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Theorem fourierdlem54 45686
Description: Given a partition 𝑄 and an arbitrary interval [𝐶, 𝐷], a partition 𝑆 on [𝐶, 𝐷] is built such that it preserves any periodic function piecewise continuous on 𝑄 will be piecewise continuous on 𝑆, with the same limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem54.t 𝑇 = (𝐵𝐴)
fourierdlem54.p 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem54.m (𝜑𝑀 ∈ ℕ)
fourierdlem54.q (𝜑𝑄 ∈ (𝑃𝑀))
fourierdlem54.c (𝜑𝐶 ∈ ℝ)
fourierdlem54.d (𝜑𝐷 ∈ ℝ)
fourierdlem54.cd (𝜑𝐶 < 𝐷)
fourierdlem54.o 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem54.h 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem54.n 𝑁 = ((♯‘𝐻) − 1)
fourierdlem54.s 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
Assertion
Ref Expression
fourierdlem54 (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
Distinct variable groups:   𝐴,𝑖,𝑚,𝑝   𝐵,𝑖,𝑚,𝑝   𝐶,𝑚,𝑝   𝑥,𝐶   𝐷,𝑚,𝑝   𝑥,𝐷   𝑓,𝐻   𝑥,𝐻   𝑖,𝑀,𝑚,𝑝   𝑓,𝑁   𝑖,𝑁,𝑚,𝑝   𝑥,𝑁,𝑖   𝑄,𝑖,𝑘   𝑄,𝑝   𝑥,𝑄,𝑘   𝑆,𝑓   𝑆,𝑖,𝑝   𝑥,𝑆   𝑇,𝑖,𝑘,𝑥   𝜑,𝑓   𝜑,𝑖,𝑘
Allowed substitution hints:   𝜑(𝑥,𝑚,𝑝)   𝐴(𝑥,𝑓,𝑘)   𝐵(𝑥,𝑓,𝑘)   𝐶(𝑓,𝑖,𝑘)   𝐷(𝑓,𝑖,𝑘)   𝑃(𝑥,𝑓,𝑖,𝑘,𝑚,𝑝)   𝑄(𝑓,𝑚)   𝑆(𝑘,𝑚)   𝑇(𝑓,𝑚,𝑝)   𝐻(𝑖,𝑘,𝑚,𝑝)   𝑀(𝑥,𝑓,𝑘)   𝑁(𝑘)   𝑂(𝑥,𝑓,𝑖,𝑘,𝑚,𝑝)

Proof of Theorem fourierdlem54
Dummy variables 𝑤 𝑦 𝑧 𝑗 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem54.n . . 3 𝑁 = ((♯‘𝐻) − 1)
2 2z 12627 . . . . . 6 2 ∈ ℤ
32a1i 11 . . . . 5 (𝜑 → 2 ∈ ℤ)
4 fourierdlem54.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ)
5 prid1g 4766 . . . . . . . . . 10 (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶, 𝐷})
6 elun1 4174 . . . . . . . . . 10 (𝐶 ∈ {𝐶, 𝐷} → 𝐶 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
74, 5, 63syl 18 . . . . . . . . 9 (𝜑𝐶 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
8 fourierdlem54.h . . . . . . . . 9 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
97, 8eleqtrrdi 2836 . . . . . . . 8 (𝜑𝐶𝐻)
109ne0d 4335 . . . . . . 7 (𝜑𝐻 ≠ ∅)
11 prfi 9348 . . . . . . . . . 10 {𝐶, 𝐷} ∈ Fin
12 fourierdlem54.p . . . . . . . . . . . . 13 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
13 fourierdlem54.m . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℕ)
14 fourierdlem54.q . . . . . . . . . . . . 13 (𝜑𝑄 ∈ (𝑃𝑀))
1512, 13, 14fourierdlem11 45644 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
1615simp1d 1139 . . . . . . . . . . 11 (𝜑𝐴 ∈ ℝ)
1715simp2d 1140 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℝ)
1815simp3d 1141 . . . . . . . . . . 11 (𝜑𝐴 < 𝐵)
19 fourierdlem54.t . . . . . . . . . . 11 𝑇 = (𝐵𝐴)
2012, 13, 14fourierdlem15 45648 . . . . . . . . . . . 12 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
21 frn 6730 . . . . . . . . . . . 12 (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) → ran 𝑄 ⊆ (𝐴[,]𝐵))
2220, 21syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
2312fourierdlem2 45635 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
2413, 23syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
2514, 24mpbid 231 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
2625simpld 493 . . . . . . . . . . . . . 14 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
27 elmapi 8868 . . . . . . . . . . . . . 14 (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
28 ffn 6723 . . . . . . . . . . . . . 14 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
2926, 27, 283syl 18 . . . . . . . . . . . . 13 (𝜑𝑄 Fn (0...𝑀))
30 fzfid 13974 . . . . . . . . . . . . 13 (𝜑 → (0...𝑀) ∈ Fin)
31 fnfi 9206 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ (0...𝑀) ∈ Fin) → 𝑄 ∈ Fin)
3229, 30, 31syl2anc 582 . . . . . . . . . . . 12 (𝜑𝑄 ∈ Fin)
33 rnfi 9361 . . . . . . . . . . . 12 (𝑄 ∈ Fin → ran 𝑄 ∈ Fin)
3432, 33syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ∈ Fin)
3525simprd 494 . . . . . . . . . . . . . 14 (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
3635simpld 493 . . . . . . . . . . . . 13 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
3736simpld 493 . . . . . . . . . . . 12 (𝜑 → (𝑄‘0) = 𝐴)
3813nnnn0d 12565 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℕ0)
39 nn0uz 12897 . . . . . . . . . . . . . . 15 0 = (ℤ‘0)
4038, 39eleqtrdi 2835 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ‘0))
41 eluzfz1 13543 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
4240, 41syl 17 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ (0...𝑀))
43 fnfvelrn 7089 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ 0 ∈ (0...𝑀)) → (𝑄‘0) ∈ ran 𝑄)
4429, 42, 43syl2anc 582 . . . . . . . . . . . 12 (𝜑 → (𝑄‘0) ∈ ran 𝑄)
4537, 44eqeltrrd 2826 . . . . . . . . . . 11 (𝜑𝐴 ∈ ran 𝑄)
4636simprd 494 . . . . . . . . . . . 12 (𝜑 → (𝑄𝑀) = 𝐵)
47 eluzfz2 13544 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
4840, 47syl 17 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (0...𝑀))
49 fnfvelrn 7089 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)) → (𝑄𝑀) ∈ ran 𝑄)
5029, 48, 49syl2anc 582 . . . . . . . . . . . 12 (𝜑 → (𝑄𝑀) ∈ ran 𝑄)
5146, 50eqeltrrd 2826 . . . . . . . . . . 11 (𝜑𝐵 ∈ ran 𝑄)
52 eqid 2725 . . . . . . . . . . 11 (abs ∘ − ) = (abs ∘ − )
53 eqid 2725 . . . . . . . . . . 11 ((ran 𝑄 × ran 𝑄) ∖ I ) = ((ran 𝑄 × ran 𝑄) ∖ I )
54 eqid 2725 . . . . . . . . . . 11 ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )) = ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I ))
55 eqid 2725 . . . . . . . . . . 11 inf(ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )), ℝ, < ) = inf(ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )), ℝ, < )
56 fourierdlem54.d . . . . . . . . . . 11 (𝜑𝐷 ∈ ℝ)
57 eqid 2725 . . . . . . . . . . 11 (topGen‘ran (,)) = (topGen‘ran (,))
58 eqid 2725 . . . . . . . . . . 11 ((topGen‘ran (,)) ↾t (𝐶[,]𝐷)) = ((topGen‘ran (,)) ↾t (𝐶[,]𝐷))
59 oveq1 7426 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥 + (𝑘 · 𝑇)) = (𝑤 + (𝑘 · 𝑇)))
6059eleq1d 2810 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6160rexbidv 3168 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6261cbvrabv 3429 . . . . . . . . . . 11 {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑤 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}
63 oveq1 7426 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑖 · 𝑇) = (𝑗 · 𝑇))
6463oveq2d 7435 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑦 + (𝑖 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
6564eleq1d 2810 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
6665anbi1d 629 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄)))
67 oveq1 7426 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑘 → (𝑙 · 𝑇) = (𝑘 · 𝑇))
6867oveq2d 7435 . . . . . . . . . . . . . . 15 (𝑙 = 𝑘 → (𝑧 + (𝑙 · 𝑇)) = (𝑧 + (𝑘 · 𝑇)))
6968eleq1d 2810 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → ((𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7069anbi2d 628 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄)))
7166, 70cbvrex2vw 3229 . . . . . . . . . . . 12 (∃𝑖 ∈ ℤ ∃𝑙 ∈ ℤ ((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ∃𝑗 ∈ ℤ ∃𝑘 ∈ ℤ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7271anbi2i 621 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑦 < 𝑧)) ∧ ∃𝑖 ∈ ℤ ∃𝑙 ∈ ℤ ((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄)) ↔ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑦 < 𝑧)) ∧ ∃𝑗 ∈ ℤ ∃𝑘 ∈ ℤ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄)))
7316, 17, 18, 19, 22, 34, 45, 51, 52, 53, 54, 55, 4, 56, 57, 58, 62, 72fourierdlem42 45675 . . . . . . . . . 10 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ∈ Fin)
74 unfi 9197 . . . . . . . . . 10 (({𝐶, 𝐷} ∈ Fin ∧ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ∈ Fin) → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ∈ Fin)
7511, 73, 74sylancr 585 . . . . . . . . 9 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ∈ Fin)
768, 75eqeltrid 2829 . . . . . . . 8 (𝜑𝐻 ∈ Fin)
77 hashnncl 14361 . . . . . . . 8 (𝐻 ∈ Fin → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
7876, 77syl 17 . . . . . . 7 (𝜑 → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
7910, 78mpbird 256 . . . . . 6 (𝜑 → (♯‘𝐻) ∈ ℕ)
8079nnzd 12618 . . . . 5 (𝜑 → (♯‘𝐻) ∈ ℤ)
81 fourierdlem54.cd . . . . . . . . 9 (𝜑𝐶 < 𝐷)
824, 81ltned 11382 . . . . . . . 8 (𝜑𝐶𝐷)
83 hashprg 14390 . . . . . . . . 9 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶𝐷 ↔ (♯‘{𝐶, 𝐷}) = 2))
844, 56, 83syl2anc 582 . . . . . . . 8 (𝜑 → (𝐶𝐷 ↔ (♯‘{𝐶, 𝐷}) = 2))
8582, 84mpbid 231 . . . . . . 7 (𝜑 → (♯‘{𝐶, 𝐷}) = 2)
8685eqcomd 2731 . . . . . 6 (𝜑 → 2 = (♯‘{𝐶, 𝐷}))
87 ssun1 4170 . . . . . . . . 9 {𝐶, 𝐷} ⊆ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
8887a1i 11 . . . . . . . 8 (𝜑 → {𝐶, 𝐷} ⊆ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
8988, 8sseqtrrdi 4028 . . . . . . 7 (𝜑 → {𝐶, 𝐷} ⊆ 𝐻)
90 hashssle 44818 . . . . . . 7 ((𝐻 ∈ Fin ∧ {𝐶, 𝐷} ⊆ 𝐻) → (♯‘{𝐶, 𝐷}) ≤ (♯‘𝐻))
9176, 89, 90syl2anc 582 . . . . . 6 (𝜑 → (♯‘{𝐶, 𝐷}) ≤ (♯‘𝐻))
9286, 91eqbrtrd 5171 . . . . 5 (𝜑 → 2 ≤ (♯‘𝐻))
93 eluz2 12861 . . . . 5 ((♯‘𝐻) ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ (♯‘𝐻) ∈ ℤ ∧ 2 ≤ (♯‘𝐻)))
943, 80, 92, 93syl3anbrc 1340 . . . 4 (𝜑 → (♯‘𝐻) ∈ (ℤ‘2))
95 uz2m1nn 12940 . . . 4 ((♯‘𝐻) ∈ (ℤ‘2) → ((♯‘𝐻) − 1) ∈ ℕ)
9694, 95syl 17 . . 3 (𝜑 → ((♯‘𝐻) − 1) ∈ ℕ)
971, 96eqeltrid 2829 . 2 (𝜑𝑁 ∈ ℕ)
98 prssg 4824 . . . . . . . . . . . . 13 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐶, 𝐷} ⊆ ℝ))
994, 56, 98syl2anc 582 . . . . . . . . . . . 12 (𝜑 → ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐶, 𝐷} ⊆ ℝ))
1004, 56, 99mpbi2and 710 . . . . . . . . . . 11 (𝜑 → {𝐶, 𝐷} ⊆ ℝ)
101 ssrab2 4073 . . . . . . . . . . . 12 {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ (𝐶[,]𝐷)
1024, 56iccssred 13446 . . . . . . . . . . . 12 (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
103101, 102sstrid 3988 . . . . . . . . . . 11 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ ℝ)
104100, 103unssd 4184 . . . . . . . . . 10 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ⊆ ℝ)
1058, 104eqsstrid 4025 . . . . . . . . 9 (𝜑𝐻 ⊆ ℝ)
106 fourierdlem54.s . . . . . . . . 9 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
10776, 105, 106, 1fourierdlem36 45669 . . . . . . . 8 (𝜑𝑆 Isom < , < ((0...𝑁), 𝐻))
108 df-isom 6558 . . . . . . . 8 (𝑆 Isom < , < ((0...𝑁), 𝐻) ↔ (𝑆:(0...𝑁)–1-1-onto𝐻 ∧ ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦))))
109107, 108sylib 217 . . . . . . 7 (𝜑 → (𝑆:(0...𝑁)–1-1-onto𝐻 ∧ ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦))))
110109simpld 493 . . . . . 6 (𝜑𝑆:(0...𝑁)–1-1-onto𝐻)
111 f1of 6838 . . . . . 6 (𝑆:(0...𝑁)–1-1-onto𝐻𝑆:(0...𝑁)⟶𝐻)
112110, 111syl 17 . . . . 5 (𝜑𝑆:(0...𝑁)⟶𝐻)
113112, 105fssd 6740 . . . 4 (𝜑𝑆:(0...𝑁)⟶ℝ)
114 reex 11231 . . . . 5 ℝ ∈ V
115 ovex 7452 . . . . . 6 (0...𝑁) ∈ V
116115a1i 11 . . . . 5 (𝜑 → (0...𝑁) ∈ V)
117 elmapg 8858 . . . . 5 ((ℝ ∈ V ∧ (0...𝑁) ∈ V) → (𝑆 ∈ (ℝ ↑m (0...𝑁)) ↔ 𝑆:(0...𝑁)⟶ℝ))
118114, 116, 117sylancr 585 . . . 4 (𝜑 → (𝑆 ∈ (ℝ ↑m (0...𝑁)) ↔ 𝑆:(0...𝑁)⟶ℝ))
119113, 118mpbird 256 . . 3 (𝜑𝑆 ∈ (ℝ ↑m (0...𝑁)))
120 df-f1o 6556 . . . . . . . . . . 11 (𝑆:(0...𝑁)–1-1-onto𝐻 ↔ (𝑆:(0...𝑁)–1-1𝐻𝑆:(0...𝑁)–onto𝐻))
121110, 120sylib 217 . . . . . . . . . 10 (𝜑 → (𝑆:(0...𝑁)–1-1𝐻𝑆:(0...𝑁)–onto𝐻))
122121simprd 494 . . . . . . . . 9 (𝜑𝑆:(0...𝑁)–onto𝐻)
123 dffo3 7111 . . . . . . . . 9 (𝑆:(0...𝑁)–onto𝐻 ↔ (𝑆:(0...𝑁)⟶𝐻 ∧ ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦)))
124122, 123sylib 217 . . . . . . . 8 (𝜑 → (𝑆:(0...𝑁)⟶𝐻 ∧ ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦)))
125124simprd 494 . . . . . . 7 (𝜑 → ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦))
126 eqeq1 2729 . . . . . . . . . 10 ( = 𝐶 → ( = (𝑆𝑦) ↔ 𝐶 = (𝑆𝑦)))
127 eqcom 2732 . . . . . . . . . 10 (𝐶 = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐶)
128126, 127bitrdi 286 . . . . . . . . 9 ( = 𝐶 → ( = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐶))
129128rexbidv 3168 . . . . . . . 8 ( = 𝐶 → (∃𝑦 ∈ (0...𝑁) = (𝑆𝑦) ↔ ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶))
130129rspcv 3602 . . . . . . 7 (𝐶𝐻 → (∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦) → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶))
1319, 125, 130sylc 65 . . . . . 6 (𝜑 → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶)
132 fveq2 6896 . . . . . . . . . . . . . 14 (𝑦 = 0 → (𝑆𝑦) = (𝑆‘0))
133132eqcomd 2731 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑆‘0) = (𝑆𝑦))
134133adantl 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) = (𝑆𝑦))
135 simplr 767 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆𝑦) = 𝐶)
136134, 135eqtrd 2765 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) = 𝐶)
1374ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → 𝐶 ∈ ℝ)
138136, 137eqeltrd 2825 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
139138, 136eqled 11349 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
1401393adantl2 1164 . . . . . . . 8 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
1414rexrd 11296 . . . . . . . . . . . . . . . . 17 (𝜑𝐶 ∈ ℝ*)
14256rexrd 11296 . . . . . . . . . . . . . . . . 17 (𝜑𝐷 ∈ ℝ*)
1434, 56, 81ltled 11394 . . . . . . . . . . . . . . . . 17 (𝜑𝐶𝐷)
144 lbicc2 13476 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*𝐶𝐷) → 𝐶 ∈ (𝐶[,]𝐷))
145141, 142, 143, 144syl3anc 1368 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ (𝐶[,]𝐷))
146 ubicc2 13477 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*𝐶𝐷) → 𝐷 ∈ (𝐶[,]𝐷))
147141, 142, 143, 146syl3anc 1368 . . . . . . . . . . . . . . . 16 (𝜑𝐷 ∈ (𝐶[,]𝐷))
148 prssg 4824 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) → ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) ↔ {𝐶, 𝐷} ⊆ (𝐶[,]𝐷)))
149145, 147, 148syl2anc 582 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) ↔ {𝐶, 𝐷} ⊆ (𝐶[,]𝐷)))
150145, 147, 149mpbi2and 710 . . . . . . . . . . . . . . 15 (𝜑 → {𝐶, 𝐷} ⊆ (𝐶[,]𝐷))
151101a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ (𝐶[,]𝐷))
152150, 151unssd 4184 . . . . . . . . . . . . . 14 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ⊆ (𝐶[,]𝐷))
1538, 152eqsstrid 4025 . . . . . . . . . . . . 13 (𝜑𝐻 ⊆ (𝐶[,]𝐷))
154 nnm1nn0 12546 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐻) ∈ ℕ → ((♯‘𝐻) − 1) ∈ ℕ0)
15579, 154syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((♯‘𝐻) − 1) ∈ ℕ0)
1561, 155eqeltrid 2829 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ0)
157156, 39eleqtrdi 2835 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘0))
158 eluzfz1 13543 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
159157, 158syl 17 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ (0...𝑁))
160112, 159ffvelcdmd 7094 . . . . . . . . . . . . 13 (𝜑 → (𝑆‘0) ∈ 𝐻)
161153, 160sseldd 3977 . . . . . . . . . . . 12 (𝜑 → (𝑆‘0) ∈ (𝐶[,]𝐷))
162102, 161sseldd 3977 . . . . . . . . . . 11 (𝜑 → (𝑆‘0) ∈ ℝ)
163162adantr 479 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
1641633ad2antl1 1182 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
1654adantr 479 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑦 = 0) → 𝐶 ∈ ℝ)
1661653ad2antl1 1182 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝐶 ∈ ℝ)
167 elfzelz 13536 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑁) → 𝑦 ∈ ℤ)
168167zred 12699 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 𝑦 ∈ ℝ)
169168adantr 479 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 𝑦 ∈ ℝ)
170 elfzle1 13539 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 0 ≤ 𝑦)
171170adantr 479 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 0 ≤ 𝑦)
172 neqne 2937 . . . . . . . . . . . . . 14 𝑦 = 0 → 𝑦 ≠ 0)
173172adantl 480 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 𝑦 ≠ 0)
174169, 171, 173ne0gt0d 11383 . . . . . . . . . . . 12 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 0 < 𝑦)
1751743ad2antl2 1183 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 0 < 𝑦)
176 simpl1 1188 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝜑)
177 simpl2 1189 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝑦 ∈ (0...𝑁))
178109simprd 494 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)))
179 breq1 5152 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
180 fveq2 6896 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → (𝑆𝑥) = (𝑆‘0))
181180breq1d 5159 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆‘0) < (𝑆𝑦)))
182179, 181bibi12d 344 . . . . . . . . . . . . . . . 16 (𝑥 = 0 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦))))
183182ralbidv 3167 . . . . . . . . . . . . . . 15 (𝑥 = 0 → (∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦))))
184183rspcv 3602 . . . . . . . . . . . . . 14 (0 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) → ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦))))
185159, 178, 184sylc 65 . . . . . . . . . . . . 13 (𝜑 → ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦)))
186185r19.21bi 3238 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦)))
187176, 177, 186syl2anc 582 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦)))
188175, 187mpbid 231 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) < (𝑆𝑦))
189 simpl3 1190 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆𝑦) = 𝐶)
190188, 189breqtrd 5175 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) < 𝐶)
191164, 166, 190ltled 11394 . . . . . . . 8 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
192140, 191pm2.61dan 811 . . . . . . 7 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) → (𝑆‘0) ≤ 𝐶)
193192rexlimdv3a 3148 . . . . . 6 (𝜑 → (∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶 → (𝑆‘0) ≤ 𝐶))
194131, 193mpd 15 . . . . 5 (𝜑 → (𝑆‘0) ≤ 𝐶)
195 elicc2 13424 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘0) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷)))
1964, 56, 195syl2anc 582 . . . . . . 7 (𝜑 → ((𝑆‘0) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷)))
197161, 196mpbid 231 . . . . . 6 (𝜑 → ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷))
198197simp2d 1140 . . . . 5 (𝜑𝐶 ≤ (𝑆‘0))
199162, 4letri3d 11388 . . . . 5 (𝜑 → ((𝑆‘0) = 𝐶 ↔ ((𝑆‘0) ≤ 𝐶𝐶 ≤ (𝑆‘0))))
200194, 198, 199mpbir2and 711 . . . 4 (𝜑 → (𝑆‘0) = 𝐶)
201 eluzfz2 13544 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
202157, 201syl 17 . . . . . . . . 9 (𝜑𝑁 ∈ (0...𝑁))
203112, 202ffvelcdmd 7094 . . . . . . . 8 (𝜑 → (𝑆𝑁) ∈ 𝐻)
204153, 203sseldd 3977 . . . . . . 7 (𝜑 → (𝑆𝑁) ∈ (𝐶[,]𝐷))
205 elicc2 13424 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆𝑁) ∈ (𝐶[,]𝐷) ↔ ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷)))
2064, 56, 205syl2anc 582 . . . . . . 7 (𝜑 → ((𝑆𝑁) ∈ (𝐶[,]𝐷) ↔ ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷)))
207204, 206mpbid 231 . . . . . 6 (𝜑 → ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷))
208207simp3d 1141 . . . . 5 (𝜑 → (𝑆𝑁) ≤ 𝐷)
209 prid2g 4767 . . . . . . . . 9 (𝐷 ∈ ℝ → 𝐷 ∈ {𝐶, 𝐷})
210 elun1 4174 . . . . . . . . 9 (𝐷 ∈ {𝐶, 𝐷} → 𝐷 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
21156, 209, 2103syl 18 . . . . . . . 8 (𝜑𝐷 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
212211, 8eleqtrrdi 2836 . . . . . . 7 (𝜑𝐷𝐻)
213 eqeq1 2729 . . . . . . . . . 10 ( = 𝐷 → ( = (𝑆𝑦) ↔ 𝐷 = (𝑆𝑦)))
214 eqcom 2732 . . . . . . . . . 10 (𝐷 = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐷)
215213, 214bitrdi 286 . . . . . . . . 9 ( = 𝐷 → ( = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐷))
216215rexbidv 3168 . . . . . . . 8 ( = 𝐷 → (∃𝑦 ∈ (0...𝑁) = (𝑆𝑦) ↔ ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷))
217216rspcv 3602 . . . . . . 7 (𝐷𝐻 → (∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦) → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷))
218212, 125, 217sylc 65 . . . . . 6 (𝜑 → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷)
219214biimpri 227 . . . . . . . . 9 ((𝑆𝑦) = 𝐷𝐷 = (𝑆𝑦))
2202193ad2ant3 1132 . . . . . . . 8 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → 𝐷 = (𝑆𝑦))
221113ffvelcdmda 7093 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑦) ∈ ℝ)
222102, 204sseldd 3977 . . . . . . . . . . 11 (𝜑 → (𝑆𝑁) ∈ ℝ)
223222adantr 479 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑁) ∈ ℝ)
224168adantl 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑦 ∈ ℝ)
225 elfzel2 13534 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
226225zred 12699 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
227226adantl 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
228 elfzle2 13540 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑁) → 𝑦𝑁)
229228adantl 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑦𝑁)
230224, 227, 229lensymd 11397 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...𝑁)) → ¬ 𝑁 < 𝑦)
231 breq1 5152 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑁 → (𝑥 < 𝑦𝑁 < 𝑦))
232 fveq2 6896 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑁 → (𝑆𝑥) = (𝑆𝑁))
233232breq1d 5159 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑁 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆𝑁) < (𝑆𝑦)))
234231, 233bibi12d 344 . . . . . . . . . . . . . . 15 (𝑥 = 𝑁 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦))))
235234ralbidv 3167 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦))))
236235rspcv 3602 . . . . . . . . . . . . 13 (𝑁 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) → ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦))))
237202, 178, 236sylc 65 . . . . . . . . . . . 12 (𝜑 → ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦)))
238237r19.21bi 3238 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦)))
239230, 238mtbid 323 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → ¬ (𝑆𝑁) < (𝑆𝑦))
240221, 223, 239nltled 11396 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑦) ≤ (𝑆𝑁))
2412403adant3 1129 . . . . . . . 8 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → (𝑆𝑦) ≤ (𝑆𝑁))
242220, 241eqbrtrd 5171 . . . . . . 7 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → 𝐷 ≤ (𝑆𝑁))
243242rexlimdv3a 3148 . . . . . 6 (𝜑 → (∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷𝐷 ≤ (𝑆𝑁)))
244218, 243mpd 15 . . . . 5 (𝜑𝐷 ≤ (𝑆𝑁))
245222, 56letri3d 11388 . . . . 5 (𝜑 → ((𝑆𝑁) = 𝐷 ↔ ((𝑆𝑁) ≤ 𝐷𝐷 ≤ (𝑆𝑁))))
246208, 244, 245mpbir2and 711 . . . 4 (𝜑 → (𝑆𝑁) = 𝐷)
247 elfzoelz 13667 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
248247zred 12699 . . . . . . . 8 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
249248ltp1d 12177 . . . . . . 7 (𝑖 ∈ (0..^𝑁) → 𝑖 < (𝑖 + 1))
250249adantl 480 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 < (𝑖 + 1))
251178adantr 479 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑁)) → ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)))
252 elfzofz 13683 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0...𝑁))
253252adantl 480 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0...𝑁))
254 fzofzp1 13765 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
255254adantl 480 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
256 breq1 5152 . . . . . . . . . 10 (𝑥 = 𝑖 → (𝑥 < 𝑦𝑖 < 𝑦))
257 fveq2 6896 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑆𝑥) = (𝑆𝑖))
258257breq1d 5159 . . . . . . . . . 10 (𝑥 = 𝑖 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆𝑖) < (𝑆𝑦)))
259256, 258bibi12d 344 . . . . . . . . 9 (𝑥 = 𝑖 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (𝑖 < 𝑦 ↔ (𝑆𝑖) < (𝑆𝑦))))
260 breq2 5153 . . . . . . . . . 10 (𝑦 = (𝑖 + 1) → (𝑖 < 𝑦𝑖 < (𝑖 + 1)))
261 fveq2 6896 . . . . . . . . . . 11 (𝑦 = (𝑖 + 1) → (𝑆𝑦) = (𝑆‘(𝑖 + 1)))
262261breq2d 5161 . . . . . . . . . 10 (𝑦 = (𝑖 + 1) → ((𝑆𝑖) < (𝑆𝑦) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1))))
263260, 262bibi12d 344 . . . . . . . . 9 (𝑦 = (𝑖 + 1) → ((𝑖 < 𝑦 ↔ (𝑆𝑖) < (𝑆𝑦)) ↔ (𝑖 < (𝑖 + 1) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1)))))
264259, 263rspc2v 3617 . . . . . . . 8 ((𝑖 ∈ (0...𝑁) ∧ (𝑖 + 1) ∈ (0...𝑁)) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) → (𝑖 < (𝑖 + 1) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1)))))
265253, 255, 264syl2anc 582 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) → (𝑖 < (𝑖 + 1) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1)))))
266251, 265mpd 15 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 < (𝑖 + 1) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1))))
267250, 266mpbid 231 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑆𝑖) < (𝑆‘(𝑖 + 1)))
268267ralrimiva 3135 . . . 4 (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1)))
269200, 246, 268jca31 513 . . 3 (𝜑 → (((𝑆‘0) = 𝐶 ∧ (𝑆𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1))))
270 fourierdlem54.o . . . . 5 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
271270fourierdlem2 45635 . . . 4 (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂𝑁) ↔ (𝑆 ∈ (ℝ ↑m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1))))))
27297, 271syl 17 . . 3 (𝜑 → (𝑆 ∈ (𝑂𝑁) ↔ (𝑆 ∈ (ℝ ↑m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1))))))
273119, 269, 272mpbir2and 711 . 2 (𝜑𝑆 ∈ (𝑂𝑁))
27497, 273, 107jca31 513 1 (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cdif 3941  cun 3942  wss 3944  c0 4322  {cpr 4632   class class class wbr 5149  cmpt 5232   I cid 5575   × cxp 5676  ran crn 5679  cres 5680  ccom 5682  cio 6499   Fn wfn 6544  wf 6545  1-1wf1 6546  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549   Isom wiso 6550  (class class class)co 7419  m cmap 8845  Fincfn 8964  infcinf 9466  cr 11139  0cc0 11140  1c1 11141   + caddc 11143   · cmul 11145  *cxr 11279   < clt 11280  cle 11281  cmin 11476  cn 12245  2c2 12300  0cn0 12505  cz 12591  cuz 12855  (,)cioo 13359  [,]cicc 13362  ...cfz 13519  ..^cfzo 13662  chash 14325  abscabs 15217  t crest 17405  topGenctg 17422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9436  df-sup 9467  df-inf 9468  df-oi 9535  df-dju 9926  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-xnn0 12578  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-ioo 13363  df-icc 13366  df-fz 13520  df-fzo 13663  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-rest 17407  df-topgen 17428  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-top 22840  df-topon 22857  df-bases 22893  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-lp 23084  df-cmp 23335
This theorem is referenced by:  fourierdlem63  45695  fourierdlem64  45696  fourierdlem65  45697  fourierdlem79  45711  fourierdlem89  45721  fourierdlem90  45722  fourierdlem91  45723  fourierdlem100  45732  fourierdlem107  45739  fourierdlem109  45741  fourierdlem112  45744
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