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Theorem fourierdlem15 43338
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 43325 . . . . . . 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 235 . . . . 5 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simpld 498 . . . 4 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
8 reex 10820 . . . . . 6 ℝ ∈ V
98a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
10 ovex 7246 . . . . . 6 (0...𝑀) ∈ V
1110a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
129, 11elmapd 8522 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
137, 12mpbid 235 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
14 ffn 6545 . . 3 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
1513, 14syl 17 . 2 (𝜑𝑄 Fn (0...𝑀))
166simprd 499 . . . . . . . . 9 (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
1716simpld 498 . . . . . . . 8 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
1817simpld 498 . . . . . . 7 (𝜑 → (𝑄‘0) = 𝐴)
19 nnnn0 12097 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
20 nn0uz 12476 . . . . . . . . . . 11 0 = (ℤ‘0)
2119, 20eleqtrdi 2848 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ‘0))
222, 21syl 17 . . . . . . . . 9 (𝜑𝑀 ∈ (ℤ‘0))
23 eluzfz1 13119 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
2422, 23syl 17 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
2513, 24ffvelrnd 6905 . . . . . . 7 (𝜑 → (𝑄‘0) ∈ ℝ)
2618, 25eqeltrrd 2839 . . . . . 6 (𝜑𝐴 ∈ ℝ)
2726adantr 484 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
2817simprd 499 . . . . . . 7 (𝜑 → (𝑄𝑀) = 𝐵)
29 eluzfz2 13120 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
3022, 29syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
3113, 30ffvelrnd 6905 . . . . . . 7 (𝜑 → (𝑄𝑀) ∈ ℝ)
3228, 31eqeltrrd 2839 . . . . . 6 (𝜑𝐵 ∈ ℝ)
3332adantr 484 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
3413ffvelrnda 6904 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
3518eqcomd 2743 . . . . . . 7 (𝜑𝐴 = (𝑄‘0))
3635adantr 484 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37 elfzuz 13108 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ‘0))
3837adantl 485 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ‘0))
3913ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40 0zd 12188 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41 elfzel2 13110 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
4241adantr 484 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43 elfzelz 13112 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
4443adantl 485 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45 elfzle1 13115 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
4645adantl 485 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
4743zred 12282 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
4847adantl 485 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49 elfzelz 13112 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
5049zred 12282 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
5150adantr 484 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
5241zred 12282 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
5352adantr 484 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54 elfzle2 13116 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
5554adantl 485 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑖)
56 elfzle2 13116 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖𝑀)
5756adantr 484 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖𝑀)
5848, 51, 53, 55, 57letrd 10989 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑀)
5940, 42, 44, 46, 58elfzd 13103 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6059adantll 714 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6139, 60ffvelrnd 6905 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄𝑗) ∈ ℝ)
62 simpll 767 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
63 elfzle1 13115 . . . . . . . . . . 11 (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
6463adantl 485 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65 elfzelz 13112 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
6665zred 12282 . . . . . . . . . . . 12 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℝ)
6766adantl 485 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℝ)
6850adantr 484 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ∈ ℝ)
6952adantr 484 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℝ)
70 peano2rem 11145 . . . . . . . . . . . . 13 (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
7168, 70syl 17 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72 elfzle2 13116 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
7372adantl 485 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
7468ltm1d 11764 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
7567, 71, 68, 73, 74lelttrd 10990 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑖)
7656adantr 484 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖𝑀)
7767, 68, 69, 75, 76ltletrd 10992 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
7865adantl 485 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
79 0zd 12188 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
8041adantr 484 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
81 elfzo 13245 . . . . . . . . . . 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 484 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86 elfzofz 13258 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
8786adantl 485 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
8885, 87ffvelrnd 6905 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) ∈ ℝ)
89 fzofzp1 13339 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
9089adantl 485 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
9185, 90ffvelrnd 6905 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92 eleq1w 2820 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9392anbi2d 632 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
94 fveq2 6717 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
95 oveq1 7220 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
9695fveq2d 6721 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9794, 96breq12d 5066 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑄𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄𝑗) < (𝑄‘(𝑗 + 1))))
9893, 97imbi12d 348 . . . . . . . . . 10 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))))
9916simprd 499 . . . . . . . . . . 11 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
10099r19.21bi 3130 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
10198, 100chvarvv 2007 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))
10288, 91, 101ltled 10980 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
10362, 84, 102syl2anc 587 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
10438, 61, 103monoord 13606 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄𝑖))
10536, 104eqbrtrd 5075 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄𝑖))
106 elfzuz3 13109 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ𝑖))
107106adantl 485 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ𝑖))
10813ad2antrr 726 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109 fz0fzelfz0 13218 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
110109adantll 714 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
111108, 110ffvelrnd 6905 . . . . . . 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 12188 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
11441ad2antlr 727 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
115 elfzelz 13112 . . . . . . . . . . 11 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116115adantl 485 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117 0red 10836 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
11850adantr 484 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
119115zred 12282 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120119adantl 485 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121 elfzle1 13115 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122121adantr 484 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123 elfzle1 13115 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖𝑗)
124123adantl 485 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖𝑗)
125117, 118, 120, 122, 124letrd 10989 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126125adantll 714 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
127119adantl 485 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
1282nnred 11845 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℝ)
129128adantr 484 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130 1red 10834 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131129, 130resubcld 11260 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132 elfzle2 13116 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133132adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134129ltm1d 11764 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135127, 131, 129, 133, 134lelttrd 10990 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136127, 129, 135ltled 10980 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
137136adantlr 715 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
138113, 114, 116, 126, 137elfzd 13103 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139112, 138ffvelrnd 6905 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ∈ ℝ)
140116peano2zd 12285 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ ℤ)
141119adantl 485 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
142 1red 10834 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143 0le1 11355 . . . . . . . . . . . 12 0 ≤ 1
144143a1i 11 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145141, 142, 126, 144addge0d 11408 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146127, 131, 130, 133leadd1dd 11446 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
1472nncnd 11846 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℂ)
148147adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149 1cnd 10828 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150148, 149npcand 11193 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151146, 150breqtrd 5079 . . . . . . . . . . 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 13103 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154112, 153ffvelrnd 6905 . . . . . . . 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 587 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))
160139, 154, 159ltled 10980 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
161107, 111, 160monoord 13606 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ (𝑄𝑀))
16228adantr 484 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑀) = 𝐵)
163161, 162breqtrd 5079 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ 𝐵)
16427, 33, 34, 105, 163eliccd 42717 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ (𝐴[,]𝐵))
165164ralrimiva 3105 . . 3 (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄𝑖) ∈ (𝐴[,]𝐵))
166 fnfvrnss 6937 . . 3 ((𝑄 Fn (0...𝑀) ∧ ∀𝑖 ∈ (0...𝑀)(𝑄𝑖) ∈ (𝐴[,]𝐵)) → ran 𝑄 ⊆ (𝐴[,]𝐵))
16715, 165, 166syl2anc 587 . 2 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
168 df-f 6384 . 2 (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 ⊆ (𝐴[,]𝐵)))
16915, 167, 168sylanbrc 586 1 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  {crab 3065  Vcvv 3408  wss 3866   class class class wbr 5053  cmpt 5135  ran crn 5552   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  m cmap 8508  cc 10727  cr 10728  0cc0 10729  1c1 10730   + caddc 10732   < clt 10867  cle 10868  cmin 11062  cn 11830  0cn0 12090  cz 12176  cuz 12438  [,]cicc 12938  ...cfz 13095  ..^cfzo 13238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-icc 12942  df-fz 13096  df-fzo 13239
This theorem is referenced by:  fourierdlem38  43361  fourierdlem50  43372  fourierdlem54  43376  fourierdlem63  43385  fourierdlem65  43387  fourierdlem69  43391  fourierdlem70  43392  fourierdlem74  43396  fourierdlem75  43397  fourierdlem76  43398  fourierdlem79  43401  fourierdlem81  43403  fourierdlem84  43406  fourierdlem85  43407  fourierdlem88  43410  fourierdlem89  43411  fourierdlem90  43412  fourierdlem91  43413  fourierdlem92  43414  fourierdlem93  43415  fourierdlem100  43422  fourierdlem101  43423  fourierdlem103  43425  fourierdlem104  43426  fourierdlem107  43429  fourierdlem111  43433  fourierdlem112  43434  fourierdlem113  43435
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