fourierdlem15 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem15 46078

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 46065	. . . . . . 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 11244	. . . . . 6 ⊢ ℝ ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
10		ovex 7464	. . . . . 6 ⊢ (0...𝑀) ∈ V
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
12	9, 11	elmapd 8879	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
13	7, 12	mpbid 232	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
14		ffn 6737	. . 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 12531	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
20		nn0uz 12918	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
21	19, 20	eleqtrdi 2849	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ_≥‘0))
22	2, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
23		eluzfz1 13568	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
24	22, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑀))
25	13, 24	ffvelcdmd 7105	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
26	18, 25	eqeltrrd 2840	. . . . . 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 13569	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
30	22, 29	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
31	13, 30	ffvelcdmd 7105	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
32	28, 31	eqeltrrd 2840	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
33	32	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
34	13	ffvelcdmda 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
35	18	eqcomd 2741	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
36	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37		elfzuz 13557	. . . . . . . 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 12623	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41		elfzel2 13559	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43		elfzelz 13561	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
44	43	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45		elfzle1 13564	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
47	43	zred 12720	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
48	47	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49		elfzelz 13561	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
50	49	zred 12720	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
52	41	zred 12720	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
53	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54		elfzle2 13565	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖)
55	54	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑖)
56		elfzle2 13565	. . . . . . . . . . . 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 11416	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑀)
59	40, 42, 44, 46, 58	elfzd 13552	. . . . . . . . 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 7105	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄‘𝑗) ∈ ℝ)
62		simpll 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
63		elfzle1 13564	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
64	63	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65		elfzelz 13561	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
66	65	zred 12720	. . . . . . . . . . . 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 11574	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
71	68, 70	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72		elfzle2 13565	. . . . . . . . . . . . 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 12198	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
75	67, 71, 68, 73, 74	lelttrd 11417	. . . . . . . . . . 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 11419	. . . . . . . . . 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 12623	. . . . . . . . . . 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 13698	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
82	78, 79, 80, 81	syl3anc 1370	. . . . . . . . . 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 13712	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
87	86	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
88	85, 87	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ∈ ℝ)
89		fzofzp1 13800	. . . . . . . . . . 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 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92		eleq1w 2822	. . . . . . . . . . . 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 6907	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
95		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
96	95	fveq2d 6911	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
97	94, 96	breq12d 5161	. . . . . . . . . . 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 3249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
101	98, 100	chvarvv 1996	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
102	88, 91, 101	ltled 11407	. . . . . . . 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 14070	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄‘𝑖))
105	36, 104	eqbrtrd 5170	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄‘𝑖))
106		elfzuz3 13558	. . . . . . . 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 13671	. . . . . . . . 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 7105	. . . . . . 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 12623	. . . . . . . . . 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 13561	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116	115	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117		0red 11262	. . . . . . . . . . . 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 12720	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120	119	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121		elfzle1 13564	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122	121	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123		elfzle1 13564	. . . . . . . . . . . . 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 11416	. . . . . . . . . . 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 12279	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
129	128	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130		1red 11260	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131	129, 130	resubcld 11689	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132		elfzle2 13565	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133	132	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134	129	ltm1d 12198	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135	127, 131, 129, 133, 134	lelttrd 11417	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136	127, 129, 135	ltled 11407	. . . . . . . . . . 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 13552	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139	112, 138	ffvelcdmd 7105	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ∈ ℝ)
140	116	peano2zd 12723	. . . . . . . . . 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 11260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143		0le1 11784	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
144	143	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145	141, 142, 126, 144	addge0d 11837	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146	127, 131, 130, 133	leadd1dd 11875	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
147	2	nncnd 12280	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
148	147	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149		1cnd 11254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150	148, 149	npcand 11622	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151	146, 150	breqtrd 5174	. . . . . . . . . . 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 13552	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154	112, 153	ffvelcdmd 7105	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
155		simpll 767	. . . . . . . . 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 1370	. . . . . . . . . 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 11407	. . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑀) = 𝐵)
163	161, 162	breqtrd 5174	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ 𝐵)
164	27, 33, 34, 105, 163	eliccd 45457	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
165	164	ralrimiva 3144	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵))
166		fnfvrnss 7141	. . 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 6567	. 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 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 [,]cicc 13387 ...cfz 13544 ..^cfzo 13691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-icc 13391 df-fz 13545 df-fzo 13692

This theorem is referenced by: fourierdlem38 46101 fourierdlem50 46112 fourierdlem54 46116 fourierdlem63 46125 fourierdlem65 46127 fourierdlem69 46131 fourierdlem70 46132 fourierdlem74 46136 fourierdlem75 46137 fourierdlem76 46138 fourierdlem79 46141 fourierdlem81 46143 fourierdlem84 46146 fourierdlem85 46147 fourierdlem88 46150 fourierdlem89 46151 fourierdlem90 46152 fourierdlem91 46153 fourierdlem92 46154 fourierdlem93 46155 fourierdlem100 46162 fourierdlem101 46163 fourierdlem103 46165 fourierdlem104 46166 fourierdlem107 46169 fourierdlem111 46173 fourierdlem112 46174 fourierdlem113 46175

W3C validator