fourierswlem - Mathbox for Glauco Siliprandi

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Theorem fourierswlem 46585

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 484	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → 2 ∥ (𝑋 / π))
2		2z 12535	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
3	2	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → 2 ∈ ℤ)
4		fourierswlem.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 ∈ ℝ
5		pirp 26438	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ⁺
6		mod0 13808	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℝ ∧ π ∈ ℝ⁺) → ((𝑋 mod π) = 0 ↔ (𝑋 / π) ∈ ℤ))
7	4, 5, 6	mp2an 693	. . . . . . . . . . . . 13 ⊢ ((𝑋 mod π) = 0 ↔ (𝑋 / π) ∈ ℤ)
8	7	biimpi 216	. . . . . . . . . . . 12 ⊢ ((𝑋 mod π) = 0 → (𝑋 / π) ∈ ℤ)
9	8	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 / π) ∈ ℤ)
10		divides 16193	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ (𝑋 / π) ∈ ℤ) → (2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π)))
11	3, 9, 10	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π)))
12	1, 11	mpbid 232	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π))
13		2cnd 12235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → 2 ∈ ℂ)
14		picn 26435	. . . . . . . . . . . . . . . . . . . 20 ⊢ π ∈ ℂ
15	14	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → π ∈ ℂ)
16		zcn 12505	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
17	13, 15, 16	mulassd 11167	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((2 · π) · 𝑘) = (2 · (π · 𝑘)))
18	15, 16	mulcld 11164	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (π · 𝑘) ∈ ℂ)
19	13, 18	mulcomd 11165	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → (2 · (π · 𝑘)) = ((π · 𝑘) · 2))
20	17, 19	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((2 · π) · 𝑘) = ((π · 𝑘) · 2))
21	20	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((2 · π) · 𝑘) = ((π · 𝑘) · 2))
22	15, 16, 13	mulassd 11167	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((π · 𝑘) · 2) = (π · (𝑘 · 2)))
23	22	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((π · 𝑘) · 2) = (π · (𝑘 · 2)))
24		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 · 2) = (𝑋 / π) → (𝑘 · 2) = (𝑋 / π))
25	24	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 · 2) = (𝑋 / π) → (𝑋 / π) = (𝑘 · 2))
26	25	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / π) = (𝑘 · 2))
27	4	recni 11158	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑋 ∈ ℂ
28	27	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑋 ∈ ℂ)
29	14	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → π ∈ ℂ)
30	16	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑘 ∈ ℂ)
31		2cnd 12235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 2 ∈ ℂ)
32	30, 31	mulcld 11164	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑘 · 2) ∈ ℂ)
33		pire 26434	. . . . . . . . . . . . . . . . . . . 20 ⊢ π ∈ ℝ
34		pipos 26436	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < π
35	33, 34	gt0ne0ii 11685	. . . . . . . . . . . . . . . . . . 19 ⊢ π ≠ 0
36	35	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → π ≠ 0)
37	28, 29, 32, 36	divmuld 11951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((𝑋 / π) = (𝑘 · 2) ↔ (π · (𝑘 · 2)) = 𝑋))
38	26, 37	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (π · (𝑘 · 2)) = 𝑋)
39	21, 23, 38	3eqtrrd 2777	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑋 = ((2 · π) · 𝑘))
40		fourierswlem.t	. . . . . . . . . . . . . . . 16 ⊢ 𝑇 = (2 · π)
41	40	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑇 = (2 · π))
42	39, 41	oveq12d 7386	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) = (((2 · π) · 𝑘) / (2 · π)))
43	13, 15	mulcld 11164	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (2 · π) ∈ ℂ)
44		2ne0 12261	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
45	44	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → 2 ≠ 0)
46	35	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → π ≠ 0)
47	13, 15, 45, 46	mulne0d 11801	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (2 · π) ≠ 0)
48	16, 43, 47	divcan3d 11934	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (((2 · π) · 𝑘) / (2 · π)) = 𝑘)
49	48	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (((2 · π) · 𝑘) / (2 · π)) = 𝑘)
50	42, 49	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) = 𝑘)
51		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑘 ∈ ℤ)
52	50, 51	eqeltrd 2837	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) ∈ ℤ)
53	52	ex 412	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ → ((𝑘 · 2) = (𝑋 / π) → (𝑋 / 𝑇) ∈ ℤ))
54	53	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑘 ∈ ℤ → ((𝑘 · 2) = (𝑋 / π) → (𝑋 / 𝑇) ∈ ℤ)))
55	54	rexlimdv 3137	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π) → (𝑋 / 𝑇) ∈ ℤ))
56	12, 55	mpd 15	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 / 𝑇) ∈ ℤ)
57		2re 12231	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
58	57, 33	remulcli 11160	. . . . . . . . . . 11 ⊢ (2 · π) ∈ ℝ
59	40, 58	eqeltri 2833	. . . . . . . . . 10 ⊢ 𝑇 ∈ ℝ
60		2pos 12260	. . . . . . . . . . . 12 ⊢ 0 < 2
61	57, 33, 60, 34	mulgt0ii 11278	. . . . . . . . . . 11 ⊢ 0 < (2 · π)
62	61, 40	breqtrri 5127	. . . . . . . . . 10 ⊢ 0 < 𝑇
63	59, 62	elrpii 12920	. . . . . . . . 9 ⊢ 𝑇 ∈ ℝ⁺
64		mod0 13808	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → ((𝑋 mod 𝑇) = 0 ↔ (𝑋 / 𝑇) ∈ ℤ))
65	4, 63, 64	mp2an 693	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 ↔ (𝑋 / 𝑇) ∈ ℤ)
66	56, 65	sylibr 234	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 mod 𝑇) = 0)
67	66	orcd 874	. . . . . 6 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
68		odd2np1 16280	. . . . . . . . . 10 ⊢ ((𝑋 / π) ∈ ℤ → (¬ 2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π)))
69	7, 68	sylbi 217	. . . . . . . . 9 ⊢ ((𝑋 mod π) = 0 → (¬ 2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π)))
70	69	biimpa 476	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π))
71	13, 16	mulcld 11164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (2 · 𝑘) ∈ ℂ)
72	71	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (2 · 𝑘) ∈ ℂ)
73		1cnd 11139	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 1 ∈ ℂ)
74	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π ∈ ℂ)
75	72, 73, 74	adddird 11169	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) + 1) · π) = (((2 · 𝑘) · π) + (1 · π)))
76	13, 16	mulcomd 11165	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (2 · 𝑘) = (𝑘 · 2))
77	76	oveq1d 7383	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) · π) = ((𝑘 · 2) · π))
78	16, 13, 15	mulassd 11167	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((𝑘 · 2) · π) = (𝑘 · (2 · π)))
79	40	eqcomi 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · π) = 𝑇
80	79	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (2 · π) = 𝑇)
81	80	oveq2d 7384	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → (𝑘 · (2 · π)) = (𝑘 · 𝑇))
82	77, 78, 81	3eqtrd 2776	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) · π) = (𝑘 · 𝑇))
83	14	mullidi 11149	. . . . . . . . . . . . . . . . . 18 ⊢ (1 · π) = π
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (1 · π) = π)
85	82, 84	oveq12d 7386	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (((2 · 𝑘) · π) + (1 · π)) = ((𝑘 · 𝑇) + π))
86	85	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) · π) + (1 · π)) = ((𝑘 · 𝑇) + π))
87	40, 43	eqeltrid 2841	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → 𝑇 ∈ ℂ)
88	16, 87	mulcld 11164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (𝑘 · 𝑇) ∈ ℂ)
89	88, 15	addcomd 11347	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → ((𝑘 · 𝑇) + π) = (π + (𝑘 · 𝑇)))
90	89	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((𝑘 · 𝑇) + π) = (π + (𝑘 · 𝑇)))
91	75, 86, 90	3eqtrrd 2777	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π + (𝑘 · 𝑇)) = (((2 · 𝑘) + 1) · π))
92		peano2cn 11317	. . . . . . . . . . . . . . . . 17 ⊢ ((2 · 𝑘) ∈ ℂ → ((2 · 𝑘) + 1) ∈ ℂ)
93	71, 92	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) + 1) ∈ ℂ)
94	93, 15	mulcomd 11165	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (((2 · 𝑘) + 1) · π) = (π · ((2 · 𝑘) + 1)))
95	94	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) + 1) · π) = (π · ((2 · 𝑘) + 1)))
96		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (((2 · 𝑘) + 1) = (𝑋 / π) → ((2 · 𝑘) + 1) = (𝑋 / π))
97	96	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 / π) = ((2 · 𝑘) + 1))
98	97	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (𝑋 / π) = ((2 · 𝑘) + 1))
99	27	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 𝑋 ∈ ℂ)
100	93	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((2 · 𝑘) + 1) ∈ ℂ)
101	35	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π ≠ 0)
102	99, 74, 100, 101	divmuld 11951	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((𝑋 / π) = ((2 · 𝑘) + 1) ↔ (π · ((2 · 𝑘) + 1)) = 𝑋))
103	98, 102	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π · ((2 · 𝑘) + 1)) = 𝑋)
104	91, 95, 103	3eqtrrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 𝑋 = (π + (𝑘 · 𝑇)))
105	104	oveq1d 7383	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (𝑋 mod 𝑇) = ((π + (𝑘 · 𝑇)) mod 𝑇))
106		modcyc 13838	. . . . . . . . . . . . . 14 ⊢ ((π ∈ ℝ ∧ 𝑇 ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → ((π + (𝑘 · 𝑇)) mod 𝑇) = (π mod 𝑇))
107	33, 63, 106	mp3an12 1454	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤ → ((π + (𝑘 · 𝑇)) mod 𝑇) = (π mod 𝑇))
108	107	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((π + (𝑘 · 𝑇)) mod 𝑇) = (π mod 𝑇))
109	33	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π ∈ ℝ)
110	63	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 𝑇 ∈ ℝ⁺)
111		0re 11146	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
112	111, 33, 34	ltleii 11268	. . . . . . . . . . . . . 14 ⊢ 0 ≤ π
113	112	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 0 ≤ π)
114		2timesgt 45647	. . . . . . . . . . . . . . . 16 ⊢ (π ∈ ℝ⁺ → π < (2 · π))
115	5, 114	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ π < (2 · π)
116	115, 40	breqtrri 5127	. . . . . . . . . . . . . 14 ⊢ π < 𝑇
117	116	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π < 𝑇)
118		modid 13828	. . . . . . . . . . . . 13 ⊢ (((π ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) ∧ (0 ≤ π ∧ π < 𝑇)) → (π mod 𝑇) = π)
119	109, 110, 113, 117, 118	syl22anc 839	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π mod 𝑇) = π)
120	105, 108, 119	3eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (𝑋 mod 𝑇) = π)
121	120	ex 412	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → (((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 mod 𝑇) = π))
122	121	a1i 11	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → (𝑘 ∈ ℤ → (((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 mod 𝑇) = π)))
123	122	rexlimdv 3137	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → (∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 mod 𝑇) = π))
124	70, 123	mpd 15	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → (𝑋 mod 𝑇) = π)
125	124	olcd 875	. . . . . 6 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
126	67, 125	pm2.61dan 813	. . . . 5 ⊢ ((𝑋 mod π) = 0 → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
127		0xr 11191	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
128	33	rexri 11202	. . . . . . . 8 ⊢ π ∈ ℝ^*
129		iocgtlb 45859	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (0(,]π)) → 0 < (𝑋 mod 𝑇))
130	127, 128, 129	mp3an12 1454	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → 0 < (𝑋 mod 𝑇))
131	130	gt0ne0d 11713	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → (𝑋 mod 𝑇) ≠ 0)
132	131	neneqd 2938	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → ¬ (𝑋 mod 𝑇) = 0)
133		pm2.53 852	. . . . . 6 ⊢ (((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π) → (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) = π))
134	133	imp 406	. . . . 5 ⊢ ((((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) = π)
135	126, 132, 134	syl2anr 598	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) = π)
136	127	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → 0 ∈ ℝ^*)
137	128	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → π ∈ ℝ^*)
138		modcl 13805	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) ∈ ℝ)
139	4, 63, 138	mp2an 693	. . . . . . . . . . . . . 14 ⊢ (𝑋 mod 𝑇) ∈ ℝ
140	139	rexri 11202	. . . . . . . . . . . . 13 ⊢ (𝑋 mod 𝑇) ∈ ℝ^*
141	140	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ∈ ℝ^*)
142		id 22	. . . . . . . . . . . . 13 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) = π)
143	34, 142	breqtrrid 5138	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → 0 < (𝑋 mod 𝑇))
144	33	eqlei2 11256	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ≤ π)
145	136, 137, 141, 143, 144	eliocd 45864	. . . . . . . . . . 11 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ∈ (0(,]π))
146	145	iftrued 4489	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = π → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
147	146	adantl 481	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
148		oveq1 7375	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → (𝑥 mod 𝑇) = (𝑋 mod 𝑇))
149	148	breq1d 5110	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → ((𝑥 mod 𝑇) < π ↔ (𝑋 mod 𝑇) < π))
150	149	ifbid 4505	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → if((𝑥 mod 𝑇) < π, 1, -1) = if((𝑋 mod 𝑇) < π, 1, -1))
151		fourierswlem.f	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if((𝑥 mod 𝑇) < π, 1, -1))
152		1ex 11140	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
153		negex 11390	. . . . . . . . . . . . . 14 ⊢ -1 ∈ V
154	152, 153	ifex 4532	. . . . . . . . . . . . 13 ⊢ if((𝑋 mod 𝑇) < π, 1, -1) ∈ V
155	150, 151, 154	fvmpt 6949	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℝ → (𝐹‘𝑋) = if((𝑋 mod 𝑇) < π, 1, -1))
156	4, 155	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹‘𝑋) = if((𝑋 mod 𝑇) < π, 1, -1)
157	139	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑋 mod 𝑇) < π → (𝑋 mod 𝑇) ∈ ℝ)
158		id 22	. . . . . . . . . . . . . 14 ⊢ ((𝑋 mod 𝑇) < π → (𝑋 mod 𝑇) < π)
159	157, 158	ltned 11281	. . . . . . . . . . . . 13 ⊢ ((𝑋 mod 𝑇) < π → (𝑋 mod 𝑇) ≠ π)
160	159	necon2bi 2963	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → ¬ (𝑋 mod 𝑇) < π)
161	160	iffalsed 4492	. . . . . . . . . . 11 ⊢ ((𝑋 mod 𝑇) = π → if((𝑋 mod 𝑇) < π, 1, -1) = -1)
162	156, 161	eqtrid 2784	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = π → (𝐹‘𝑋) = -1)
163	162	adantl 481	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (𝐹‘𝑋) = -1)
164	147, 163	oveq12d 7386	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (1 + -1))
165		1pneg1e0 12271	. . . . . . . 8 ⊢ (1 + -1) = 0
166	164, 165	eqtrdi 2788	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = 0)
167	166	oveq1d 7383	. . . . . 6 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = (0 / 2))
168	167	adantll 715	. . . . 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))
169		2cn 12232	. . . . . . 7 ⊢ 2 ∈ ℂ
170	169, 44	div0i 11887	. . . . . 6 ⊢ (0 / 2) = 0
171	170	a1i 11	. . . . 5 ⊢ ((((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) ∧ (𝑋 mod 𝑇) = π) → (0 / 2) = 0)
172		fourierswlem.y	. . . . . . 7 ⊢ 𝑌 = if((𝑋 mod π) = 0, 0, (𝐹‘𝑋))
173		iftrue 4487	. . . . . . 7 ⊢ ((𝑋 mod π) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = 0)
174	172, 173	eqtr2id 2785	. . . . . 6 ⊢ ((𝑋 mod π) = 0 → 0 = 𝑌)
175	174	ad2antlr 728	. . . . 5 ⊢ ((((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) ∧ (𝑋 mod 𝑇) = π) → 0 = 𝑌)
176	168, 171, 175	3eqtrrd 2777	. . . 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))
177	135, 176	mpdan 688	. . 3 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
178		iftrue 4487	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
179	178	adantr 480	. . . . . 6 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
180	139	a1i 11	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) ∈ ℝ)
181	33	a1i 11	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → π ∈ ℝ)
182		iocleub 45860	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (0(,]π)) → (𝑋 mod 𝑇) ≤ π)
183	127, 128, 182	mp3an12 1454	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → (𝑋 mod 𝑇) ≤ π)
184	183	adantr 480	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) ≤ π)
185		ax-1cn 11096	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
186	185, 14	mulcomi 11152	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 · π) = (π · 1)
187	83, 186	eqtr3i 2762	. . . . . . . . . . . . . . . . . 18 ⊢ π = (π · 1)
188	187	oveq1i 7378	. . . . . . . . . . . . . . . . 17 ⊢ (π + (π · (2 · (⌊‘(𝑋 / 𝑇))))) = ((π · 1) + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
189	169, 14	mulcomi 11152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2 · π) = (π · 2)
190	40, 189	eqtri 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑇 = (π · 2)
191	190	oveq1i 7378	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) = ((π · 2) · (⌊‘(𝑋 / 𝑇)))
192	111, 62	gtneii 11257	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑇 ≠ 0
193	4, 59, 192	redivcli 11920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 / 𝑇) ∈ ℝ
194		flcl 13727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 / 𝑇) ∈ ℝ → (⌊‘(𝑋 / 𝑇)) ∈ ℤ)
195	193, 194	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⌊‘(𝑋 / 𝑇)) ∈ ℤ
196		zcn 12505	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⌊‘(𝑋 / 𝑇)) ∈ ℤ → (⌊‘(𝑋 / 𝑇)) ∈ ℂ)
197	195, 196	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⌊‘(𝑋 / 𝑇)) ∈ ℂ
198	14, 169, 197	mulassi 11155	. . . . . . . . . . . . . . . . . . 19 ⊢ ((π · 2) · (⌊‘(𝑋 / 𝑇))) = (π · (2 · (⌊‘(𝑋 / 𝑇))))
199	191, 198	eqtri 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) = (π · (2 · (⌊‘(𝑋 / 𝑇))))
200	199	oveq2i 7379	. . . . . . . . . . . . . . . . 17 ⊢ (π + (𝑇 · (⌊‘(𝑋 / 𝑇)))) = (π + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
201	169, 197	mulcli 11151	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ
202	14, 185, 201	adddii 11156	. . . . . . . . . . . . . . . . 17 ⊢ (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) = ((π · 1) + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
203	188, 200, 202	3eqtr4ri 2771	. . . . . . . . . . . . . . . 16 ⊢ (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) = (π + (𝑇 · (⌊‘(𝑋 / 𝑇))))
204	203	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (π = (𝑋 mod 𝑇) → (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) = (π + (𝑇 · (⌊‘(𝑋 / 𝑇)))))
205		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (π = (𝑋 mod 𝑇) → π = (𝑋 mod 𝑇))
206		modval 13803	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))))
207	4, 63, 206	mp2an 693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 mod 𝑇) = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇))))
208	205, 207	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ (π = (𝑋 mod 𝑇) → π = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))))
209	208	oveq1d 7383	. . . . . . . . . . . . . . 15 ⊢ (π = (𝑋 mod 𝑇) → (π + (𝑇 · (⌊‘(𝑋 / 𝑇)))) = ((𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))) + (𝑇 · (⌊‘(𝑋 / 𝑇)))))
210	27	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (π = (𝑋 mod 𝑇) → 𝑋 ∈ ℂ)
211	59	recni 11158	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 ∈ ℂ
212	211, 197	mulcli 11151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ
213	212	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (π = (𝑋 mod 𝑇) → (𝑇 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ)
214	210, 213	npcand 11508	. . . . . . . . . . . . . . 15 ⊢ (π = (𝑋 mod 𝑇) → ((𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))) + (𝑇 · (⌊‘(𝑋 / 𝑇)))) = 𝑋)
215	204, 209, 214	3eqtrrd 2777	. . . . . . . . . . . . . 14 ⊢ (π = (𝑋 mod 𝑇) → 𝑋 = (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))))
216	215	oveq1d 7383	. . . . . . . . . . . . 13 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) = ((π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) / π))
217	185, 201	addcli 11150	. . . . . . . . . . . . . 14 ⊢ (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℂ
218	217, 14, 35	divcan3i 11899	. . . . . . . . . . . . 13 ⊢ ((π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) / π) = (1 + (2 · (⌊‘(𝑋 / 𝑇))))
219	216, 218	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) = (1 + (2 · (⌊‘(𝑋 / 𝑇)))))
220		1z 12533	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
221		zmulcl 12552	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℤ ∧ (⌊‘(𝑋 / 𝑇)) ∈ ℤ) → (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ)
222	2, 195, 221	mp2an 693	. . . . . . . . . . . . . 14 ⊢ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ
223		zaddcl 12543	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ) → (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ)
224	220, 222, 223	mp2an 693	. . . . . . . . . . . . 13 ⊢ (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ
225	224	a1i 11	. . . . . . . . . . . 12 ⊢ (π = (𝑋 mod 𝑇) → (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ)
226	219, 225	eqeltrd 2837	. . . . . . . . . . 11 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) ∈ ℤ)
227	226, 7	sylibr 234	. . . . . . . . . 10 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 mod π) = 0)
228	227	necon3bi 2959	. . . . . . . . 9 ⊢ (¬ (𝑋 mod π) = 0 → π ≠ (𝑋 mod 𝑇))
229	228	adantl 481	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → π ≠ (𝑋 mod 𝑇))
230	180, 181, 184, 229	leneltd 11299	. . . . . . 7 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) < π)
231		iftrue 4487	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) < π → if((𝑋 mod 𝑇) < π, 1, -1) = 1)
232	156, 231	eqtrid 2784	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) < π → (𝐹‘𝑋) = 1)
233	230, 232	syl 17	. . . . . 6 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝐹‘𝑋) = 1)
234	179, 233	oveq12d 7386	. . . . 5 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (1 + 1))
235	234	oveq1d 7383	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = ((1 + 1) / 2))
236		1p1e2 12277	. . . . . . 7 ⊢ (1 + 1) = 2
237	236	oveq1i 7378	. . . . . 6 ⊢ ((1 + 1) / 2) = (2 / 2)
238		2div2e1 12293	. . . . . 6 ⊢ (2 / 2) = 1
239	237, 238	eqtr2i 2761	. . . . 5 ⊢ 1 = ((1 + 1) / 2)
240	233, 239	eqtr2di 2789	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → ((1 + 1) / 2) = (𝐹‘𝑋))
241		iffalse 4490	. . . . . 6 ⊢ (¬ (𝑋 mod π) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = (𝐹‘𝑋))
242	172, 241	eqtr2id 2785	. . . . 5 ⊢ (¬ (𝑋 mod π) = 0 → (𝐹‘𝑋) = 𝑌)
243	242	adantl 481	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝐹‘𝑋) = 𝑌)
244	235, 240, 243	3eqtrrd 2777	. . 3 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
245	177, 244	pm2.61dan 813	. 2 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
246	131	necon2bi 2963	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → ¬ (𝑋 mod 𝑇) ∈ (0(,]π))
247	246	iffalsed 4492	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = -1)
248		id 22	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) = 0)
249	248, 34	eqbrtrdi 5139	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) < π)
250	249	iftrued 4489	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod 𝑇) < π, 1, -1) = 1)
251	156, 250	eqtrid 2784	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → (𝐹‘𝑋) = 1)
252	247, 251	oveq12d 7386	. . . . . 6 ⊢ ((𝑋 mod 𝑇) = 0 → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (-1 + 1))
253	252	oveq1d 7383	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = ((-1 + 1) / 2))
254		neg1cn 12142	. . . . . . . . 9 ⊢ -1 ∈ ℂ
255	185, 254, 165	addcomli 11337	. . . . . . . 8 ⊢ (-1 + 1) = 0
256	255	oveq1i 7378	. . . . . . 7 ⊢ ((-1 + 1) / 2) = (0 / 2)
257	256, 170	eqtri 2760	. . . . . 6 ⊢ ((-1 + 1) / 2) = 0
258	257	a1i 11	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → ((-1 + 1) / 2) = 0)
259	40	oveq2i 7379	. . . . . . . . . . . . 13 ⊢ (𝑋 / 𝑇) = (𝑋 / (2 · π))
260		2cnne0 12362	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
261	14, 35	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (π ∈ ℂ ∧ π ≠ 0)
262		divdiv1 11864	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (π ∈ ℂ ∧ π ≠ 0)) → ((𝑋 / 2) / π) = (𝑋 / (2 · π)))
263	27, 260, 261, 262	mp3an 1464	. . . . . . . . . . . . 13 ⊢ ((𝑋 / 2) / π) = (𝑋 / (2 · π))
264	27, 169, 14, 44, 35	divdiv32i 11908	. . . . . . . . . . . . 13 ⊢ ((𝑋 / 2) / π) = ((𝑋 / π) / 2)
265	259, 263, 264	3eqtr2i 2766	. . . . . . . . . . . 12 ⊢ (𝑋 / 𝑇) = ((𝑋 / π) / 2)
266	265	oveq2i 7379	. . . . . . . . . . 11 ⊢ (2 · (𝑋 / 𝑇)) = (2 · ((𝑋 / π) / 2))
267	27, 14, 35	divcli 11895	. . . . . . . . . . . 12 ⊢ (𝑋 / π) ∈ ℂ
268	267, 169, 44	divcan2i 11896	. . . . . . . . . . 11 ⊢ (2 · ((𝑋 / π) / 2)) = (𝑋 / π)
269	266, 268	eqtr2i 2761	. . . . . . . . . 10 ⊢ (𝑋 / π) = (2 · (𝑋 / 𝑇))
270	2	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 / 𝑇) ∈ ℤ → 2 ∈ ℤ)
271		id 22	. . . . . . . . . . 11 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (𝑋 / 𝑇) ∈ ℤ)
272	270, 271	zmulcld 12614	. . . . . . . . . 10 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (2 · (𝑋 / 𝑇)) ∈ ℤ)
273	269, 272	eqeltrid 2841	. . . . . . . . 9 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (𝑋 / π) ∈ ℤ)
274	65, 273	sylbi 217	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 / π) ∈ ℤ)
275	274, 7	sylibr 234	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod π) = 0)
276	275	iftrued 4489	. . . . . 6 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = 0)
277	172, 276	eqtr2id 2785	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → 0 = 𝑌)
278	253, 258, 277	3eqtrrd 2777	. . . 4 ⊢ ((𝑋 mod 𝑇) = 0 → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
279	278	adantl 481	. . 3 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod 𝑇) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
280	128	a1i 11	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → π ∈ ℝ^*)
281	59	rexri 11202	. . . . . 6 ⊢ 𝑇 ∈ ℝ^*
282	281	a1i 11	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → 𝑇 ∈ ℝ^*)
283	139	a1i 11	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ ℝ)
284		pm4.56 991	. . . . . . . 8 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) ↔ ¬ ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
285	284	biimpi 216	. . . . . . 7 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → ¬ ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
286		olc 869	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) = 0 → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
287	286	adantl 481	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ (𝑋 mod 𝑇) = 0) → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
288	127	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → 0 ∈ ℝ^*)
289	128	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → π ∈ ℝ^*)
290	140	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ ℝ^*)
291		0red 11147	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 ∈ ℝ)
292	139	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ∈ ℝ)
293		modge0 13811	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → 0 ≤ (𝑋 mod 𝑇))
294	4, 63, 293	mp2an 693	. . . . . . . . . . . . 13 ⊢ 0 ≤ (𝑋 mod 𝑇)
295	294	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 ≤ (𝑋 mod 𝑇))
296		neqne 2941	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ≠ 0)
297	291, 292, 295, 296	leneltd 11299	. . . . . . . . . . 11 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 < (𝑋 mod 𝑇))
298	297	adantl 481	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → 0 < (𝑋 mod 𝑇))
299		simpl 482	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
300	288, 289, 290, 298, 299	eliocd 45864	. . . . . . . . 9 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ (0(,]π))
301	300	orcd 874	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
302	287, 301	pm2.61dan 813	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ≤ π → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
303	285, 302	nsyl 140	. . . . . 6 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → ¬ (𝑋 mod 𝑇) ≤ π)
304	33	a1i 11	. . . . . . 7 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → π ∈ ℝ)
305	304, 283	ltnled 11292	. . . . . 6 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (π < (𝑋 mod 𝑇) ↔ ¬ (𝑋 mod 𝑇) ≤ π))
306	303, 305	mpbird 257	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → π < (𝑋 mod 𝑇))
307		modlt 13812	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) < 𝑇)
308	4, 63, 307	mp2an 693	. . . . . 6 ⊢ (𝑋 mod 𝑇) < 𝑇
309	308	a1i 11	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) < 𝑇)
310	280, 282, 283, 306, 309	eliood 45855	. . . 4 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ (π(,)𝑇))
311	127	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → 0 ∈ ℝ^*)
312	33	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → π ∈ ℝ)
313	140	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (𝑋 mod 𝑇) ∈ ℝ^*)
314		ioogtlb 45852	. . . . . . . . . 10 ⊢ ((π ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (π(,)𝑇)) → π < (𝑋 mod 𝑇))
315	128, 281, 314	mp3an12 1454	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → π < (𝑋 mod 𝑇))
316	311, 312, 313, 315	gtnelioc 45848	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) ∈ (0(,]π))
317	316	iffalsed 4492	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = -1)
318	139	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (𝑋 mod 𝑇) ∈ ℝ)
319	312, 318, 315	ltnsymd 11294	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) < π)
320	319	iffalsed 4492	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → if((𝑋 mod 𝑇) < π, 1, -1) = -1)
321	156, 320	eqtrid 2784	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (𝐹‘𝑋) = -1)
322	317, 321	oveq12d 7386	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (-1 + -1))
323	322	oveq1d 7383	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = ((-1 + -1) / 2))
324		df-2 12220	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
325	324	negeqi 11385	. . . . . . . . 9 ⊢ -2 = -(1 + 1)
326	185, 185	negdii 11477	. . . . . . . . 9 ⊢ -(1 + 1) = (-1 + -1)
327	325, 326	eqtr2i 2761	. . . . . . . 8 ⊢ (-1 + -1) = -2
328	327	oveq1i 7378	. . . . . . 7 ⊢ ((-1 + -1) / 2) = (-2 / 2)
329		divneg 11845	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(2 / 2) = (-2 / 2))
330	169, 169, 44, 329	mp3an 1464	. . . . . . 7 ⊢ -(2 / 2) = (-2 / 2)
331	238	negeqi 11385	. . . . . . 7 ⊢ -(2 / 2) = -1
332	328, 330, 331	3eqtr2i 2766	. . . . . 6 ⊢ ((-1 + -1) / 2) = -1
333	332	a1i 11	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ((-1 + -1) / 2) = -1)
334	172	a1i 11	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → 𝑌 = if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)))
335	312, 318	ltnled 11292	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (π < (𝑋 mod 𝑇) ↔ ¬ (𝑋 mod 𝑇) ≤ π))
336	315, 335	mpbid 232	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) ≤ π)
337	248, 112	eqbrtrdi 5139	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ≤ π)
338	337	adantl 481	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
339	126	orcanai 1005	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) = π)
340	339, 144	syl 17	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
341	338, 340	pm2.61dan 813	. . . . . . . 8 ⊢ ((𝑋 mod π) = 0 → (𝑋 mod 𝑇) ≤ π)
342	336, 341	nsyl 140	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod π) = 0)
343	342	iffalsed 4492	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = (𝐹‘𝑋))
344	334, 343, 321	3eqtrrd 2777	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → -1 = 𝑌)
345	323, 333, 344	3eqtrrd 2777	. . . 4 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
346	310, 345	syl 17	. . 3 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
347	279, 346	pm2.61dan 813	. 2 ⊢ (¬ (𝑋 mod 𝑇) ∈ (0(,]π) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
348	245, 347	pm2.61i 182	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 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 ifcif 4481 class class class wbr 5100 ↦ cmpt 5181 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 − cmin 11376 -cneg 11377 / cdiv 11806 2c2 12212 ℤcz 12500 ℝ⁺crp 12917 (,)cioo 13273 (,]cioc 13274 ⌊cfl 13722 mod cmo 13801 πcpi 16001 ∥ cdvds 16191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-pi 16007 df-dvds 16192 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836

This theorem is referenced by: fouriersw 46586

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