fourierdlem11 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem11		Structured version Visualization version GIF version

Theorem fourierdlem11 40335

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 40326	. . . . . . . 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 222	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simprd 479	. . . . 5 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
8	7	simpld 475	. . . 4 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
9	8	simpld 475	. . 3 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
10	6	simpld 475	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)))
11		elmapi 7879	. . . . 5 ⊢ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
13		0red 10041	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
14	13	leidd 10594	. . . . 5 ⊢ (𝜑 → 0 ≤ 0)
15	2	nnred 11035	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
16	2	nngt0d 11064	. . . . . 6 ⊢ (𝜑 → 0 < 𝑀)
17	13, 15, 16	ltled 10185	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑀)
18		0zd 11389	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℤ)
19	2	nnzd 11481	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
20		elfz 12332	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ∈ (0...𝑀) ↔ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
21	18, 18, 19, 20	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (0 ∈ (0...𝑀) ↔ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
22	14, 17, 21	mpbir2and 957	. . . 4 ⊢ (𝜑 → 0 ∈ (0...𝑀))
23	12, 22	ffvelrnd 6360	. . 3 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
24	9, 23	eqeltrrd 2702	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
25	8	simprd 479	. . 3 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
26	15	leidd 10594	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
27		elfz 12332	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (0...𝑀) ↔ (0 ≤ 𝑀 ∧ 𝑀 ≤ 𝑀)))
28	19, 18, 19, 27	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (0...𝑀) ↔ (0 ≤ 𝑀 ∧ 𝑀 ≤ 𝑀)))
29	17, 26, 28	mpbir2and 957	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
30	12, 29	ffvelrnd 6360	. . 3 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
31	25, 30	eqeltrrd 2702	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
32		0le1 10551	. . . . . 6 ⊢ 0 ≤ 1
33	32	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≤ 1)
34	2	nnge1d 11063	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
35		1zzd 11408	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
36		elfz 12332	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 ∈ (0...𝑀) ↔ (0 ≤ 1 ∧ 1 ≤ 𝑀)))
37	35, 18, 19, 36	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (1 ∈ (0...𝑀) ↔ (0 ≤ 1 ∧ 1 ≤ 𝑀)))
38	33, 34, 37	mpbir2and 957	. . . 4 ⊢ (𝜑 → 1 ∈ (0...𝑀))
39	12, 38	ffvelrnd 6360	. . 3 ⊢ (𝜑 → (𝑄‘1) ∈ ℝ)
40		elfzo 12472	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ∈ (0..^𝑀) ↔ (0 ≤ 0 ∧ 0 < 𝑀)))
41	18, 18, 19, 40	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → (0 ∈ (0..^𝑀) ↔ (0 ≤ 0 ∧ 0 < 𝑀)))
42	14, 16, 41	mpbir2and 957	. . . . 5 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
43		0re 10040	. . . . . 6 ⊢ 0 ∈ ℝ
44		eleq1 2689	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑖 ∈ (0..^𝑀) ↔ 0 ∈ (0..^𝑀)))
45	44	anbi2d 740	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 0 ∈ (0..^𝑀))))
46		fveq2 6191	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
47		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
48	47	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑄‘(𝑖 + 1)) = (𝑄‘(0 + 1)))
49	46, 48	breq12d 4666	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘0) < (𝑄‘(0 + 1))))
50	45, 49	imbi12d 334	. . . . . . 7 ⊢ (𝑖 = 0 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1)))))
51	7	simprd 479	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
52	51	r19.21bi 2932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
53	50, 52	vtoclg 3266	. . . . . 6 ⊢ (0 ∈ ℝ → ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1))))
54	43, 53	ax-mp 5	. . . . 5 ⊢ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1)))
55	42, 54	mpdan 702	. . . 4 ⊢ (𝜑 → (𝑄‘0) < (𝑄‘(0 + 1)))
56		0p1e1 11132	. . . . . 6 ⊢ (0 + 1) = 1
57	56	a1i 11	. . . . 5 ⊢ (𝜑 → (0 + 1) = 1)
58	57	fveq2d 6195	. . . 4 ⊢ (𝜑 → (𝑄‘(0 + 1)) = (𝑄‘1))
59	55, 9, 58	3brtr3d 4684	. . 3 ⊢ (𝜑 → 𝐴 < (𝑄‘1))
60		nnuz 11723	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
61	2, 60	syl6eleq 2711	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
62	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
63		0red 10041	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
64		elfzelz 12342	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
65	64	zred 11482	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
66		1red 10055	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
67		0lt1 10550	. . . . . . . . . . 11 ⊢ 0 < 1
68	67	a1i 11	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 0 < 1)
69		elfzle1 12344	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
70	63, 66, 65, 68, 69	ltletrd 10197	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 0 < 𝑖)
71	63, 65, 70	ltled 10185	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
72		elfzle2 12345	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
73		0zd 11389	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℤ)
74		elfzel2 12340	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
75		elfz 12332	. . . . . . . . 9 ⊢ ((𝑖 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 ∈ (0...𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
76	64, 73, 74, 75	syl3anc 1326	. . . . . . . 8 ⊢ (𝑖 ∈ (1...𝑀) → (𝑖 ∈ (0...𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
77	71, 72, 76	mpbir2and 957	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (0...𝑀))
78	77	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (0...𝑀))
79	62, 78	ffvelrnd 6360	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
80	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
81		0red 10041	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 ∈ ℝ)
82		elfzelz 12342	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℤ)
83	82	zred 11482	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℝ)
84		1red 10055	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 ∈ ℝ)
85	67	a1i 11	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 < 1)
86		elfzle1 12344	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 ≤ 𝑖)
87	81, 84, 83, 85, 86	ltletrd 10197	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 < 𝑖)
88	81, 83, 87	ltled 10185	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 ≤ 𝑖)
89	88	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ≤ 𝑖)
90	83	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℝ)
91	15	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℝ)
92		peano2rem 10348	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
93	91, 92	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
94		elfzle2 12345	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ≤ (𝑀 − 1))
95	94	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ≤ (𝑀 − 1))
96	91	ltm1d 10956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
97	90, 93, 91, 95, 96	lelttrd 10195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 < 𝑀)
98	90, 91, 97	ltled 10185	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ≤ 𝑀)
99	82	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℤ)
100		0zd 11389	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ∈ ℤ)
101	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℤ)
102	99, 100, 101, 75	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (0...𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)))
103	89, 98, 102	mpbir2and 957	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (0...𝑀))
104	80, 103	ffvelrnd 6360	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) ∈ ℝ)
105		0red 10041	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ∈ ℝ)
106		peano2re 10209	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
107	90, 106	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ ℝ)
108		1red 10055	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 1 ∈ ℝ)
109	67	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 < 1)
110	83, 106	syl 17	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → (𝑖 + 1) ∈ ℝ)
111	83	ltp1d 10954	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 < (𝑖 + 1))
112	84, 83, 110, 86, 111	lelttrd 10195	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 < (𝑖 + 1))
113	112	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 1 < (𝑖 + 1))
114	105, 108, 107, 109, 113	lttrd 10198	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 < (𝑖 + 1))
115	105, 107, 114	ltled 10185	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ≤ (𝑖 + 1))
116	90, 93, 108, 95	leadd1dd 10641	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ≤ ((𝑀 − 1) + 1))
117	2	nncnd 11036	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℂ)
118		1cnd 10056	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
119	117, 118	npcand 10396	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
120	119	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
121	116, 120	breqtrd 4679	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ≤ 𝑀)
122	99	peano2zd 11485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ ℤ)
123		elfz 12332	. . . . . . . . 9 ⊢ (((𝑖 + 1) ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑖 + 1) ∈ (0...𝑀) ↔ (0 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ 𝑀)))
124	122, 100, 101, 123	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ((𝑖 + 1) ∈ (0...𝑀) ↔ (0 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ 𝑀)))
125	115, 121, 124	mpbir2and 957	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ (0...𝑀))
126	80, 125	ffvelrnd 6360	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
127		elfzo 12472	. . . . . . . . 9 ⊢ ((𝑖 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 ∈ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 < 𝑀)))
128	99, 100, 101, 127	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 < 𝑀)))
129	89, 97, 128	mpbir2and 957	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (0..^𝑀))
130	129, 52	syldan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
131	104, 126, 130	ltled 10185	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1)))
132	61, 79, 131	monoord 12831	. . . 4 ⊢ (𝜑 → (𝑄‘1) ≤ (𝑄‘𝑀))
133	132, 25	breqtrd 4679	. . 3 ⊢ (𝜑 → (𝑄‘1) ≤ 𝐵)
134	24, 39, 31, 59, 133	ltletrd 10197	. 2 ⊢ (𝜑 → 𝐴 < 𝐵)
135	24, 31, 134	3jca 1242	1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466

This theorem is referenced by: fourierdlem37 40361 fourierdlem54 40377 fourierdlem63 40386 fourierdlem64 40387 fourierdlem65 40388 fourierdlem69 40392 fourierdlem79 40402 fourierdlem89 40412 fourierdlem90 40413 fourierdlem91 40414 fourierdlem107 40430 fourierdlem109 40432

W3C validator