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Theorem fourierdlem15 46137
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 46124 . . . . . . 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 232 . . . . 5 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simpld 494 . . . 4 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
8 reex 11246 . . . . . 6 ℝ ∈ V
98a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
10 ovex 7464 . . . . . 6 (0...𝑀) ∈ V
1110a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
129, 11elmapd 8880 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
137, 12mpbid 232 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
14 ffn 6736 . . 3 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
1513, 14syl 17 . 2 (𝜑𝑄 Fn (0...𝑀))
166simprd 495 . . . . . . . . 9 (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
1716simpld 494 . . . . . . . 8 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
1817simpld 494 . . . . . . 7 (𝜑 → (𝑄‘0) = 𝐴)
19 nnnn0 12533 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
20 nn0uz 12920 . . . . . . . . . . 11 0 = (ℤ‘0)
2119, 20eleqtrdi 2851 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ‘0))
222, 21syl 17 . . . . . . . . 9 (𝜑𝑀 ∈ (ℤ‘0))
23 eluzfz1 13571 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
2422, 23syl 17 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
2513, 24ffvelcdmd 7105 . . . . . . 7 (𝜑 → (𝑄‘0) ∈ ℝ)
2618, 25eqeltrrd 2842 . . . . . 6 (𝜑𝐴 ∈ ℝ)
2726adantr 480 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
2817simprd 495 . . . . . . 7 (𝜑 → (𝑄𝑀) = 𝐵)
29 eluzfz2 13572 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
3022, 29syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
3113, 30ffvelcdmd 7105 . . . . . . 7 (𝜑 → (𝑄𝑀) ∈ ℝ)
3228, 31eqeltrrd 2842 . . . . . 6 (𝜑𝐵 ∈ ℝ)
3332adantr 480 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
3413ffvelcdmda 7104 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
3518eqcomd 2743 . . . . . . 7 (𝜑𝐴 = (𝑄‘0))
3635adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37 elfzuz 13560 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ‘0))
3837adantl 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ‘0))
3913ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40 0zd 12625 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41 elfzel2 13562 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
4241adantr 480 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43 elfzelz 13564 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
4443adantl 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45 elfzle1 13567 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
4645adantl 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
4743zred 12722 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
4847adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49 elfzelz 13564 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
5049zred 12722 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
5150adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
5241zred 12722 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
5352adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54 elfzle2 13568 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
5554adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑖)
56 elfzle2 13568 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖𝑀)
5756adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖𝑀)
5848, 51, 53, 55, 57letrd 11418 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑀)
5940, 42, 44, 46, 58elfzd 13555 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6059adantll 714 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6139, 60ffvelcdmd 7105 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄𝑗) ∈ ℝ)
62 simpll 767 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
63 elfzle1 13567 . . . . . . . . . . 11 (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
6463adantl 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65 elfzelz 13564 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
6665zred 12722 . . . . . . . . . . . 12 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℝ)
6766adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℝ)
6850adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ∈ ℝ)
6952adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℝ)
70 peano2rem 11576 . . . . . . . . . . . . 13 (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
7168, 70syl 17 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72 elfzle2 13568 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
7372adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
7468ltm1d 12200 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
7567, 71, 68, 73, 74lelttrd 11419 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑖)
7656adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖𝑀)
7767, 68, 69, 75, 76ltletrd 11421 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
7865adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
79 0zd 12625 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
8041adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
81 elfzo 13701 . . . . . . . . . . 11 ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
8278, 79, 80, 81syl3anc 1373 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
8364, 77, 82mpbir2and 713 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
8483adantll 714 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
8513adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86 elfzofz 13715 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
8786adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
8885, 87ffvelcdmd 7105 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) ∈ ℝ)
89 fzofzp1 13803 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
9089adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
9185, 90ffvelcdmd 7105 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92 eleq1w 2824 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9392anbi2d 630 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
94 fveq2 6906 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
95 oveq1 7438 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
9695fveq2d 6910 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9794, 96breq12d 5156 . . . . . . . . . . 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 495 . . . . . . . . . . 11 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
10099r19.21bi 3251 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
10198, 100chvarvv 1998 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))
10288, 91, 101ltled 11409 . . . . . . . 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 14073 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄𝑖))
10536, 104eqbrtrd 5165 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄𝑖))
106 elfzuz3 13561 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ𝑖))
107106adantl 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ𝑖))
10813ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109 fz0fzelfz0 13674 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
110109adantll 714 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
111108, 110ffvelcdmd 7105 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → (𝑄𝑗) ∈ ℝ)
11213ad2antrr 726 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
113 0zd 12625 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
11441ad2antlr 727 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
115 elfzelz 13564 . . . . . . . . . . 11 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116115adantl 481 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117 0red 11264 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
11850adantr 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
119115zred 12722 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120119adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121 elfzle1 13567 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122121adantr 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123 elfzle1 13567 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖𝑗)
124123adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖𝑗)
125117, 118, 120, 122, 124letrd 11418 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126125adantll 714 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
127119adantl 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
1282nnred 12281 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℝ)
129128adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130 1red 11262 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131129, 130resubcld 11691 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132 elfzle2 13568 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133132adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134129ltm1d 12200 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135127, 131, 129, 133, 134lelttrd 11419 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136127, 129, 135ltled 11409 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
137136adantlr 715 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
138113, 114, 116, 126, 137elfzd 13555 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139112, 138ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ∈ ℝ)
140116peano2zd 12725 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ ℤ)
141119adantl 481 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
142 1red 11262 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143 0le1 11786 . . . . . . . . . . . 12 0 ≤ 1
144143a1i 11 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145141, 142, 126, 144addge0d 11839 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146127, 131, 130, 133leadd1dd 11877 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
1472nncnd 12282 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℂ)
148147adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149 1cnd 11256 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150148, 149npcand 11624 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151146, 150breqtrd 5169 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
152151adantlr 715 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
153113, 114, 140, 145, 152elfzd 13555 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154112, 153ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
155 simpll 767 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝜑)
156135adantlr 715 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
157116, 113, 114, 81syl3anc 1373 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
158126, 156, 157mpbir2and 713 . . . . . . . . 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 11409 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
161107, 111, 160monoord 14073 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ (𝑄𝑀))
16228adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑀) = 𝐵)
163161, 162breqtrd 5169 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ 𝐵)
16427, 33, 34, 105, 163eliccd 45517 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ (𝐴[,]𝐵))
165164ralrimiva 3146 . . 3 (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄𝑖) ∈ (𝐴[,]𝐵))
166 fnfvrnss 7141 . . 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 6565 . 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 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  cn 12266  0cn0 12526  cz 12613  cuz 12878  [,]cicc 13390  ...cfz 13547  ..^cfzo 13694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-icc 13394  df-fz 13548  df-fzo 13695
This theorem is referenced by:  fourierdlem38  46160  fourierdlem50  46171  fourierdlem54  46175  fourierdlem63  46184  fourierdlem65  46186  fourierdlem69  46190  fourierdlem70  46191  fourierdlem74  46195  fourierdlem75  46196  fourierdlem76  46197  fourierdlem79  46200  fourierdlem81  46202  fourierdlem84  46205  fourierdlem85  46206  fourierdlem88  46209  fourierdlem89  46210  fourierdlem90  46211  fourierdlem91  46212  fourierdlem92  46213  fourierdlem93  46214  fourierdlem100  46221  fourierdlem101  46222  fourierdlem103  46224  fourierdlem104  46225  fourierdlem107  46228  fourierdlem111  46232  fourierdlem112  46233  fourierdlem113  46234
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