . . . . . 12 ⊢ (𝑋 / π) ∈ ℂ
267	266, 168, 43	divcan2i 11896	. . . . . . . . . . 11 ⊢ (2 · ((𝑋 / π) / 2)) = (𝑋 / π)
268	265, 267	eqtr2i 2764	. . . . . . . . . 10 ⊢ (𝑋 / π) = (2 · (𝑋 / 𝑇))
269	2	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 / 𝑇) ∈ ℤ → 2 ∈ ℤ)
270		id 22	. . . . . . . . . . 11 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (𝑋 / 𝑇) ∈ ℤ)
271	269, 270	zmulcld 12637	. . . . . . . . . 10 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (2 · (𝑋 / 𝑇)) ∈ ℤ)
272	268, 271	eqeltrid 2844	. . . . . . . . 9 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (𝑋 / π) ∈ ℤ)
273	64, 272	sylbi 218	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 / π) ∈ ℤ)
274	273, 7	sylibr 235	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod π) = 0)
275	274	iftrued 4469	. . . . . 6 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = 0)
276	171, 275	eqtr2id 2788	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → 0 = 𝑌)
277	252, 257, 276	3eqtrrd 2780	. . . 4 ⊢ ((𝑋 mod 𝑇) = 0 → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
278	277	adantl 482	. . 3 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod 𝑇) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
279	127	a1i 11	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → π ∈ ℝ^*)
280	58	rexri 11201	. . . . . 6 ⊢ 𝑇 ∈ ℝ^*
281	280	a1i 11	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → 𝑇 ∈ ℝ^*)
282	138	a1i 11	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ ℝ)
283		pm4.56 996	. . . . . . . 8 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) ↔ ¬ ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
284	283	biimpi 217	. . . . . . 7 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → ¬ ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
285		olc 874	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) = 0 → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
286	285	adantl 482	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ (𝑋 mod 𝑇) = 0) → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
287	126	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → 0 ∈ ℝ^*)
288	127	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → π ∈ ℝ^*)
289	139	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ ℝ^*)
290		0red 11145	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 ∈ ℝ)
291	138	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ∈ ℝ)
292		modge0 13836	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → 0 ≤ (𝑋 mod 𝑇))
293	4, 62, 292	mp2an 698	. . . . . . . . . . . . 13 ⊢ 0 ≤ (𝑋 mod 𝑇)
294	293	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 ≤ (𝑋 mod 𝑇))
295		neqne 2943	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ≠ 0)
296	290, 291, 294, 295	leneltd 11298	. . . . . . . . . . 11 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 < (𝑋 mod 𝑇))
297	296	adantl 482	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → 0 < (𝑋 mod 𝑇))
298		simpl 483	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
299	287, 288, 289, 297, 298	eliocd 45959	. . . . . . . . 9 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ (0(,]π))
300	299	orcd 879	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
301	286, 300	pm2.61dan 818	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ≤ π → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
302	284, 301	nsyl 140	. . . . . 6 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → ¬ (𝑋 mod 𝑇) ≤ π)
303	32	a1i 11	. . . . . . 7 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → π ∈ ℝ)
304	303, 282	ltnled 11291	. . . . . 6 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (π < (𝑋 mod 𝑇) ↔ ¬ (𝑋 mod 𝑇) ≤ π))
305	302, 304	mpbird 258	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → π < (𝑋 mod 𝑇))
306		modlt 13837	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) < 𝑇)
307	4, 62, 306	mp2an 698	. . . . . 6 ⊢ (𝑋 mod 𝑇) < 𝑇
308	307	a1i 11	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) < 𝑇)
309	279, 281, 282, 305, 308	eliood 45950	. . . 4 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ (π(,)𝑇))
310	126	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → 0 ∈ ℝ^*)
311	32	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → π ∈ ℝ)
312	139	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (𝑋 mod 𝑇) ∈ ℝ^*)
313		ioogtlb 45947	. . . . . . . . . 10 ⊢ ((π ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (π(,)𝑇)) → π < (𝑋 mod 𝑇))
314	127, 280, 313	mp3an12 1459	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → π < (𝑋 mod 𝑇))
315	310, 311, 312, 314	gtnelioc 45943	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) ∈ (0(,]π))
316	315	iffalsed 4472	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = -1)
317	138	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (𝑋 mod 𝑇) ∈ ℝ)
318	311, 317, 314	ltnsymd 11293	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) < π)
319	318	iffalsed 4472	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → if((𝑋 mod 𝑇) < π, 1, -1) = -1)
320	155, 319	eqtrid 2787	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (𝐹‘𝑋) = -1)
321	316, 320	oveq12d 7381	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (-1 + -1))
322	321	oveq1d 7378	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = ((-1 + -1) / 2))
323		df-2 12242	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
324	323	negeqi 11384	. . . . . . . . 9 ⊢ -2 = -(1 + 1)
325	184, 184	negdii 11476	. . . . . . . . 9 ⊢ -(1 + 1) = (-1 + -1)
326	324, 325	eqtr2i 2764	. . . . . . . 8 ⊢ (-1 + -1) = -2
327	326	oveq1i 7373	. . . . . . 7 ⊢ ((-1 + -1) / 2) = (-2 / 2)
328		divneg 11844	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(2 / 2) = (-2 / 2))
329	168, 168, 43, 328	mp3an 1469	. . . . . . 7 ⊢ -(2 / 2) = (-2 / 2)
330	237	negeqi 11384	. . . . . . 7 ⊢ -(2 / 2) = -1
331	327, 329, 330	3eqtr2i 2769	. . . . . 6 ⊢ ((-1 + -1) / 2) = -1
332	331	a1i 11	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ((-1 + -1) / 2) = -1)
333	171	a1i 11	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → 𝑌 = if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)))
334	311, 317	ltnled 11291	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (π < (𝑋 mod 𝑇) ↔ ¬ (𝑋 mod 𝑇) ≤ π))
335	314, 334	mpbid 233	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) ≤ π)
336	247, 111	eqbrtrdi 5118	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ≤ π)
337	336	adantl 482	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
338	125	orcanai 1010	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) = π)
339	338, 143	syl 17	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
340	337, 339	pm2.61dan 818	. . . . . . . 8 ⊢ ((𝑋 mod π) = 0 → (𝑋 mod 𝑇) ≤ π)
341	335, 340	nsyl 140	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod π) = 0)
342	341	iffalsed 4472	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = (𝐹‘𝑋))
343	333, 342, 320	3eqtrrd 2780	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → -1 = 𝑌)
344	322, 332, 343	3eqtrrd 2780	. . . 4 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
345	309, 344	syl 17	. . 3 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
346	278, 345	pm2.61dan 818	. 2 ⊢ (¬ (𝑋 mod 𝑇) ∈ (0(,]π) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
347	244, 346	pm2.61i 183	1 ⊢ 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∃wrex 3064 ifcif 4461 class class class wbr 5079 ↦ cmpt 5160 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 − cmin 11375 -cneg 11376 / cdiv 11805 2c2 12234 ℤcz 12522 ℝ⁺crp 12940 (,)cioo 13296 (,]cioc 13297 ⌊cfl 13747 mod cmo 13826 πcpi 16029 ∥ cdvds 16219

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030 df-sin 16032 df-cos 16033 df-pi 16035 df-dvds 16220 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859

This theorem is referenced by: fouriersw 46681

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