fourierdlem15 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem15 46819

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 46806	. . . . . . 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 18	. . . . . 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 235	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simpld 499	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
8		reex 11192	. . . . . 6 ⊢ ℝ ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
10		ovex 7445	. . . . . 6 ⊢ (0...𝑀) ∈ V
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
12	9, 11	elmapd 8838	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
13	7, 12	mpbid 235	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
14		ffn 6707	. . 3 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
15	13, 14	syl 18	. 2 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
16	6	simprd 500	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
17	16	simpld 499	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
18	17	simpld 499	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
19		nnnn0 12512	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
20		nn0uz 12901	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
21	19, 20	eleqtrdi 2873	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ_≥‘0))
22	2, 21	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
23		eluzfz1 13560	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
24	22, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑀))
25	13, 24	ffvelcdmd 7082	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
26	18, 25	eqeltrrd 2864	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
27	26	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
28	17	simprd 500	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
29		eluzfz2 13561	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
30	22, 29	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
31	13, 30	ffvelcdmd 7082	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
32	28, 31	eqeltrrd 2864	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
33	32	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
34	13	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
35	18	eqcomd 2769	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
36	35	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37		elfzuz 13549	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ_≥‘0))
38	37	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ_≥‘0))
39	13	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40		0zd 12604	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41		elfzel2 13551	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
42	41	adantr 485	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43		elfzelz 13553	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
44	43	adantl 486	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45		elfzle1 13556	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
46	45	adantl 486	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
47	43	zred 12701	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
48	47	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49		elfzelz 13553	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
50	49	zred 12701	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
51	50	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
52	41	zred 12701	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
53	52	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54		elfzle2 13557	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖)
55	54	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑖)
56		elfzle2 13557	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ≤ 𝑀)
57	56	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ≤ 𝑀)
58	48, 51, 53, 55, 57	letrd 11368	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑀)
59	40, 42, 44, 46, 58	elfzd 13544	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
60	59	adantll 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
61	39, 60	ffvelcdmd 7082	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄‘𝑗) ∈ ℝ)
62		simpll 778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
63		elfzle1 13556	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
64	63	adantl 486	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65		elfzelz 13553	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
66	65	zred 12701	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℝ)
67	66	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℝ)
68	50	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ∈ ℝ)
69	52	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℝ)
70		peano2rem 11526	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
71	68, 70	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72		elfzle2 13557	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
73	72	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
74	68	ltm1d 12148	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
75	67, 71, 68, 73, 74	lelttrd 11369	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑖)
76	56	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ≤ 𝑀)
77	67, 68, 69, 75, 76	ltletrd 11371	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
78	65	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
79		0zd 12604	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
80	41	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
81		elfzo 13691	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
82	78, 79, 80, 81	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
83	64, 77, 82	mpbir2and 725	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
84	83	adantll 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
85	13	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86		elfzofz 13706	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
87	86	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
88	85, 87	ffvelcdmd 7082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ∈ ℝ)
89		fzofzp1 13795	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
90	89	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
91	85, 90	ffvelcdmd 7082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92		eleq1w 2846	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
93	92	anbi2d 641	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0..^𝑀))))
94		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
95		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
96	95	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
97	94, 96	breq12d 5123	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑗) < (𝑄‘(𝑗 + 1))))
98	93, 97	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))))
99	16	simprd 500	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
100	99	r19.21bi 3257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
101	98, 100	chvarvv 2019	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
102	88, 91, 101	ltled 11359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
103	62, 84, 102	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
104	38, 61, 103	monoord 14070	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄‘𝑖))
105	36, 104	eqbrtrd 5134	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄‘𝑖))
106		elfzuz3 13550	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ_≥‘𝑖))
107	106	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑖))
108	13	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109		fz0fzelfz0 13664	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
110	109	adantll 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
111	108, 110	ffvelcdmd 7082	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
112	13	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
113		0zd 12604	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
114	41	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
115		elfzelz 13553	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116	115	adantl 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117		0red 11212	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
118	50	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
119	115	zred 12701	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120	119	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121		elfzle1 13556	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122	121	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123		elfzle1 13556	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖 ≤ 𝑗)
124	123	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ≤ 𝑗)
125	117, 118, 120, 122, 124	letrd 11368	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126	125	adantll 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
127	119	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
128	2	nnred 12249	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
129	128	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130		1red 11210	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131	129, 130	resubcld 11643	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132		elfzle2 13557	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133	132	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134	129	ltm1d 12148	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135	127, 131, 129, 133, 134	lelttrd 11369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136	127, 129, 135	ltled 11359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ 𝑀)
137	136	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ 𝑀)
138	113, 114, 116, 126, 137	elfzd 13544	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139	112, 138	ffvelcdmd 7082	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ∈ ℝ)
140	116	peano2zd 12704	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ ℤ)
141	119	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
142		1red 11210	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143		0le1 11738	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
144	143	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145	141, 142, 126, 144	addge0d 11791	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146	127, 131, 130, 133	leadd1dd 11829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
147	2	nncnd 12250	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
148	147	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149		1cnd 11203	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150	148, 149	npcand 11574	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151	146, 150	breqtrd 5138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
152	151	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
153	113, 114, 140, 145, 152	elfzd 13544	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154	112, 153	ffvelcdmd 7082	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
155		simpll 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝜑)
156	135	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
157	116, 113, 114, 81	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
158	126, 156, 157	mpbir2and 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0..^𝑀))
159	155, 158, 101	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
160	139, 154, 159	ltled 11359	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
161	107, 111, 160	monoord 14070	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
162	28	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑀) = 𝐵)
163	161, 162	breqtrd 5138	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ 𝐵)
164	27, 33, 34, 105, 163	eliccd 46203	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
165	164	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵))
166		fnfvrnss 7118	. . 3 ⊢ ((𝑄 Fn (0...𝑀) ∧ ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵)) → ran 𝑄 ⊆ (𝐴[,]𝐵))
167	15, 165, 166	syl2anc 595	. 2 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
168		df-f 6542	. 2 ⊢ (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 ⊆ (𝐴[,]𝐵)))
169	15, 167, 168	sylanbrc 594	1 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 {crab 3416 Vcvv 3455 ⊆ wss 3906 class class class wbr 5110 ↦ cmpt 5193 ran crn 5664 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ↑_m cmap 8825 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 < clt 11244 ≤ cle 11245 − cmin 11442 ℕcn 12234 ℕ₀cn0 12505 ℤcz 12592 ℤ_≥cuz 12863 [,]cicc 13376 ...cfz 13536 ..^cfzo 13684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-n0 12506 df-z 12593 df-uz 12864 df-icc 13380 df-fz 13537 df-fzo 13685

This theorem is referenced by: fourierdlem38 46842 fourierdlem50 46853 fourierdlem54 46857 fourierdlem63 46866 fourierdlem65 46868 fourierdlem69 46872 fourierdlem70 46873 fourierdlem74 46877 fourierdlem75 46878 fourierdlem76 46879 fourierdlem79 46882 fourierdlem81 46884 fourierdlem84 46887 fourierdlem85 46888 fourierdlem88 46891 fourierdlem89 46892 fourierdlem90 46893 fourierdlem91 46894 fourierdlem92 46895 fourierdlem93 46896 fourierdlem100 46903 fourierdlem101 46904 fourierdlem103 46906 fourierdlem104 46907 fourierdlem107 46910 fourierdlem111 46914 fourierdlem112 46915 fourierdlem113 46916

W3C validator