fourierdlem11 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem11 43549

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 43540	. . . . . . . 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 495	. . . . 5 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
8	7	simpld 494	. . . 4 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
9	8	simpld 494	. . 3 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
10	6	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
11		elmapi 8595	. . . . 5 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
13		0zd 12261	. . . . 5 ⊢ (𝜑 → 0 ∈ ℤ)
14	2	nnzd 12354	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
15		0red 10909	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
16	15	leidd 11471	. . . . 5 ⊢ (𝜑 → 0 ≤ 0)
17	2	nnred 11918	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
18	2	nngt0d 11952	. . . . . 6 ⊢ (𝜑 → 0 < 𝑀)
19	15, 17, 18	ltled 11053	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑀)
20	13, 14, 13, 16, 19	elfzd 13176	. . . 4 ⊢ (𝜑 → 0 ∈ (0...𝑀))
21	12, 20	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
22	9, 21	eqeltrrd 2840	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
23	8	simprd 495	. . 3 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
24	17	leidd 11471	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
25	13, 14, 14, 19, 24	elfzd 13176	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
26	12, 25	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
27	23, 26	eqeltrrd 2840	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
28		1zzd 12281	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
29		0le1 11428	. . . . . 6 ⊢ 0 ≤ 1
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≤ 1)
31	2	nnge1d 11951	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
32	13, 14, 28, 30, 31	elfzd 13176	. . . 4 ⊢ (𝜑 → 1 ∈ (0...𝑀))
33	12, 32	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝑄‘1) ∈ ℝ)
34		elfzo 13318	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ∈ (0..^𝑀) ↔ (0 ≤ 0 ∧ 0 < 𝑀)))
35	13, 13, 14, 34	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (0 ∈ (0..^𝑀) ↔ (0 ≤ 0 ∧ 0 < 𝑀)))
36	16, 18, 35	mpbir2and 709	. . . . 5 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
37		0re 10908	. . . . . 6 ⊢ 0 ∈ ℝ
38		eleq1 2826	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑖 ∈ (0..^𝑀) ↔ 0 ∈ (0..^𝑀)))
39	38	anbi2d 628	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 0 ∈ (0..^𝑀))))
40		fveq2 6756	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
41		oveq1 7262	. . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
42	41	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘(𝑖 + 1)) = (𝑄‘(0 + 1)))
43	40, 42	breq12d 5083	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘0) < (𝑄‘(0 + 1))))
44	39, 43	imbi12d 344	. . . . . . 7 ⊢ (𝑖 = 0 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1)))))
45	7	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
46	45	r19.21bi 3132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
47	44, 46	vtoclg 3495	. . . . . 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 683	. . . 4 ⊢ (𝜑 → (𝑄‘0) < (𝑄‘(0 + 1)))
50		0p1e1 12025	. . . . . 6 ⊢ (0 + 1) = 1
51	50	a1i 11	. . . . 5 ⊢ (𝜑 → (0 + 1) = 1)
52	51	fveq2d 6760	. . . 4 ⊢ (𝜑 → (𝑄‘(0 + 1)) = (𝑄‘1))
53	49, 9, 52	3brtr3d 5101	. . 3 ⊢ (𝜑 → 𝐴 < (𝑄‘1))
54		nnuz 12550	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
55	2, 54	eleqtrdi 2849	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
56	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
57		0zd 12261	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℤ)
58		elfzel2 13183	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
59		elfzelz 13185	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
60		0red 10909	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
61	59	zred 12355	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
62		1red 10907	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
63		0lt1 11427	. . . . . . . . . . 11 ⊢ 0 < 1
64	63	a1i 11	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 0 < 1)
65		elfzle1 13188	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
66	60, 62, 61, 64, 65	ltletrd 11065	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 0 < 𝑖)
67	60, 61, 66	ltled 11053	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
68		elfzle2 13189	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
69	57, 58, 59, 67, 68	elfzd 13176	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (0...𝑀))
70	69	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (0...𝑀))
71	56, 70	ffvelrnd 6944	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
72	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
73		0zd 12261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ∈ ℤ)
74	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℤ)
75		elfzelz 13185	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℤ)
76	75	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℤ)
77		0red 10909	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 ∈ ℝ)
78	75	zred 12355	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℝ)
79		1red 10907	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 ∈ ℝ)
80	63	a1i 11	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 < 1)
81		elfzle1 13188	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 ≤ 𝑖)
82	77, 79, 78, 80, 81	ltletrd 11065	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 < 𝑖)
83	77, 78, 82	ltled 11053	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 ≤ 𝑖)
84	83	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ≤ 𝑖)
85	78	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℝ)
86	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℝ)
87		peano2rem 11218	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
88	86, 87	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
89		elfzle2 13189	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ≤ (𝑀 − 1))
90	89	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ≤ (𝑀 − 1))
91	86	ltm1d 11837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
92	85, 88, 86, 90, 91	lelttrd 11063	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 < 𝑀)
93	85, 86, 92	ltled 11053	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ≤ 𝑀)
94	73, 74, 76, 84, 93	elfzd 13176	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (0...𝑀))
95	72, 94	ffvelrnd 6944	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) ∈ ℝ)
96	76	peano2zd 12358	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ ℤ)
97		0red 10909	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ∈ ℝ)
98		peano2re 11078	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
99	85, 98	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ ℝ)
100		1red 10907	. . . . . . . . . 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 11835	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 < (𝑖 + 1))
104	79, 78, 102, 81, 103	lelttrd 11063	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 < (𝑖 + 1))
105	104	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 1 < (𝑖 + 1))
106	97, 100, 99, 101, 105	lttrd 11066	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 < (𝑖 + 1))
107	97, 99, 106	ltled 11053	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ≤ (𝑖 + 1))
108	85, 88, 100, 90	leadd1dd 11519	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ≤ ((𝑀 − 1) + 1))
109	2	nncnd 11919	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℂ)
110		1cnd 10901	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
111	109, 110	npcand 11266	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
112	111	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
113	108, 112	breqtrd 5096	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ≤ 𝑀)
114	73, 74, 96, 107, 113	elfzd 13176	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ (0...𝑀))
115	72, 114	ffvelrnd 6944	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
116		elfzo 13318	. . . . . . . . 9 ⊢ ((𝑖 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 ∈ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 < 𝑀)))
117	76, 73, 74, 116	syl3anc 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 < 𝑀)))
118	84, 92, 117	mpbir2and 709	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (0..^𝑀))
119	118, 46	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
120	95, 115, 119	ltled 11053	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1)))
121	55, 71, 120	monoord 13681	. . . 4 ⊢ (𝜑 → (𝑄‘1) ≤ (𝑄‘𝑀))
122	121, 23	breqtrd 5096	. . 3 ⊢ (𝜑 → (𝑄‘1) ≤ 𝐵)
123	22, 33, 27, 53, 122	ltletrd 11065	. 2 ⊢ (𝜑 → 𝐴 < 𝐵)
124	22, 27, 123	3jca 1126	1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 ℤcz 12249 ℤ_≥cuz 12511 ...cfz 13168 ..^cfzo 13311

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312

This theorem is referenced by: fourierdlem37 43575 fourierdlem54 43591 fourierdlem63 43600 fourierdlem64 43601 fourierdlem65 43602 fourierdlem69 43606 fourierdlem79 43616 fourierdlem89 43626 fourierdlem90 43627 fourierdlem91 43628 fourierdlem107 43644 fourierdlem109 43646

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