fourierswlem - Mathbox for Glauco Siliprandi

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Theorem fourierswlem 47058

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 490	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → 2 ∥ (𝑋 / π))
2		2z 12650	. . . . . . . . . . . 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 26699	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ⁺
6		mod0 13937	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℝ ∧ π ∈ ℝ⁺) → ((𝑋 mod π) = 0 ↔ (𝑋 / π) ∈ ℤ))
7	4, 5, 6	mp2an 705	. . . . . . . . . . . 12 ⊢ ((𝑋 mod π) = 0 ↔ (𝑋 / π) ∈ ℤ)
8	7	birani 509	. . . . . . . . . . 11 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 / π) ∈ ℤ)
9		divides 16344	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ (𝑋 / π) ∈ ℤ) → (2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π)))
10	3, 8, 9	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π)))
11	1, 10	mpbid 235	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π))
12		2cnd 12343	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → 2 ∈ ℂ)
13		picn 26694	. . . . . . . . . . . . . . . . . . . 20 ⊢ π ∈ ℂ
14	13	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → π ∈ ℂ)
15		zcn 12620	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
16	12, 14, 15	mulassd 11256	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((2 · π) · 𝑘) = (2 · (π · 𝑘)))
17	14, 15	mulcld 11253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (π · 𝑘) ∈ ℂ)
18	12, 17	mulcomd 11254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → (2 · (π · 𝑘)) = ((π · 𝑘) · 2))
19	16, 18	eqtrd 2795	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((2 · π) · 𝑘) = ((π · 𝑘) · 2))
20	19	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((2 · π) · 𝑘) = ((π · 𝑘) · 2))
21	14, 15, 12	mulassd 11256	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((π · 𝑘) · 2) = (π · (𝑘 · 2)))
22	21	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((π · 𝑘) · 2) = (π · (𝑘 · 2)))
23		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 · 2) = (𝑋 / π) → (𝑘 · 2) = (𝑋 / π))
24	23	eqcomd 2766	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 · 2) = (𝑋 / π) → (𝑋 / π) = (𝑘 · 2))
25	24	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / π) = (𝑘 · 2))
26	4	recni 11247	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑘 ∈ ℂ)
30		2cnd 12343	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 2 ∈ ℂ)
31	29, 30	mulcld 11253	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑘 · 2) ∈ ℂ)
32		pire 26692	. . . . . . . . . . . . . . . . . . . 20 ⊢ π ∈ ℝ
33		pipos 26696	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < π
34	32, 33	gt0ne0ii 11774	. . . . . . . . . . . . . . . . . . 19 ⊢ π ≠ 0
35	34	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → π ≠ 0)
36	27, 28, 31, 35	divmuld 12037	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → ((𝑋 / π) = (𝑘 · 2) ↔ (π · (𝑘 · 2)) = 𝑋))
37	25, 36	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (π · (𝑘 · 2)) = 𝑋)
38	20, 22, 37	3eqtrrd 2800	. . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) = (((2 · π) · 𝑘) / (2 · π)))
42	12, 14	mulcld 11253	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (2 · π) ∈ ℂ)
43		2ne0 12371	. . . . . . . . . . . . . . . . . 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 11890	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (2 · π) ≠ 0)
47	15, 42, 46	divcan3d 12020	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (((2 · π) · 𝑘) / (2 · π)) = 𝑘)
48	47	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (((2 · π) · 𝑘) / (2 · π)) = 𝑘)
49	41, 48	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) = 𝑘)
50		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → 𝑘 ∈ ℤ)
51	49, 50	eqeltrd 2860	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 · 2) = (𝑋 / π)) → (𝑋 / 𝑇) ∈ ℤ)
52	51	ex 418	. . . . . . . . . . 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 3161	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑋 / π) → (𝑋 / 𝑇) ∈ ℤ))
55	11, 54	mpd 16	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 / 𝑇) ∈ ℤ)
56		2re 12339	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
57	56, 32	remulcli 11249	. . . . . . . . . . 11 ⊢ (2 · π) ∈ ℝ
58	39, 57	eqeltri 2856	. . . . . . . . . 10 ⊢ 𝑇 ∈ ℝ
59		2pos 12369	. . . . . . . . . . . 12 ⊢ 0 < 2
60	56, 32, 59, 33	mulgt0ii 11367	. . . . . . . . . . 11 ⊢ 0 < (2 · π)
61	60, 39	breqtrri 5132	. . . . . . . . . 10 ⊢ 0 < 𝑇
62	58, 61	elrpii 13045	. . . . . . . . 9 ⊢ 𝑇 ∈ ℝ⁺
63		mod0 13937	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → ((𝑋 mod 𝑇) = 0 ↔ (𝑋 / 𝑇) ∈ ℤ))
64	4, 62, 63	mp2an 705	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 ↔ (𝑋 / 𝑇) ∈ ℤ)
65	55, 64	sylibr 237	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → (𝑋 mod 𝑇) = 0)
66	65	orcd 887	. . . . . 6 ⊢ (((𝑋 mod π) = 0 ∧ 2 ∥ (𝑋 / π)) → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
67		odd2np1 16431	. . . . . . . . . 10 ⊢ ((𝑋 / π) ∈ ℤ → (¬ 2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π)))
68	7, 67	sylbi 220	. . . . . . . . 9 ⊢ ((𝑋 mod π) = 0 → (¬ 2 ∥ (𝑋 / π) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π)))
69	68	biimpa 482	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π))
70	12, 15	mulcld 11253	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (2 · 𝑘) ∈ ℂ)
71	70	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (2 · 𝑘) ∈ ℂ)
72		1cnd 11226	. . . . . . . . . . . . . . . 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 11258	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) + 1) · π) = (((2 · 𝑘) · π) + (1 · π)))
75	12, 15	mulcomd 11254	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (2 · 𝑘) = (𝑘 · 2))
76	75	oveq1d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) · π) = ((𝑘 · 2) · π))
77	15, 12, 14	mulassd 11256	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → ((𝑘 · 2) · π) = (𝑘 · (2 · π)))
78	39	eqcomi 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · π) = 𝑇
79	78	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℤ → (2 · π) = 𝑇)
80	79	oveq2d 7429	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → (𝑘 · (2 · π)) = (𝑘 · 𝑇))
81	76, 77, 80	3eqtrd 2799	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) · π) = (𝑘 · 𝑇))
82	13	mullidi 11238	. . . . . . . . . . . . . . . . . 18 ⊢ (1 · π) = π
83	82	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (1 · π) = π)
84	81, 83	oveq12d 7431	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (((2 · 𝑘) · π) + (1 · π)) = ((𝑘 · 𝑇) + π))
85	84	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) · π) + (1 · π)) = ((𝑘 · 𝑇) + π))
86	39, 42	eqeltrid 2864	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → 𝑇 ∈ ℂ)
87	15, 86	mulcld 11253	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (𝑘 · 𝑇) ∈ ℂ)
88	87, 14	addcomd 11436	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → ((𝑘 · 𝑇) + π) = (π + (𝑘 · 𝑇)))
89	88	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((𝑘 · 𝑇) + π) = (π + (𝑘 · 𝑇)))
90	74, 85, 89	3eqtrrd 2800	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π + (𝑘 · 𝑇)) = (((2 · 𝑘) + 1) · π))
91		peano2cn 11406	. . . . . . . . . . . . . . . . 17 ⊢ ((2 · 𝑘) ∈ ℂ → ((2 · 𝑘) + 1) ∈ ℂ)
92	70, 91	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → ((2 · 𝑘) + 1) ∈ ℂ)
93	92, 14	mulcomd 11254	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (((2 · 𝑘) + 1) · π) = (π · ((2 · 𝑘) + 1)))
94	93	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (((2 · 𝑘) + 1) · π) = (π · ((2 · 𝑘) + 1)))
95		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (((2 · 𝑘) + 1) = (𝑋 / π) → ((2 · 𝑘) + 1) = (𝑋 / π))
96	95	eqcomd 2766	. . . . . . . . . . . . . . . 16 ⊢ (((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 / π) = ((2 · 𝑘) + 1))
97	96	adantl 487	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 12037	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → ((𝑋 / π) = ((2 · 𝑘) + 1) ↔ (π · ((2 · 𝑘) + 1)) = 𝑋))
102	97, 101	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π · ((2 · 𝑘) + 1)) = 𝑋)
103	90, 94, 102	3eqtrrd 2800	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 𝑋 = (π + (𝑘 · 𝑇)))
104	103	oveq1d 7428	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (𝑋 mod 𝑇) = ((π + (𝑘 · 𝑇)) mod 𝑇))
105		modcyc 13967	. . . . . . . . . . . . . 14 ⊢ ((π ∈ ℝ ∧ 𝑇 ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → ((π + (𝑘 · 𝑇)) mod 𝑇) = (π mod 𝑇))
106	32, 62, 105	mp3an12 1480	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤ → ((π + (𝑘 · 𝑇)) mod 𝑇) = (π mod 𝑇))
107	106	adantr 486	. . . . . . . . . . . 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 11234	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
111	110, 32, 33	ltleii 11357	. . . . . . . . . . . . . 14 ⊢ 0 ≤ π
112	111	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → 0 ≤ π)
113		2timesgt 46121	. . . . . . . . . . . . . . . 16 ⊢ (π ∈ ℝ⁺ → π < (2 · π))
114	5, 113	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ π < (2 · π)
115	114, 39	breqtrri 5132	. . . . . . . . . . . . . 14 ⊢ π < 𝑇
116	115	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → π < 𝑇)
117		modid 13957	. . . . . . . . . . . . 13 ⊢ (((π ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) ∧ (0 ≤ π ∧ π < 𝑇)) → (π mod 𝑇) = π)
118	108, 109, 112, 116, 117	syl22anc 852	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (π mod 𝑇) = π)
119	104, 107, 118	3eqtrd 2799	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = (𝑋 / π)) → (𝑋 mod 𝑇) = π)
120	119	ex 418	. . . . . . . . . 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 3161	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → (∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (𝑋 / π) → (𝑋 mod 𝑇) = π))
123	69, 122	mpd 16	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → (𝑋 mod 𝑇) = π)
124	123	olcd 888	. . . . . 6 ⊢ (((𝑋 mod π) = 0 ∧ ¬ 2 ∥ (𝑋 / π)) → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
125	66, 124	pm2.61dan 825	. . . . 5 ⊢ ((𝑋 mod π) = 0 → ((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π))
126		0xr 11280	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
127	32	rexri 11291	. . . . . . . 8 ⊢ π ∈ ℝ^*
128		iocgtlb 46332	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (0(,]π)) → 0 < (𝑋 mod 𝑇))
129	126, 127, 128	mp3an12 1480	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → 0 < (𝑋 mod 𝑇))
130	129	gt0ne0d 11802	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → (𝑋 mod 𝑇) ≠ 0)
131	130	neneqd 2960	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → ¬ (𝑋 mod 𝑇) = 0)
132		pm2.53 865	. . . . . 6 ⊢ (((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π) → (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) = π))
133	132	imp 412	. . . . 5 ⊢ ((((𝑋 mod 𝑇) = 0 ∨ (𝑋 mod 𝑇) = π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) = π)
134	125, 131, 133	syl2anr 609	. . . 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 13934	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) ∈ ℝ)
138	4, 62, 137	mp2an 705	. . . . . . . . . . . . . 14 ⊢ (𝑋 mod 𝑇) ∈ ℝ
139	138	rexri 11291	. . . . . . . . . . . . 13 ⊢ (𝑋 mod 𝑇) ∈ ℝ^*
140	139	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ∈ ℝ^*)
141		id 23	. . . . . . . . . . . . 13 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) = π)
142	33, 141	breqtrrid 5143	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → 0 < (𝑋 mod 𝑇))
143	32	eqlei2 11345	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ≤ π)
144	135, 136, 140, 142, 143	eliocd 46337	. . . . . . . . . . 11 ⊢ ((𝑋 mod 𝑇) = π → (𝑋 mod 𝑇) ∈ (0(,]π))
145	144	iftrued 4490	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = π → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
146	145	adantl 487	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
147		oveq1 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → (𝑥 mod 𝑇) = (𝑋 mod 𝑇))
148	147	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → ((𝑥 mod 𝑇) < π ↔ (𝑋 mod 𝑇) < π))
149	148	ifbid 4506	. . . . . . . . . . . . 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 11227	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
152		negex 11479	. . . . . . . . . . . . . 14 ⊢ -1 ∈ V
153	151, 152	ifex 4533	. . . . . . . . . . . . 13 ⊢ if((𝑋 mod 𝑇) < π, 1, -1) ∈ V
154	149, 150, 153	fvmpt 6986	. . . . . . . . . . . 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 23	. . . . . . . . . . . . . 14 ⊢ ((𝑋 mod 𝑇) < π → (𝑋 mod 𝑇) < π)
158	156, 157	ltned 11370	. . . . . . . . . . . . 13 ⊢ ((𝑋 mod 𝑇) < π → (𝑋 mod 𝑇) ≠ π)
159	158	necon2bi 2985	. . . . . . . . . . . 12 ⊢ ((𝑋 mod 𝑇) = π → ¬ (𝑋 mod 𝑇) < π)
160	159	iffalsed 4493	. . . . . . . . . . 11 ⊢ ((𝑋 mod 𝑇) = π → if((𝑋 mod 𝑇) < π, 1, -1) = -1)
161	155, 160	eqtrid 2807	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = π → (𝐹‘𝑋) = -1)
162	161	adantl 487	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (𝐹‘𝑋) = -1)
163	146, 162	oveq12d 7431	. . . . . . . 8 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (1 + -1))
164		1pneg1e0 12382	. . . . . . . 8 ⊢ (1 + -1) = 0
165	163, 164	eqtrdi 2811	. . . . . . 7 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = π) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = 0)
166	165	oveq1d 7428	. . . . . 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 727	. . . . 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 12340	. . . . . . 7 ⊢ 2 ∈ ℂ
169	168, 43	div0i 11973	. . . . . 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 4488	. . . . . . 7 ⊢ ((𝑋 mod π) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = 0)
173	171, 172	eqtr2id 2808	. . . . . 6 ⊢ ((𝑋 mod π) = 0 → 0 = 𝑌)
174	173	ad2antlr 740	. . . . 5 ⊢ ((((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) ∧ (𝑋 mod 𝑇) = π) → 0 = 𝑌)
175	167, 170, 174	3eqtrrd 2800	. . . 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 700	. . 3 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ (𝑋 mod π) = 0) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
177		iftrue 4488	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = 1)
178	177	adantr 486	. . . . . 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 46333	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (0(,]π)) → (𝑋 mod 𝑇) ≤ π)
182	126, 127, 181	mp3an12 1480	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → (𝑋 mod 𝑇) ≤ π)
183	182	adantr 486	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) ≤ π)
184		ax-1cn 11182	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
185	184, 13	mulcomi 11241	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 · π) = (π · 1)
186	82, 185	eqtr3i 2785	. . . . . . . . . . . . . . . . . 18 ⊢ π = (π · 1)
187	186	oveq1i 7423	. . . . . . . . . . . . . . . . 17 ⊢ (π + (π · (2 · (⌊‘(𝑋 / 𝑇))))) = ((π · 1) + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
188	168, 13	mulcomi 11241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2 · π) = (π · 2)
189	39, 188	eqtri 2783	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑇 = (π · 2)
190	189	oveq1i 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) = ((π · 2) · (⌊‘(𝑋 / 𝑇)))
191	110, 61	gtneii 11346	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑇 ≠ 0
192	4, 58, 191	redivcli 12006	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 / 𝑇) ∈ ℝ
193		flcl 13856	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 / 𝑇) ∈ ℝ → (⌊‘(𝑋 / 𝑇)) ∈ ℤ)
194	192, 193	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⌊‘(𝑋 / 𝑇)) ∈ ℤ
195		zcn 12620	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⌊‘(𝑋 / 𝑇)) ∈ ℤ → (⌊‘(𝑋 / 𝑇)) ∈ ℂ)
196	194, 195	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⌊‘(𝑋 / 𝑇)) ∈ ℂ
197	13, 168, 196	mulassi 11244	. . . . . . . . . . . . . . . . . . 19 ⊢ ((π · 2) · (⌊‘(𝑋 / 𝑇))) = (π · (2 · (⌊‘(𝑋 / 𝑇))))
198	190, 197	eqtri 2783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) = (π · (2 · (⌊‘(𝑋 / 𝑇))))
199	198	oveq2i 7424	. . . . . . . . . . . . . . . . 17 ⊢ (π + (𝑇 · (⌊‘(𝑋 / 𝑇)))) = (π + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
200	168, 196	mulcli 11240	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ
201	13, 184, 200	adddii 11245	. . . . . . . . . . . . . . . . 17 ⊢ (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) = ((π · 1) + (π · (2 · (⌊‘(𝑋 / 𝑇)))))
202	187, 199, 201	3eqtr4ri 2794	. . . . . . . . . . . . . . . 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 23	. . . . . . . . . . . . . . . . 17 ⊢ (π = (𝑋 mod 𝑇) → π = (𝑋 mod 𝑇))
205		modval 13932	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))))
206	4, 62, 205	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 mod 𝑇) = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇))))
207	204, 206	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ (π = (𝑋 mod 𝑇) → π = (𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))))
208	207	oveq1d 7428	. . . . . . . . . . . . . . 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 11247	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 ∈ ℂ
211	210, 196	mulcli 11240	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ
212	211	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (π = (𝑋 mod 𝑇) → (𝑇 · (⌊‘(𝑋 / 𝑇))) ∈ ℂ)
213	209, 212	npcand 11597	. . . . . . . . . . . . . . 15 ⊢ (π = (𝑋 mod 𝑇) → ((𝑋 − (𝑇 · (⌊‘(𝑋 / 𝑇)))) + (𝑇 · (⌊‘(𝑋 / 𝑇)))) = 𝑋)
214	203, 208, 213	3eqtrrd 2800	. . . . . . . . . . . . . 14 ⊢ (π = (𝑋 mod 𝑇) → 𝑋 = (π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))))
215	214	oveq1d 7428	. . . . . . . . . . . . 13 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) = ((π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) / π))
216	184, 200	addcli 11239	. . . . . . . . . . . . . 14 ⊢ (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℂ
217	216, 13, 34	divcan3i 11985	. . . . . . . . . . . . 13 ⊢ ((π · (1 + (2 · (⌊‘(𝑋 / 𝑇))))) / π) = (1 + (2 · (⌊‘(𝑋 / 𝑇))))
218	215, 217	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) = (1 + (2 · (⌊‘(𝑋 / 𝑇)))))
219		1z 12648	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
220		zmulcl 12667	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℤ ∧ (⌊‘(𝑋 / 𝑇)) ∈ ℤ) → (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ)
221	2, 194, 220	mp2an 705	. . . . . . . . . . . . . 14 ⊢ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ
222		zaddcl 12658	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ (2 · (⌊‘(𝑋 / 𝑇))) ∈ ℤ) → (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ)
223	219, 221, 222	mp2an 705	. . . . . . . . . . . . 13 ⊢ (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ
224	223	a1i 11	. . . . . . . . . . . 12 ⊢ (π = (𝑋 mod 𝑇) → (1 + (2 · (⌊‘(𝑋 / 𝑇)))) ∈ ℤ)
225	218, 224	eqeltrd 2860	. . . . . . . . . . 11 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 / π) ∈ ℤ)
226	225, 7	sylibr 237	. . . . . . . . . 10 ⊢ (π = (𝑋 mod 𝑇) → (𝑋 mod π) = 0)
227	226	necon3bi 2981	. . . . . . . . 9 ⊢ (¬ (𝑋 mod π) = 0 → π ≠ (𝑋 mod 𝑇))
228	227	adantl 487	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → π ≠ (𝑋 mod 𝑇))
229	179, 180, 183, 228	leneltd 11388	. . . . . . 7 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝑋 mod 𝑇) < π)
230		iftrue 4488	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) < π → if((𝑋 mod 𝑇) < π, 1, -1) = 1)
231	155, 230	eqtrid 2807	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) < π → (𝐹‘𝑋) = 1)
232	229, 231	syl 18	. . . . . 6 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝐹‘𝑋) = 1)
233	178, 232	oveq12d 7431	. . . . 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 7428	. . . 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 12388	. . . . . . 7 ⊢ (1 + 1) = 2
236	235	oveq1i 7423	. . . . . 6 ⊢ ((1 + 1) / 2) = (2 / 2)
237		2div2e1 12405	. . . . . 6 ⊢ (2 / 2) = 1
238	236, 237	eqtr2i 2784	. . . . 5 ⊢ 1 = ((1 + 1) / 2)
239	232, 238	eqtr2di 2812	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → ((1 + 1) / 2) = (𝐹‘𝑋))
240		iffalse 4491	. . . . . 6 ⊢ (¬ (𝑋 mod π) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = (𝐹‘𝑋))
241	171, 240	eqtr2id 2808	. . . . 5 ⊢ (¬ (𝑋 mod π) = 0 → (𝐹‘𝑋) = 𝑌)
242	241	adantl 487	. . . 4 ⊢ (((𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod π) = 0) → (𝐹‘𝑋) = 𝑌)
243	234, 239, 242	3eqtrrd 2800	. . 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 825	. 2 ⊢ ((𝑋 mod 𝑇) ∈ (0(,]π) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
245	130	necon2bi 2985	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → ¬ (𝑋 mod 𝑇) ∈ (0(,]π))
246	245	iffalsed 4493	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) = -1)
247		id 23	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) = 0)
248	247, 33	eqbrtrdi 5144	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) < π)
249	248	iftrued 4490	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod 𝑇) < π, 1, -1) = 1)
250	155, 249	eqtrid 2807	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → (𝐹‘𝑋) = 1)
251	246, 250	oveq12d 7431	. . . . . 6 ⊢ ((𝑋 mod 𝑇) = 0 → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (-1 + 1))
252	251	oveq1d 7428	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = ((-1 + 1) / 2))
253		neg1cn 12227	. . . . . . . . 9 ⊢ -1 ∈ ℂ
254	184, 253, 164	addcomli 11426	. . . . . . . 8 ⊢ (-1 + 1) = 0
255	254	oveq1i 7423	. . . . . . 7 ⊢ ((-1 + 1) / 2) = (0 / 2)
256	255, 169	eqtri 2783	. . . . . 6 ⊢ ((-1 + 1) / 2) = 0
257	256	a1i 11	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → ((-1 + 1) / 2) = 0)
258	39	oveq2i 7424	. . . . . . . . . . . . 13 ⊢ (𝑋 / 𝑇) = (𝑋 / (2 · π))
259		2cnne0 12477	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
260	13, 34	pm3.2i 476	. . . . . . . . . . . . . 14 ⊢ (π ∈ ℂ ∧ π ≠ 0)
261		divdiv1 11950	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (π ∈ ℂ ∧ π ≠ 0)) → ((𝑋 / 2) / π) = (𝑋 / (2 · π)))
262	26, 259, 260, 261	mp3an 1490	. . . . . . . . . . . . 13 ⊢ ((𝑋 / 2) / π) = (𝑋 / (2 · π))
263	26, 168, 13, 43, 34	divdiv32i 11994	. . . . . . . . . . . . 13 ⊢ ((𝑋 / 2) / π) = ((𝑋 / π) / 2)
264	258, 262, 263	3eqtr2i 2789	. . . . . . . . . . . 12 ⊢ (𝑋 / 𝑇) = ((𝑋 / π) / 2)
265	264	oveq2i 7424	. . . . . . . . . . 11 ⊢ (2 · (𝑋 / 𝑇)) = (2 · ((𝑋 / π) / 2))
266	26, 13, 34	divcli 11981	. . . . . . . . . . . 12 ⊢ (𝑋 / π) ∈ ℂ
267	266, 168, 43	divcan2i 11982	. . . . . . . . . . 11 ⊢ (2 · ((𝑋 / π) / 2)) = (𝑋 / π)
268	265, 267	eqtr2i 2784	. . . . . . . . . 10 ⊢ (𝑋 / π) = (2 · (𝑋 / 𝑇))
269	2	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 / 𝑇) ∈ ℤ → 2 ∈ ℤ)
270		id 23	. . . . . . . . . . 11 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (𝑋 / 𝑇) ∈ ℤ)
271	269, 270	zmulcld 12731	. . . . . . . . . 10 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (2 · (𝑋 / 𝑇)) ∈ ℤ)
272	268, 271	eqeltrid 2864	. . . . . . . . 9 ⊢ ((𝑋 / 𝑇) ∈ ℤ → (𝑋 / π) ∈ ℤ)
273	64, 272	sylbi 220	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 / π) ∈ ℤ)
274	273, 7	sylibr 237	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod π) = 0)
275	274	iftrued 4490	. . . . . 6 ⊢ ((𝑋 mod 𝑇) = 0 → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = 0)
276	171, 275	eqtr2id 2808	. . . . 5 ⊢ ((𝑋 mod 𝑇) = 0 → 0 = 𝑌)
277	252, 257, 276	3eqtrrd 2800	. . . 4 ⊢ ((𝑋 mod 𝑇) = 0 → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
278	277	adantl 487	. . 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 11291	. . . . . 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 1004	. . . . . . . 8 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) ↔ ¬ ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
284	283	biimpi 219	. . . . . . 7 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → ¬ ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
285		olc 882	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) = 0 → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
286	285	adantl 487	. . . . . . . 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 11235	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 ∈ ℝ)
291	138	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ∈ ℝ)
292		modge0 13940	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → 0 ≤ (𝑋 mod 𝑇))
293	4, 62, 292	mp2an 705	. . . . . . . . . . . . 13 ⊢ 0 ≤ (𝑋 mod 𝑇)
294	293	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 ≤ (𝑋 mod 𝑇))
295		neqne 2963	. . . . . . . . . . . 12 ⊢ (¬ (𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ≠ 0)
296	290, 291, 294, 295	leneltd 11388	. . . . . . . . . . 11 ⊢ (¬ (𝑋 mod 𝑇) = 0 → 0 < (𝑋 mod 𝑇))
297	296	adantl 487	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → 0 < (𝑋 mod 𝑇))
298		simpl 488	. . . . . . . . . 10 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
299	287, 288, 289, 297, 298	eliocd 46337	. . . . . . . . 9 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ∈ (0(,]π))
300	299	orcd 887	. . . . . . . 8 ⊢ (((𝑋 mod 𝑇) ≤ π ∧ ¬ (𝑋 mod 𝑇) = 0) → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
301	286, 300	pm2.61dan 825	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ≤ π → ((𝑋 mod 𝑇) ∈ (0(,]π) ∨ (𝑋 mod 𝑇) = 0))
302	284, 301	nsyl 141	. . . . . 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 11381	. . . . . 6 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → (π < (𝑋 mod 𝑇) ↔ ¬ (𝑋 mod 𝑇) ≤ π))
305	302, 304	mpbird 260	. . . . 5 ⊢ ((¬ (𝑋 mod 𝑇) ∈ (0(,]π) ∧ ¬ (𝑋 mod 𝑇) = 0) → π < (𝑋 mod 𝑇))
306		modlt 13941	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ⁺) → (𝑋 mod 𝑇) < 𝑇)
307	4, 62, 306	mp2an 705	. . . . . 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 46328	. . . 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 46325	. . . . . . . . . 10 ⊢ ((π ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ (𝑋 mod 𝑇) ∈ (π(,)𝑇)) → π < (𝑋 mod 𝑇))
314	127, 280, 313	mp3an12 1480	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → π < (𝑋 mod 𝑇))
315	310, 311, 312, 314	gtnelioc 46321	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) ∈ (0(,]π))
316	315	iffalsed 4493	. . . . . . 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 11383	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) < π)
319	318	iffalsed 4493	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → if((𝑋 mod 𝑇) < π, 1, -1) = -1)
320	155, 319	eqtrid 2807	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (𝐹‘𝑋) = -1)
321	316, 320	oveq12d 7431	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) = (-1 + -1))
322	321	oveq1d 7428	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2) = ((-1 + -1) / 2))
323		df-2 12327	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
324	323	negeqi 11474	. . . . . . . . 9 ⊢ -2 = -(1 + 1)
325	184, 184	negdii 11566	. . . . . . . . 9 ⊢ -(1 + 1) = (-1 + -1)
326	324, 325	eqtr2i 2784	. . . . . . . 8 ⊢ (-1 + -1) = -2
327	326	oveq1i 7423	. . . . . . 7 ⊢ ((-1 + -1) / 2) = (-2 / 2)
328		divneg 11930	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(2 / 2) = (-2 / 2))
329	168, 168, 43, 328	mp3an 1490	. . . . . . 7 ⊢ -(2 / 2) = (-2 / 2)
330	237	negeqi 11474	. . . . . . 7 ⊢ -(2 / 2) = -1
331	327, 329, 330	3eqtr2i 2789	. . . . . 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 11381	. . . . . . . . 9 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → (π < (𝑋 mod 𝑇) ↔ ¬ (𝑋 mod 𝑇) ≤ π))
335	314, 334	mpbid 235	. . . . . . . 8 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod 𝑇) ≤ π)
336	247, 111	eqbrtrdi 5144	. . . . . . . . . 10 ⊢ ((𝑋 mod 𝑇) = 0 → (𝑋 mod 𝑇) ≤ π)
337	336	adantl 487	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
338	125	orcanai 1018	. . . . . . . . . 10 ⊢ (((𝑋 mod π) = 0 ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) = π)
339	338, 143	syl 18	. . . . . . . . 9 ⊢ (((𝑋 mod π) = 0 ∧ ¬ (𝑋 mod 𝑇) = 0) → (𝑋 mod 𝑇) ≤ π)
340	337, 339	pm2.61dan 825	. . . . . . . 8 ⊢ ((𝑋 mod π) = 0 → (𝑋 mod 𝑇) ≤ π)
341	335, 340	nsyl 141	. . . . . . 7 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → ¬ (𝑋 mod π) = 0)
342	341	iffalsed 4493	. . . . . 6 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → if((𝑋 mod π) = 0, 0, (𝐹‘𝑋)) = (𝐹‘𝑋))
343	333, 342, 320	3eqtrrd 2800	. . . . 5 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → -1 = 𝑌)
344	322, 332, 343	3eqtrrd 2800	. . . 4 ⊢ ((𝑋 mod 𝑇) ∈ (π(,)𝑇) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
345	309, 344	syl 18	. . 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 825	. 2 ⊢ (¬ (𝑋 mod 𝑇) ∈ (0(,]π) → 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2))
347	244, 346	pm2.61i 184	1 ⊢ 𝑌 = ((if((𝑋 mod 𝑇) ∈ (0(,]π), 1, -1) + (𝐹‘𝑋)) / 2)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∃wrex 3086 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 − cmin 11465 -cneg 11466 / cdiv 11895 2c2 12319 ℤcz 12615 ℝ⁺crp 13042 (,)cioo 13398 (,]cioc 13399 ⌊cfl 13851 mod cmo 13930 πcpi 16152 ∥ cdvds 16342

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16153 df-sin 16155 df-cos 16156 df-pi 16158 df-dvds 16343 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-limc 26093 df-dv 26094

This theorem is used by: fouriersw 47059

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