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Theorem fourierdlem15 40657
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 40644 . . . . . . 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 222 . . . . 5 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simpld 474 . . . 4 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
8 reex 10065 . . . . . 6 ℝ ∈ V
98a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
10 ovex 6718 . . . . . 6 (0...𝑀) ∈ V
1110a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
129, 11elmapd 7913 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
137, 12mpbid 222 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
14 ffn 6083 . . 3 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
1513, 14syl 17 . 2 (𝜑𝑄 Fn (0...𝑀))
166simprd 478 . . . . . . . . 9 (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
1716simpld 474 . . . . . . . 8 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
1817simpld 474 . . . . . . 7 (𝜑 → (𝑄‘0) = 𝐴)
19 nnnn0 11337 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
20 nn0uz 11760 . . . . . . . . . . 11 0 = (ℤ‘0)
2119, 20syl6eleq 2740 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ‘0))
222, 21syl 17 . . . . . . . . 9 (𝜑𝑀 ∈ (ℤ‘0))
23 eluzfz1 12386 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
2422, 23syl 17 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
2513, 24ffvelrnd 6400 . . . . . . 7 (𝜑 → (𝑄‘0) ∈ ℝ)
2618, 25eqeltrrd 2731 . . . . . 6 (𝜑𝐴 ∈ ℝ)
2726adantr 480 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
2817simprd 478 . . . . . . 7 (𝜑 → (𝑄𝑀) = 𝐵)
29 eluzfz2 12387 . . . . . . . . 9 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
3022, 29syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
3113, 30ffvelrnd 6400 . . . . . . 7 (𝜑 → (𝑄𝑀) ∈ ℝ)
3228, 31eqeltrrd 2731 . . . . . 6 (𝜑𝐵 ∈ ℝ)
3332adantr 480 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
3413ffvelrnda 6399 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
3518eqcomd 2657 . . . . . . 7 (𝜑𝐴 = (𝑄‘0))
3635adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37 elfzuz 12376 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ‘0))
3837adantl 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ‘0))
3913ad2antrr 762 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40 elfzle1 12382 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
4140adantl 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
42 elfzelz 12380 . . . . . . . . . . . . 13 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
4342zred 11520 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
4443adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
45 elfzelz 12380 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
4645zred 11520 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
4746adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
48 elfzel2 12378 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
4948zred 11520 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
5049adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
51 elfzle2 12383 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
5251adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑖)
53 elfzle2 12383 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖𝑀)
5453adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖𝑀)
5544, 47, 50, 52, 54letrd 10232 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗𝑀)
5642adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
57 0zd 11427 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
5848adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
59 elfz 12370 . . . . . . . . . . 11 ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0...𝑀) ↔ (0 ≤ 𝑗𝑗𝑀)))
6056, 57, 58, 59syl3anc 1366 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → (𝑗 ∈ (0...𝑀) ↔ (0 ≤ 𝑗𝑗𝑀)))
6141, 55, 60mpbir2and 977 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6261adantll 750 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
6339, 62ffvelrnd 6400 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄𝑗) ∈ ℝ)
64 simpll 805 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
65 elfzle1 12382 . . . . . . . . . . 11 (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
6665adantl 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
67 elfzelz 12380 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
6867zred 11520 . . . . . . . . . . . 12 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℝ)
6968adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℝ)
7046adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ∈ ℝ)
7149adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℝ)
72 peano2rem 10386 . . . . . . . . . . . . 13 (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
7370, 72syl 17 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
74 elfzle2 12383 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
7574adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
7670ltm1d 10994 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
7769, 73, 70, 75, 76lelttrd 10233 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑖)
7853adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖𝑀)
7969, 70, 71, 77, 78ltletrd 10235 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
8067adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
81 0zd 11427 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
8248adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
83 elfzo 12511 . . . . . . . . . . 11 ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
8480, 81, 82, 83syl3anc 1366 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
8566, 79, 84mpbir2and 977 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
8685adantll 750 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
8713adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
88 elfzofz 12524 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
8988adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
9087, 89ffvelrnd 6400 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) ∈ ℝ)
91 fzofzp1 12605 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
9291adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
9387, 92ffvelrnd 6400 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
94 eleq1 2718 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9594anbi2d 740 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
96 fveq2 6229 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
97 oveq1 6697 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
9897fveq2d 6233 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9996, 98breq12d 4698 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑄𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄𝑗) < (𝑄‘(𝑗 + 1))))
10095, 99imbi12d 333 . . . . . . . . . 10 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))))
10116simprd 478 . . . . . . . . . . 11 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
102101r19.21bi 2961 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
103100, 102chvarv 2299 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))
10490, 93, 103ltled 10223 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
10564, 86, 104syl2anc 694 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
10638, 63, 105monoord 12871 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄𝑖))
10736, 106eqbrtrd 4707 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄𝑖))
108 elfzuz3 12377 . . . . . . . 8 (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ𝑖))
109108adantl 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ𝑖))
11013ad2antrr 762 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
111 fz0fzelfz0 12484 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
112111adantll 750 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
113110, 112ffvelrnd 6400 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → (𝑄𝑗) ∈ ℝ)
11413ad2antrr 762 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
115 0red 10079 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
11646adantr 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
117 elfzelz 12380 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
118117zred 11520 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
119118adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
120 elfzle1 12382 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
121120adantr 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
122 elfzle1 12382 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖𝑗)
123122adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖𝑗)
124115, 116, 119, 121, 123letrd 10232 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
125124adantll 750 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126118adantl 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
1272nnred 11073 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℝ)
128127adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
129 1red 10093 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
130128, 129resubcld 10496 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
131 elfzle2 12383 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
132131adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
133128ltm1d 10994 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
134126, 130, 128, 132, 133lelttrd 10233 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
135126, 128, 134ltled 10223 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
136135adantlr 751 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗𝑀)
137117adantl 481 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
138 0zd 11427 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
13948ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
140137, 138, 139, 59syl3anc 1366 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0...𝑀) ↔ (0 ≤ 𝑗𝑗𝑀)))
141125, 136, 140mpbir2and 977 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
142114, 141ffvelrnd 6400 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ∈ ℝ)
143118adantl 481 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
144 1red 10093 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
145 0le1 10589 . . . . . . . . . . . 12 0 ≤ 1
146145a1i 11 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
147143, 144, 125, 146addge0d 10641 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
148126, 130, 129, 132leadd1dd 10679 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
1492nncnd 11074 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℂ)
150149adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
151 1cnd 10094 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
152150, 151npcand 10434 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
153148, 152breqtrd 4711 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
154153adantlr 751 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
155137peano2zd 11523 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ ℤ)
156 elfz 12370 . . . . . . . . . . 11 (((𝑗 + 1) ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑗 + 1) ∈ (0...𝑀) ↔ (0 ≤ (𝑗 + 1) ∧ (𝑗 + 1) ≤ 𝑀)))
157155, 138, 139, 156syl3anc 1366 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑗 + 1) ∈ (0...𝑀) ↔ (0 ≤ (𝑗 + 1) ∧ (𝑗 + 1) ≤ 𝑀)))
158147, 154, 157mpbir2and 977 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
159114, 158ffvelrnd 6400 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
160 simpll 805 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝜑)
161134adantlr 751 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
162137, 138, 139, 83syl3anc 1366 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗𝑗 < 𝑀)))
163125, 161, 162mpbir2and 977 . . . . . . . . 9 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0..^𝑀))
164160, 163, 103syl2anc 694 . . . . . . . 8 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) < (𝑄‘(𝑗 + 1)))
165142, 159, 164ltled 10223 . . . . . . 7 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄𝑗) ≤ (𝑄‘(𝑗 + 1)))
166109, 113, 165monoord 12871 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ (𝑄𝑀))
16728adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑀) = 𝐵)
168166, 167breqtrd 4711 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ≤ 𝐵)
16927, 33, 34, 107, 168eliccd 40044 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ (𝐴[,]𝐵))
170169ralrimiva 2995 . . 3 (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄𝑖) ∈ (𝐴[,]𝐵))
171 fnfvrnss 6430 . . 3 ((𝑄 Fn (0...𝑀) ∧ ∀𝑖 ∈ (0...𝑀)(𝑄𝑖) ∈ (𝐴[,]𝐵)) → ran 𝑄 ⊆ (𝐴[,]𝐵))
17215, 170, 171syl2anc 694 . 2 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
173 df-f 5930 . 2 (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 ⊆ (𝐴[,]𝐵)))
17415, 172, 173sylanbrc 699 1 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  wss 3607   class class class wbr 4685  cmpt 4762  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  cn 11058  0cn0 11330  cz 11415  cuz 11725  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-icc 12220  df-fz 12365  df-fzo 12505
This theorem is referenced by:  fourierdlem38  40680  fourierdlem50  40691  fourierdlem54  40695  fourierdlem63  40704  fourierdlem65  40706  fourierdlem69  40710  fourierdlem70  40711  fourierdlem74  40715  fourierdlem75  40716  fourierdlem76  40717  fourierdlem79  40720  fourierdlem81  40722  fourierdlem84  40725  fourierdlem85  40726  fourierdlem88  40729  fourierdlem89  40730  fourierdlem90  40731  fourierdlem91  40732  fourierdlem92  40733  fourierdlem93  40734  fourierdlem100  40741  fourierdlem101  40742  fourierdlem103  40744  fourierdlem104  40745  fourierdlem107  40748  fourierdlem111  40752  fourierdlem112  40753  fourierdlem113  40754
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