fourierdlem15 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem15 44838

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 44825	. . . . . . 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 231	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simpld 496	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
8		reex 11201	. . . . . 6 ⊢ ℝ ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
10		ovex 7442	. . . . . 6 ⊢ (0...𝑀) ∈ V
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
12	9, 11	elmapd 8834	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
13	7, 12	mpbid 231	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
14		ffn 6718	. . 3 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
15	13, 14	syl 17	. 2 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
16	6	simprd 497	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
17	16	simpld 496	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
18	17	simpld 496	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
19		nnnn0 12479	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
20		nn0uz 12864	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
21	19, 20	eleqtrdi 2844	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ_≥‘0))
22	2, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
23		eluzfz1 13508	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
24	22, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑀))
25	13, 24	ffvelcdmd 7088	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
26	18, 25	eqeltrrd 2835	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
27	26	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
28	17	simprd 497	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
29		eluzfz2 13509	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
30	22, 29	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
31	13, 30	ffvelcdmd 7088	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
32	28, 31	eqeltrrd 2835	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
33	32	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
34	13	ffvelcdmda 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
35	18	eqcomd 2739	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
36	35	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37		elfzuz 13497	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ_≥‘0))
38	37	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ_≥‘0))
39	13	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40		0zd 12570	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41		elfzel2 13499	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
42	41	adantr 482	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43		elfzelz 13501	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
44	43	adantl 483	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45		elfzle1 13504	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
46	45	adantl 483	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
47	43	zred 12666	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
48	47	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49		elfzelz 13501	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
50	49	zred 12666	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
51	50	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
52	41	zred 12666	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
53	52	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54		elfzle2 13505	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖)
55	54	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑖)
56		elfzle2 13505	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ≤ 𝑀)
57	56	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ≤ 𝑀)
58	48, 51, 53, 55, 57	letrd 11371	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑀)
59	40, 42, 44, 46, 58	elfzd 13492	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
60	59	adantll 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
61	39, 60	ffvelcdmd 7088	. . . . . . 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 13504	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
64	63	adantl 483	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65		elfzelz 13501	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
66	65	zred 12666	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℝ)
67	66	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℝ)
68	50	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ∈ ℝ)
69	52	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℝ)
70		peano2rem 11527	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
71	68, 70	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72		elfzle2 13505	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
73	72	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
74	68	ltm1d 12146	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
75	67, 71, 68, 73, 74	lelttrd 11372	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑖)
76	56	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ≤ 𝑀)
77	67, 68, 69, 75, 76	ltletrd 11374	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
78	65	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
79		0zd 12570	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
80	41	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
81		elfzo 13634	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
82	78, 79, 80, 81	syl3anc 1372	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
83	64, 77, 82	mpbir2and 712	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
84	83	adantll 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
85	13	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86		elfzofz 13648	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
87	86	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
88	85, 87	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ∈ ℝ)
89		fzofzp1 13729	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
90	89	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
91	85, 90	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92		eleq1w 2817	. . . . . . . . . . . 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 6892	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
95		oveq1 7416	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
96	95	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
97	94, 96	breq12d 5162	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑗) < (𝑄‘(𝑗 + 1))))
98	93, 97	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))))
99	16	simprd 497	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
100	99	r19.21bi 3249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
101	98, 100	chvarvv 2003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
102	88, 91, 101	ltled 11362	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
103	62, 84, 102	syl2anc 585	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
104	38, 61, 103	monoord 13998	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄‘𝑖))
105	36, 104	eqbrtrd 5171	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄‘𝑖))
106		elfzuz3 13498	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ_≥‘𝑖))
107	106	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑖))
108	13	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109		fz0fzelfz0 13607	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
110	109	adantll 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
111	108, 110	ffvelcdmd 7088	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
112	13	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
113		0zd 12570	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
114	41	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
115		elfzelz 13501	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116	115	adantl 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117		0red 11217	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
118	50	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
119	115	zred 12666	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120	119	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121		elfzle1 13504	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122	121	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123		elfzle1 13504	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖 ≤ 𝑗)
124	123	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ≤ 𝑗)
125	117, 118, 120, 122, 124	letrd 11371	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126	125	adantll 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
127	119	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
128	2	nnred 12227	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
129	128	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130		1red 11215	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131	129, 130	resubcld 11642	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132		elfzle2 13505	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133	132	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134	129	ltm1d 12146	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135	127, 131, 129, 133, 134	lelttrd 11372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136	127, 129, 135	ltled 11362	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ 𝑀)
137	136	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ 𝑀)
138	113, 114, 116, 126, 137	elfzd 13492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139	112, 138	ffvelcdmd 7088	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ∈ ℝ)
140	116	peano2zd 12669	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ ℤ)
141	119	adantl 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
142		1red 11215	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143		0le1 11737	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
144	143	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145	141, 142, 126, 144	addge0d 11790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146	127, 131, 130, 133	leadd1dd 11828	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
147	2	nncnd 12228	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
148	147	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149		1cnd 11209	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150	148, 149	npcand 11575	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151	146, 150	breqtrd 5175	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
152	151	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
153	113, 114, 140, 145, 152	elfzd 13492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154	112, 153	ffvelcdmd 7088	. . . . . . . 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 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
157	116, 113, 114, 81	syl3anc 1372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
158	126, 156, 157	mpbir2and 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0..^𝑀))
159	155, 158, 101	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
160	139, 154, 159	ltled 11362	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
161	107, 111, 160	monoord 13998	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
162	28	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑀) = 𝐵)
163	161, 162	breqtrd 5175	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ 𝐵)
164	27, 33, 34, 105, 163	eliccd 44217	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
165	164	ralrimiva 3147	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵))
166		fnfvrnss 7120	. . 3 ⊢ ((𝑄 Fn (0...𝑀) ∧ ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵)) → ran 𝑄 ⊆ (𝐴[,]𝐵))
167	15, 165, 166	syl2anc 585	. 2 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
168		df-f 6548	. 2 ⊢ (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 ⊆ (𝐴[,]𝐵)))
169	15, 167, 168	sylanbrc 584	1 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 Vcvv 3475 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 [,]cicc 13327 ...cfz 13484 ..^cfzo 13627

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-icc 13331 df-fz 13485 df-fzo 13628

This theorem is referenced by: fourierdlem38 44861 fourierdlem50 44872 fourierdlem54 44876 fourierdlem63 44885 fourierdlem65 44887 fourierdlem69 44891 fourierdlem70 44892 fourierdlem74 44896 fourierdlem75 44897 fourierdlem76 44898 fourierdlem79 44901 fourierdlem81 44903 fourierdlem84 44906 fourierdlem85 44907 fourierdlem88 44910 fourierdlem89 44911 fourierdlem90 44912 fourierdlem91 44913 fourierdlem92 44914 fourierdlem93 44915 fourierdlem100 44922 fourierdlem101 44923 fourierdlem103 44925 fourierdlem104 44926 fourierdlem107 44929 fourierdlem111 44933 fourierdlem112 44934 fourierdlem113 44935

W3C validator