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

Theorem fourierdlem15 44353
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 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem15.2 (𝜑𝑀 ∈ ℕ)
fourierdlem15.3 (𝜑𝑄 ∈ (𝑃𝑀))
Assertion
Ref Expression
fourierdlem15 (𝜑𝑄:(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.
StepHypRef Expression
1 fourierdlem15.3 . . . . . 6 (𝜑𝑄 ∈ (𝑃𝑀))
2 fourierdlem15.2 . . . . . . 7 (𝜑𝑀 ∈ ℕ)
3 fourierdlem15.1 . . . . . . . 8 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
43fourierdlem2 44340 . . . . . . 7 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
52, 4syl 17 . . . . . 6 (𝜑 → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
61, 5mpbid 231 . . . . 5 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simpld 495 . . . 4 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
8 reex 11142 . . . . . 6 ℝ ∈ V
98a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
10 ovex 7390 . . . . . 6 (0...𝑀) ∈ V
1110a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
129, 11elmapd 8779 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
137, 12mpbid 231 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
14 ffn 6668 . . 3 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
1513, 14syl 17 . 2 (𝜑𝑄 Fn (0...𝑀))
166simprd 496 . . . . . . . . 9 (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
1716simpld 495 . . . . . . . 8 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
1817simpld 495 . . . . . . 7 (𝜑 → (𝑄‘0) = 𝐴)
19 nnnn0 12420 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
20 nn0uz 12805 . . . . . . . . . . 11 0 = (ℤ‘0)
2119, 20eleqtrdi 2848 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ‘0))
222, 21syl 17 . . . . . . . . 9 (𝜑𝑀 ∈ (ℤ‘0))
23 eluzfz1 13448 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
2422, 23syl 17 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
2513, 24ffvelcdmd 7036 . . . . . . 7 (𝜑 → (𝑄‘0) ∈ ℝ)
2618, 25eqeltrrd 2839 . . . . . 6 (𝜑𝐴 ∈ ℝ)
2726adantr 481 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
2817simprd 496 . . . . . . 7 (𝜑 → (𝑄𝑀) = 𝐵)
29 eluzfz2 13449 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
3022, 29syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
3113, 30ffvelcdmd 7036 . . . . . . 7 (𝜑 → (𝑄𝑀) ∈ ℝ)
3228, 31eqeltrrd 2839 . . . . . 6 (𝜑𝐵 ∈ ℝ)
3332adantr 481 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
3413ffvelcdmda 7035 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
3518eqcomd 2742 . . . . . . 7 (𝜑𝐴 = (𝑄‘0))
3635adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37 elfzuz 13437 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ‘0))
3837adantl 482 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ‘0))
3913ad2antrr 724 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40 0zd 12511 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41 elfzel2 13439 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
4241adantr 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43 elfzelz 13441 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
4443adantl 482 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45 elfzle1 13444 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
4645adantl 482 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
4743zred 12607 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
4847adantl 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49 elfzelz 13441 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
5049zred 12607 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
5150adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
5241zred 12607 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
5352adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54 elfzle2 13445 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
5554adantl 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑖)
56 elfzle2 13445 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖𝑀)
5756adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖𝑀)
5848, 51, 53, 55, 57letrd 11312 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑀)
5940, 42, 44, 46, 58elfzd 13432 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6059adantll 712 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6139, 60ffvelcdmd 7036 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄𝑗) ∈ ℝ)
62 simpll 765 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
63 elfzle1 13444 . . . . . . . . . . 11 (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
6463adantl 482 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65 elfzelz 13441 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
6665zred 12607 . . . . . . . . . . . 12 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℝ)
6766adantl 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℝ)
6850adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ∈ ℝ)
6952adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℝ)
70 peano2rem 11468 . . . . . . . . . . . . 13 (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
7168, 70syl 17 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72 elfzle2 13445 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
7372adantl 482 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
7468ltm1d 12087 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
7567, 71, 68, 73, 74lelttrd 11313 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑖)
7656adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖𝑀)
7767, 68, 69, 75, 76ltletrd 11315 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
7865adantl 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
79 0zd 12511 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
8041adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
81 elfzo 13574 . . . . . . . . . . 11 ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
8278, 79, 80, 81syl3anc 1371 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
8364, 77, 82mpbir2and 711 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
8483adantll 712 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
8513adantr 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86 elfzofz 13588 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
8786adantl 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
8885, 87ffvelcdmd 7036 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) ∈ ℝ)
89 fzofzp1 13669 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
9089adantl 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
9185, 90ffvelcdmd 7036 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92 eleq1w 2820 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9392anbi2d 629 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
94 fveq2 6842 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
95 oveq1 7364 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
9695fveq2d 6846 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9794, 96breq12d 5118 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑄𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄𝑗) < (𝑄‘(𝑗 + 1))))
9893, 97imbi12d 344 . . . . . . . . . 10 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))))
9916simprd 496 . . . . . . . . . . 11 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
10099r19.21bi 3234 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
10198, 100chvarvv 2002 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))
10288, 91, 101ltled 11303 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
10362, 84, 102syl2anc 584 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
10438, 61, 103monoord 13938 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄𝑖))
10536, 104eqbrtrd 5127 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄𝑖))
106 elfzuz3 13438 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ𝑖))
107106adantl 482 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ𝑖))
10813ad2antrr 724 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109 fz0fzelfz0 13547 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
110109adantll 712 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
111108, 110ffvelcdmd 7036 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → (𝑄𝑗) ∈ ℝ)
11213ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
113 0zd 12511 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
11441ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
115 elfzelz 13441 . . . . . . . . . . 11 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116115adantl 482 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117 0red 11158 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
11850adantr 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
119115zred 12607 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120119adantl 482 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121 elfzle1 13444 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122121adantr 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123 elfzle1 13444 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖𝑗)
124123adantl 482 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖𝑗)
125117, 118, 120, 122, 124letrd 11312 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126125adantll 712 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
127119adantl 482 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
1282nnred 12168 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℝ)
129128adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130 1red 11156 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131129, 130resubcld 11583 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132 elfzle2 13445 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133132adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134129ltm1d 12087 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135127, 131, 129, 133, 134lelttrd 11313 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136127, 129, 135ltled 11303 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
137136adantlr 713 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
138113, 114, 116, 126, 137elfzd 13432 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139112, 138ffvelcdmd 7036 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ∈ ℝ)
140116peano2zd 12610 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ ℤ)
141119adantl 482 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
142 1red 11156 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143 0le1 11678 . . . . . . . . . . . 12 0 ≤ 1
144143a1i 11 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145141, 142, 126, 144addge0d 11731 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146127, 131, 130, 133leadd1dd 11769 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
1472nncnd 12169 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℂ)
148147adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149 1cnd 11150 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150148, 149npcand 11516 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151146, 150breqtrd 5131 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
152151adantlr 713 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
153113, 114, 140, 145, 152elfzd 13432 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154112, 153ffvelcdmd 7036 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
155 simpll 765 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝜑)
156135adantlr 713 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
157116, 113, 114, 81syl3anc 1371 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
158126, 156, 157mpbir2and 711 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0..^𝑀))
159155, 158, 101syl2anc 584 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))
160139, 154, 159ltled 11303 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
161107, 111, 160monoord 13938 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ (𝑄𝑀))
16228adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑀) = 𝐵)
163161, 162breqtrd 5131 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ 𝐵)
16427, 33, 34, 105, 163eliccd 43732 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ (𝐴[,]𝐵))
165164ralrimiva 3143 . . 3 (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄𝑖) ∈ (𝐴[,]𝐵))
166 fnfvrnss 7068 . . 3 ((𝑄 Fn (0...𝑀) ∧ ∀𝑖 ∈ (0...𝑀)(𝑄𝑖) ∈ (𝐴[,]𝐵)) → ran 𝑄 ⊆ (𝐴[,]𝐵))
16715, 165, 166syl2anc 584 . 2 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
168 df-f 6500 . 2 (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 ⊆ (𝐴[,]𝐵)))
16915, 167, 168sylanbrc 583 1 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  wss 3910   class class class wbr 5105  cmpt 5188  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  cn 12153  0cn0 12413  cz 12499  cuz 12763  [,]cicc 13267  ...cfz 13424  ..^cfzo 13567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-icc 13271  df-fz 13425  df-fzo 13568
This theorem is referenced by:  fourierdlem38  44376  fourierdlem50  44387  fourierdlem54  44391  fourierdlem63  44400  fourierdlem65  44402  fourierdlem69  44406  fourierdlem70  44407  fourierdlem74  44411  fourierdlem75  44412  fourierdlem76  44413  fourierdlem79  44416  fourierdlem81  44418  fourierdlem84  44421  fourierdlem85  44422  fourierdlem88  44425  fourierdlem89  44426  fourierdlem90  44427  fourierdlem91  44428  fourierdlem92  44429  fourierdlem93  44430  fourierdlem100  44437  fourierdlem101  44438  fourierdlem103  44440  fourierdlem104  44441  fourierdlem107  44444  fourierdlem111  44448  fourierdlem112  44449  fourierdlem113  44450
  Copyright terms: Public domain W3C validator