fourierdlem11 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem11 43659

Description: If there is a partition, than the lower bound is strictly less than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem11.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem11.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem11.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))

**Assertion**
Ref	Expression
fourierdlem11	⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))

Distinct variable groups: 𝐴,𝑚,𝑝 𝐵,𝑚,𝑝 𝑖,𝑀,𝑚,𝑝 𝑄,𝑖,𝑝 𝜑,𝑖 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝐴(𝑖) 𝐵(𝑖) 𝑃(𝑖,𝑚,𝑝) 𝑄(𝑚)

**Proof of Theorem fourierdlem11**
Step	Hyp	Ref	Expression
1		fourierdlem11.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
2		fourierdlem11.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem11.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
4	3	fourierdlem2 43650	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
6	1, 5	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simprd 496	. . . . 5 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
8	7	simpld 495	. . . 4 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
9	8	simpld 495	. . 3 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
10	6	simpld 495	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
11		elmapi 8637	. . . . 5 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
13		0zd 12331	. . . . 5 ⊢ (𝜑 → 0 ∈ ℤ)
14	2	nnzd 12425	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
15		0red 10978	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
16	15	leidd 11541	. . . . 5 ⊢ (𝜑 → 0 ≤ 0)
17	2	nnred 11988	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
18	2	nngt0d 12022	. . . . . 6 ⊢ (𝜑 → 0 < 𝑀)
19	15, 17, 18	ltled 11123	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑀)
20	13, 14, 13, 16, 19	elfzd 13247	. . . 4 ⊢ (𝜑 → 0 ∈ (0...𝑀))
21	12, 20	ffvelrnd 6962	. . 3 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
22	9, 21	eqeltrrd 2840	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
23	8	simprd 496	. . 3 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
24	17	leidd 11541	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
25	13, 14, 14, 19, 24	elfzd 13247	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
26	12, 25	ffvelrnd 6962	. . 3 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
27	23, 26	eqeltrrd 2840	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
28		1zzd 12351	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
29		0le1 11498	. . . . . 6 ⊢ 0 ≤ 1
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≤ 1)
31	2	nnge1d 12021	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
32	13, 14, 28, 30, 31	elfzd 13247	. . . 4 ⊢ (𝜑 → 1 ∈ (0...𝑀))
33	12, 32	ffvelrnd 6962	. . 3 ⊢ (𝜑 → (𝑄‘1) ∈ ℝ)
34		elfzo 13389	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ∈ (0..^𝑀) ↔ (0 ≤ 0 ∧ 0 < 𝑀)))
35	13, 13, 14, 34	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (0 ∈ (0..^𝑀) ↔ (0 ≤ 0 ∧ 0 < 𝑀)))
36	16, 18, 35	mpbir2and 710	. . . . 5 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
37		0re 10977	. . . . . 6 ⊢ 0 ∈ ℝ
38		eleq1 2826	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑖 ∈ (0..^𝑀) ↔ 0 ∈ (0..^𝑀)))
39	38	anbi2d 629	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 0 ∈ (0..^𝑀))))
40		fveq2 6774	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
41		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
42	41	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘(𝑖 + 1)) = (𝑄‘(0 + 1)))
43	40, 42	breq12d 5087	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘0) < (𝑄‘(0 + 1))))
44	39, 43	imbi12d 345	. . . . . . 7 ⊢ (𝑖 = 0 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1)))))
45	7	simprd 496	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
46	45	r19.21bi 3134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
47	44, 46	vtoclg 3505	. . . . . 6 ⊢ (0 ∈ ℝ → ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1))))
48	37, 47	ax-mp 5	. . . . 5 ⊢ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1)))
49	36, 48	mpdan 684	. . . 4 ⊢ (𝜑 → (𝑄‘0) < (𝑄‘(0 + 1)))
50		0p1e1 12095	. . . . . 6 ⊢ (0 + 1) = 1
51	50	a1i 11	. . . . 5 ⊢ (𝜑 → (0 + 1) = 1)
52	51	fveq2d 6778	. . . 4 ⊢ (𝜑 → (𝑄‘(0 + 1)) = (𝑄‘1))
53	49, 9, 52	3brtr3d 5105	. . 3 ⊢ (𝜑 → 𝐴 < (𝑄‘1))
54		nnuz 12621	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
55	2, 54	eleqtrdi 2849	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
56	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
57		0zd 12331	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℤ)
58		elfzel2 13254	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
59		elfzelz 13256	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
60		0red 10978	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
61	59	zred 12426	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
62		1red 10976	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
63		0lt1 11497	. . . . . . . . . . 11 ⊢ 0 < 1
64	63	a1i 11	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 0 < 1)
65		elfzle1 13259	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
66	60, 62, 61, 64, 65	ltletrd 11135	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 0 < 𝑖)
67	60, 61, 66	ltled 11123	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
68		elfzle2 13260	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
69	57, 58, 59, 67, 68	elfzd 13247	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (0...𝑀))
70	69	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (0...𝑀))
71	56, 70	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
72	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
73		0zd 12331	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ∈ ℤ)
74	14	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℤ)
75		elfzelz 13256	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℤ)
76	75	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℤ)
77		0red 10978	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 ∈ ℝ)
78	75	zred 12426	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℝ)
79		1red 10976	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 ∈ ℝ)
80	63	a1i 11	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 < 1)
81		elfzle1 13259	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 ≤ 𝑖)
82	77, 79, 78, 80, 81	ltletrd 11135	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 < 𝑖)
83	77, 78, 82	ltled 11123	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 ≤ 𝑖)
84	83	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ≤ 𝑖)
85	78	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℝ)
86	17	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℝ)
87		peano2rem 11288	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
88	86, 87	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
89		elfzle2 13260	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ≤ (𝑀 − 1))
90	89	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ≤ (𝑀 − 1))
91	86	ltm1d 11907	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
92	85, 88, 86, 90, 91	lelttrd 11133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 < 𝑀)
93	85, 86, 92	ltled 11123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ≤ 𝑀)
94	73, 74, 76, 84, 93	elfzd 13247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (0...𝑀))
95	72, 94	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) ∈ ℝ)
96	76	peano2zd 12429	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ ℤ)
97		0red 10978	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ∈ ℝ)
98		peano2re 11148	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
99	85, 98	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ ℝ)
100		1red 10976	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 1 ∈ ℝ)
101	63	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 < 1)
102	78, 98	syl 17	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → (𝑖 + 1) ∈ ℝ)
103	78	ltp1d 11905	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 < (𝑖 + 1))
104	79, 78, 102, 81, 103	lelttrd 11133	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 < (𝑖 + 1))
105	104	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 1 < (𝑖 + 1))
106	97, 100, 99, 101, 105	lttrd 11136	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 < (𝑖 + 1))
107	97, 99, 106	ltled 11123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ≤ (𝑖 + 1))
108	85, 88, 100, 90	leadd1dd 11589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ≤ ((𝑀 − 1) + 1))
109	2	nncnd 11989	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℂ)
110		1cnd 10970	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
111	109, 110	npcand 11336	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
112	111	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
113	108, 112	breqtrd 5100	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ≤ 𝑀)
114	73, 74, 96, 107, 113	elfzd 13247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ (0...𝑀))
115	72, 114	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
116		elfzo 13389	. . . . . . . . 9 ⊢ ((𝑖 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 ∈ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 < 𝑀)))
117	76, 73, 74, 116	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 < 𝑀)))
118	84, 92, 117	mpbir2and 710	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (0..^𝑀))
119	118, 46	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
120	95, 115, 119	ltled 11123	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1)))
121	55, 71, 120	monoord 13753	. . . 4 ⊢ (𝜑 → (𝑄‘1) ≤ (𝑄‘𝑀))
122	121, 23	breqtrd 5100	. . 3 ⊢ (𝜑 → (𝑄‘1) ≤ 𝐵)
123	22, 33, 27, 53, 122	ltletrd 11135	. 2 ⊢ (𝜑 → 𝐴 < 𝐵)
124	22, 27, 123	3jca 1127	1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 class class class wbr 5074 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ..^cfzo 13382

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383

This theorem is referenced by: fourierdlem37 43685 fourierdlem54 43701 fourierdlem63 43710 fourierdlem64 43711 fourierdlem65 43712 fourierdlem69 43716 fourierdlem79 43726 fourierdlem89 43736 fourierdlem90 43737 fourierdlem91 43738 fourierdlem107 43754 fourierdlem109 43756

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