fourierdlem15 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem15		Structured version Visualization version GIF version

Theorem fourierdlem15 46123

Description: The range of the partition is between its starting point and its ending point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem15.1	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem15.2	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem15.3	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))

**Assertion**
Ref	Expression
fourierdlem15	⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))

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

Proof of Theorem fourierdlem15 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem15.3	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
2		fourierdlem15.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem15.1	. . . . . . . 8 ⊢ 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 46110	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ (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 232	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simpld 494	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
8		reex 11119	. . . . . 6 ⊢ ℝ ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
10		ovex 7386	. . . . . 6 ⊢ (0...𝑀) ∈ V
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
12	9, 11	elmapd 8774	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
13	7, 12	mpbid 232	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
14		ffn 6656	. . 3 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
15	13, 14	syl 17	. 2 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
16	6	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
17	16	simpld 494	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
18	17	simpld 494	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
19		nnnn0 12410	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
20		nn0uz 12796	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
21	19, 20	eleqtrdi 2838	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ_≥‘0))
22	2, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
23		eluzfz1 13453	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
24	22, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑀))
25	13, 24	ffvelcdmd 7023	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
26	18, 25	eqeltrrd 2829	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
28	17	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
29		eluzfz2 13454	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
30	22, 29	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
31	13, 30	ffvelcdmd 7023	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
32	28, 31	eqeltrrd 2829	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
33	32	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
34	13	ffvelcdmda 7022	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
35	18	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
36	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37		elfzuz 13442	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ_≥‘0))
38	37	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ_≥‘0))
39	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40		0zd 12502	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41		elfzel2 13444	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43		elfzelz 13446	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
44	43	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45		elfzle1 13449	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
47	43	zred 12599	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
48	47	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49		elfzelz 13446	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
50	49	zred 12599	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
52	41	zred 12599	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
53	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54		elfzle2 13450	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖)
55	54	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑖)
56		elfzle2 13450	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ≤ 𝑀)
57	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ≤ 𝑀)
58	48, 51, 53, 55, 57	letrd 11292	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑀)
59	40, 42, 44, 46, 58	elfzd 13437	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
60	59	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
61	39, 60	ffvelcdmd 7023	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄‘𝑗) ∈ ℝ)
62		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
63		elfzle1 13449	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
64	63	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65		elfzelz 13446	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
66	65	zred 12599	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℝ)
67	66	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℝ)
68	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ∈ ℝ)
69	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℝ)
70		peano2rem 11450	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
71	68, 70	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72		elfzle2 13450	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
73	72	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
74	68	ltm1d 12076	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
75	67, 71, 68, 73, 74	lelttrd 11293	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑖)
76	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ≤ 𝑀)
77	67, 68, 69, 75, 76	ltletrd 11295	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
78	65	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
79		0zd 12502	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
80	41	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
81		elfzo 13583	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
82	78, 79, 80, 81	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
83	64, 77, 82	mpbir2and 713	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
84	83	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
85	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86		elfzofz 13597	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
87	86	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
88	85, 87	ffvelcdmd 7023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ∈ ℝ)
89		fzofzp1 13686	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
90	89	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
91	85, 90	ffvelcdmd 7023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92		eleq1w 2811	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
93	92	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0..^𝑀))))
94		fveq2 6826	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
95		oveq1 7360	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
96	95	fveq2d 6830	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
97	94, 96	breq12d 5108	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑗) < (𝑄‘(𝑗 + 1))))
98	93, 97	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))))
99	16	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
100	99	r19.21bi 3221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
101	98, 100	chvarvv 1989	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
102	88, 91, 101	ltled 11283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
103	62, 84, 102	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
104	38, 61, 103	monoord 13958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄‘𝑖))
105	36, 104	eqbrtrd 5117	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄‘𝑖))
106		elfzuz3 13443	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ_≥‘𝑖))
107	106	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑖))
108	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109		fz0fzelfz0 13556	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
110	109	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
111	108, 110	ffvelcdmd 7023	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
112	13	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
113		0zd 12502	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
114	41	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
115		elfzelz 13446	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116	115	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117		0red 11137	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
118	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
119	115	zred 12599	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120	119	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121		elfzle1 13449	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122	121	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123		elfzle1 13449	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖 ≤ 𝑗)
124	123	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ≤ 𝑗)
125	117, 118, 120, 122, 124	letrd 11292	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126	125	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
127	119	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
128	2	nnred 12162	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
129	128	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130		1red 11135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131	129, 130	resubcld 11567	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132		elfzle2 13450	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133	132	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134	129	ltm1d 12076	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135	127, 131, 129, 133, 134	lelttrd 11293	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136	127, 129, 135	ltled 11283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ 𝑀)
137	136	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ 𝑀)
138	113, 114, 116, 126, 137	elfzd 13437	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139	112, 138	ffvelcdmd 7023	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ∈ ℝ)
140	116	peano2zd 12602	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ ℤ)
141	119	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
142		1red 11135	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143		0le1 11662	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
144	143	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145	141, 142, 126, 144	addge0d 11715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146	127, 131, 130, 133	leadd1dd 11753	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
147	2	nncnd 12163	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
148	147	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149		1cnd 11129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150	148, 149	npcand 11498	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151	146, 150	breqtrd 5121	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
152	151	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
153	113, 114, 140, 145, 152	elfzd 13437	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154	112, 153	ffvelcdmd 7023	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
155		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝜑)
156	135	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
157	116, 113, 114, 81	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
158	126, 156, 157	mpbir2and 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0..^𝑀))
159	155, 158, 101	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
160	139, 154, 159	ltled 11283	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
161	107, 111, 160	monoord 13958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
162	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑀) = 𝐵)
163	161, 162	breqtrd 5121	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ 𝐵)
164	27, 33, 34, 105, 163	eliccd 45505	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
165	164	ralrimiva 3121	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵))
166		fnfvrnss 7059	. . 3 ⊢ ((𝑄 Fn (0...𝑀) ∧ ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵)) → ran 𝑄 ⊆ (𝐴[,]𝐵))
167	15, 165, 166	syl2anc 584	. 2 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
168		df-f 6490	. 2 ⊢ (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 ⊆ (𝐴[,]𝐵)))
169	15, 167, 168	sylanbrc 583	1 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 {crab 3396 Vcvv 3438 ⊆ wss 3905 class class class wbr 5095 ↦ cmpt 5176 ran crn 5624 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ↑_m cmap 8760 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 < clt 11168 ≤ cle 11169 − cmin 11366 ℕcn 12147 ℕ₀cn0 12403 ℤcz 12490 ℤ_≥cuz 12754 [,]cicc 13270 ...cfz 13429 ..^cfzo 13576

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-n0 12404 df-z 12491 df-uz 12755 df-icc 13274 df-fz 13430 df-fzo 13577

This theorem is referenced by: fourierdlem38 46146 fourierdlem50 46157 fourierdlem54 46161 fourierdlem63 46170 fourierdlem65 46172 fourierdlem69 46176 fourierdlem70 46177 fourierdlem74 46181 fourierdlem75 46182 fourierdlem76 46183 fourierdlem79 46186 fourierdlem81 46188 fourierdlem84 46191 fourierdlem85 46192 fourierdlem88 46195 fourierdlem89 46196 fourierdlem90 46197 fourierdlem91 46198 fourierdlem92 46199 fourierdlem93 46200 fourierdlem100 46207 fourierdlem101 46208 fourierdlem103 46210 fourierdlem104 46211 fourierdlem107 46214 fourierdlem111 46218 fourierdlem112 46219 fourierdlem113 46220

W3C validator