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Theorem fourierdlem15 43988
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 43975 . . . . . . 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 11055 . . . . . 6 ℝ ∈ V
98a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
10 ovex 7362 . . . . . 6 (0...𝑀) ∈ V
1110a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
129, 11elmapd 8692 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
137, 12mpbid 231 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
14 ffn 6645 . . 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 12333 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
20 nn0uz 12713 . . . . . . . . . . 11 0 = (ℤ‘0)
2119, 20eleqtrdi 2847 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ‘0))
222, 21syl 17 . . . . . . . . 9 (𝜑𝑀 ∈ (ℤ‘0))
23 eluzfz1 13356 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
2422, 23syl 17 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
2513, 24ffvelcdmd 7012 . . . . . . 7 (𝜑 → (𝑄‘0) ∈ ℝ)
2618, 25eqeltrrd 2838 . . . . . 6 (𝜑𝐴 ∈ ℝ)
2726adantr 481 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
2817simprd 496 . . . . . . 7 (𝜑 → (𝑄𝑀) = 𝐵)
29 eluzfz2 13357 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
3022, 29syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
3113, 30ffvelcdmd 7012 . . . . . . 7 (𝜑 → (𝑄𝑀) ∈ ℝ)
3228, 31eqeltrrd 2838 . . . . . 6 (𝜑𝐵 ∈ ℝ)
3332adantr 481 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
3413ffvelcdmda 7011 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
3518eqcomd 2742 . . . . . . 7 (𝜑𝐴 = (𝑄‘0))
3635adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37 elfzuz 13345 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ‘0))
3837adantl 482 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ‘0))
3913ad2antrr 723 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40 0zd 12424 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41 elfzel2 13347 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
4241adantr 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43 elfzelz 13349 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
4443adantl 482 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45 elfzle1 13352 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
4645adantl 482 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
4743zred 12519 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
4847adantl 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49 elfzelz 13349 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
5049zred 12519 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
5150adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
5241zred 12519 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
5352adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54 elfzle2 13353 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
5554adantl 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑖)
56 elfzle2 13353 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖𝑀)
5756adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖𝑀)
5848, 51, 53, 55, 57letrd 11225 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑀)
5940, 42, 44, 46, 58elfzd 13340 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6059adantll 711 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6139, 60ffvelcdmd 7012 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄𝑗) ∈ ℝ)
62 simpll 764 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
63 elfzle1 13352 . . . . . . . . . . 11 (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
6463adantl 482 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65 elfzelz 13349 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
6665zred 12519 . . . . . . . . . . . 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 11381 . . . . . . . . . . . . 13 (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
7168, 70syl 17 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72 elfzle2 13353 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
7372adantl 482 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
7468ltm1d 12000 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
7567, 71, 68, 73, 74lelttrd 11226 . . . . . . . . . . 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 11228 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
7865adantl 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
79 0zd 12424 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
8041adantr 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
81 elfzo 13482 . . . . . . . . . . 11 ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
8278, 79, 80, 81syl3anc 1370 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
8364, 77, 82mpbir2and 710 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
8483adantll 711 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
8513adantr 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86 elfzofz 13496 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
8786adantl 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
8885, 87ffvelcdmd 7012 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) ∈ ℝ)
89 fzofzp1 13577 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
9089adantl 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
9185, 90ffvelcdmd 7012 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92 eleq1w 2819 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9392anbi2d 629 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
94 fveq2 6819 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
95 oveq1 7336 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
9695fveq2d 6823 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9794, 96breq12d 5102 . . . . . . . . . . 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 3230 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
10198, 100chvarvv 2001 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))
10288, 91, 101ltled 11216 . . . . . . . 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 13846 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄𝑖))
10536, 104eqbrtrd 5111 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄𝑖))
106 elfzuz3 13346 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ𝑖))
107106adantl 482 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ𝑖))
10813ad2antrr 723 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109 fz0fzelfz0 13455 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
110109adantll 711 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
111108, 110ffvelcdmd 7012 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → (𝑄𝑗) ∈ ℝ)
11213ad2antrr 723 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
113 0zd 12424 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
11441ad2antlr 724 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
115 elfzelz 13349 . . . . . . . . . . 11 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116115adantl 482 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117 0red 11071 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
11850adantr 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
119115zred 12519 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120119adantl 482 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121 elfzle1 13352 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122121adantr 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123 elfzle1 13352 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖𝑗)
124123adantl 482 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖𝑗)
125117, 118, 120, 122, 124letrd 11225 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126125adantll 711 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
127119adantl 482 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
1282nnred 12081 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℝ)
129128adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130 1red 11069 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131129, 130resubcld 11496 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132 elfzle2 13353 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133132adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134129ltm1d 12000 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135127, 131, 129, 133, 134lelttrd 11226 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136127, 129, 135ltled 11216 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
137136adantlr 712 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
138113, 114, 116, 126, 137elfzd 13340 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139112, 138ffvelcdmd 7012 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ∈ ℝ)
140116peano2zd 12522 . . . . . . . . . 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 11069 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143 0le1 11591 . . . . . . . . . . . 12 0 ≤ 1
144143a1i 11 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145141, 142, 126, 144addge0d 11644 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146127, 131, 130, 133leadd1dd 11682 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
1472nncnd 12082 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℂ)
148147adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149 1cnd 11063 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150148, 149npcand 11429 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151146, 150breqtrd 5115 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
152151adantlr 712 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
153113, 114, 140, 145, 152elfzd 13340 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154112, 153ffvelcdmd 7012 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
155 simpll 764 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝜑)
156135adantlr 712 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
157116, 113, 114, 81syl3anc 1370 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
158126, 156, 157mpbir2and 710 . . . . . . . . 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 11216 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
161107, 111, 160monoord 13846 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ (𝑄𝑀))
16228adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑀) = 𝐵)
163161, 162breqtrd 5115 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ 𝐵)
16427, 33, 34, 105, 163eliccd 43367 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ (𝐴[,]𝐵))
165164ralrimiva 3139 . . 3 (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄𝑖) ∈ (𝐴[,]𝐵))
166 fnfvrnss 7044 . . 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 6477 . 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 1540  wcel 2105  wral 3061  {crab 3403  Vcvv 3441  wss 3897   class class class wbr 5089  cmpt 5172  ran crn 5615   Fn wfn 6468  wf 6469  cfv 6473  (class class class)co 7329  m cmap 8678  cc 10962  cr 10963  0cc0 10964  1c1 10965   + caddc 10967   < clt 11102  cle 11103  cmin 11298  cn 12066  0cn0 12326  cz 12412  cuz 12675  [,]cicc 13175  ...cfz 13332  ..^cfzo 13475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-er 8561  df-map 8680  df-en 8797  df-dom 8798  df-sdom 8799  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-n0 12327  df-z 12413  df-uz 12676  df-icc 13179  df-fz 13333  df-fzo 13476
This theorem is referenced by:  fourierdlem38  44011  fourierdlem50  44022  fourierdlem54  44026  fourierdlem63  44035  fourierdlem65  44037  fourierdlem69  44041  fourierdlem70  44042  fourierdlem74  44046  fourierdlem75  44047  fourierdlem76  44048  fourierdlem79  44051  fourierdlem81  44053  fourierdlem84  44056  fourierdlem85  44057  fourierdlem88  44060  fourierdlem89  44061  fourierdlem90  44062  fourierdlem91  44063  fourierdlem92  44064  fourierdlem93  44065  fourierdlem100  44072  fourierdlem101  44073  fourierdlem103  44075  fourierdlem104  44076  fourierdlem107  44079  fourierdlem111  44083  fourierdlem112  44084  fourierdlem113  44085
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