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Theorem fourierdlem54 40946
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 11656 . . . . . 6 2 ∈ ℤ
32a1i 11 . . . . 5 (𝜑 → 2 ∈ ℤ)
4 fourierdlem54.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ)
5 prid1g 4450 . . . . . . . . . 10 (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶, 𝐷})
6 elun1 3942 . . . . . . . . . 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, 8syl6eleqr 2855 . . . . . . . 8 (𝜑𝐶𝐻)
109ne0d 4086 . . . . . . 7 (𝜑𝐻 ≠ ∅)
11 prfi 8442 . . . . . . . . . 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 40904 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
1615simp1d 1172 . . . . . . . . . . 11 (𝜑𝐴 ∈ ℝ)
1715simp2d 1173 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℝ)
1815simp3d 1174 . . . . . . . . . . 11 (𝜑𝐴 < 𝐵)
19 fourierdlem54.t . . . . . . . . . . 11 𝑇 = (𝐵𝐴)
2012, 13, 14fourierdlem15 40908 . . . . . . . . . . . 12 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
21 frn 6229 . . . . . . . . . . . 12 (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) → ran 𝑄 ⊆ (𝐴[,]𝐵))
2220, 21syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
2312fourierdlem2 40895 . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
2625simpld 488 . . . . . . . . . . . . . 14 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
27 elmapi 8082 . . . . . . . . . . . . . 14 (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
28 ffn 6223 . . . . . . . . . . . . . 14 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
2926, 27, 283syl 18 . . . . . . . . . . . . 13 (𝜑𝑄 Fn (0...𝑀))
30 fzfid 12980 . . . . . . . . . . . . 13 (𝜑 → (0...𝑀) ∈ Fin)
31 fnfi 8445 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ (0...𝑀) ∈ Fin) → 𝑄 ∈ Fin)
3229, 30, 31syl2anc 579 . . . . . . . . . . . 12 (𝜑𝑄 ∈ Fin)
33 rnfi 8456 . . . . . . . . . . . 12 (𝑄 ∈ Fin → ran 𝑄 ∈ Fin)
3432, 33syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ∈ Fin)
3525simprd 489 . . . . . . . . . . . . . 14 (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
3635simpld 488 . . . . . . . . . . . . 13 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
3736simpld 488 . . . . . . . . . . . 12 (𝜑 → (𝑄‘0) = 𝐴)
3813nnnn0d 11598 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℕ0)
39 nn0uz 11922 . . . . . . . . . . . . . . 15 0 = (ℤ‘0)
4038, 39syl6eleq 2854 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ‘0))
41 eluzfz1 12555 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
4240, 41syl 17 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ (0...𝑀))
43 fnfvelrn 6546 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ 0 ∈ (0...𝑀)) → (𝑄‘0) ∈ ran 𝑄)
4429, 42, 43syl2anc 579 . . . . . . . . . . . 12 (𝜑 → (𝑄‘0) ∈ ran 𝑄)
4537, 44eqeltrrd 2845 . . . . . . . . . . 11 (𝜑𝐴 ∈ ran 𝑄)
4636simprd 489 . . . . . . . . . . . 12 (𝜑 → (𝑄𝑀) = 𝐵)
47 eluzfz2 12556 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
4840, 47syl 17 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (0...𝑀))
49 fnfvelrn 6546 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)) → (𝑄𝑀) ∈ ran 𝑄)
5029, 48, 49syl2anc 579 . . . . . . . . . . . 12 (𝜑 → (𝑄𝑀) ∈ ran 𝑄)
5146, 50eqeltrrd 2845 . . . . . . . . . . 11 (𝜑𝐵 ∈ ran 𝑄)
52 eqid 2765 . . . . . . . . . . 11 (abs ∘ − ) = (abs ∘ − )
53 eqid 2765 . . . . . . . . . . 11 ((ran 𝑄 × ran 𝑄) ∖ I ) = ((ran 𝑄 × ran 𝑄) ∖ I )
54 eqid 2765 . . . . . . . . . . 11 ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )) = ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I ))
55 eqid 2765 . . . . . . . . . . 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 2765 . . . . . . . . . . 11 (topGen‘ran (,)) = (topGen‘ran (,))
58 eqid 2765 . . . . . . . . . . 11 ((topGen‘ran (,)) ↾t (𝐶[,]𝐷)) = ((topGen‘ran (,)) ↾t (𝐶[,]𝐷))
59 oveq1 6849 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥 + (𝑘 · 𝑇)) = (𝑤 + (𝑘 · 𝑇)))
6059eleq1d 2829 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6160rexbidv 3199 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6261cbvrabv 3348 . . . . . . . . . . 11 {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑤 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}
63 oveq1 6849 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑖 · 𝑇) = (𝑗 · 𝑇))
6463oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑦 + (𝑖 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
6564eleq1d 2829 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
6665anbi1d 623 . . . . . . . . . . . . 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 6849 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑘 → (𝑙 · 𝑇) = (𝑘 · 𝑇))
6867oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑙 = 𝑘 → (𝑧 + (𝑙 · 𝑇)) = (𝑧 + (𝑘 · 𝑇)))
6968eleq1d 2829 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → ((𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7069anbi2d 622 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄)))
7166, 70cbvrex2v 3328 . . . . . . . . . . . 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 616 . . . . . . . . . . 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 40935 . . . . . . . . . 10 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ∈ Fin)
74 unfi 8434 . . . . . . . . . 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 581 . . . . . . . . 9 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ∈ Fin)
768, 75syl5eqel 2848 . . . . . . . 8 (𝜑𝐻 ∈ Fin)
77 hashnncl 13359 . . . . . . . 8 (𝐻 ∈ Fin → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
7876, 77syl 17 . . . . . . 7 (𝜑 → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
7910, 78mpbird 248 . . . . . 6 (𝜑 → (♯‘𝐻) ∈ ℕ)
8079nnzd 11728 . . . . 5 (𝜑 → (♯‘𝐻) ∈ ℤ)
81 fourierdlem54.cd . . . . . . . . 9 (𝜑𝐶 < 𝐷)
824, 81ltned 10427 . . . . . . . 8 (𝜑𝐶𝐷)
83 hashprg 13384 . . . . . . . . 9 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶𝐷 ↔ (♯‘{𝐶, 𝐷}) = 2))
844, 56, 83syl2anc 579 . . . . . . . 8 (𝜑 → (𝐶𝐷 ↔ (♯‘{𝐶, 𝐷}) = 2))
8582, 84mpbid 223 . . . . . . 7 (𝜑 → (♯‘{𝐶, 𝐷}) = 2)
8685eqcomd 2771 . . . . . 6 (𝜑 → 2 = (♯‘{𝐶, 𝐷}))
87 ssun1 3938 . . . . . . . . 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, 8syl6sseqr 3812 . . . . . . 7 (𝜑 → {𝐶, 𝐷} ⊆ 𝐻)
90 hashssle 40083 . . . . . . 7 ((𝐻 ∈ Fin ∧ {𝐶, 𝐷} ⊆ 𝐻) → (♯‘{𝐶, 𝐷}) ≤ (♯‘𝐻))
9176, 89, 90syl2anc 579 . . . . . 6 (𝜑 → (♯‘{𝐶, 𝐷}) ≤ (♯‘𝐻))
9286, 91eqbrtrd 4831 . . . . 5 (𝜑 → 2 ≤ (♯‘𝐻))
93 eluz2 11892 . . . . 5 ((♯‘𝐻) ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ (♯‘𝐻) ∈ ℤ ∧ 2 ≤ (♯‘𝐻)))
943, 80, 92, 93syl3anbrc 1443 . . . 4 (𝜑 → (♯‘𝐻) ∈ (ℤ‘2))
95 uz2m1nn 11964 . . . 4 ((♯‘𝐻) ∈ (ℤ‘2) → ((♯‘𝐻) − 1) ∈ ℕ)
9694, 95syl 17 . . 3 (𝜑 → ((♯‘𝐻) − 1) ∈ ℕ)
971, 96syl5eqel 2848 . 2 (𝜑𝑁 ∈ ℕ)
98 prssg 4504 . . . . . . . . . . . . 13 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐶, 𝐷} ⊆ ℝ))
994, 56, 98syl2anc 579 . . . . . . . . . . . 12 (𝜑 → ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐶, 𝐷} ⊆ ℝ))
1004, 56, 99mpbi2and 703 . . . . . . . . . . 11 (𝜑 → {𝐶, 𝐷} ⊆ ℝ)
101 ssrab2 3847 . . . . . . . . . . . 12 {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ (𝐶[,]𝐷)
1024, 56iccssred 40301 . . . . . . . . . . . 12 (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
103101, 102syl5ss 3772 . . . . . . . . . . 11 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ ℝ)
104100, 103unssd 3951 . . . . . . . . . 10 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ⊆ ℝ)
1058, 104syl5eqss 3809 . . . . . . . . 9 (𝜑𝐻 ⊆ ℝ)
106 fourierdlem54.s . . . . . . . . 9 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
10776, 105, 106, 1fourierdlem36 40929 . . . . . . . 8 (𝜑𝑆 Isom < , < ((0...𝑁), 𝐻))
108 df-isom 6077 . . . . . . . 8 (𝑆 Isom < , < ((0...𝑁), 𝐻) ↔ (𝑆:(0...𝑁)–1-1-onto𝐻 ∧ ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦))))
109107, 108sylib 209 . . . . . . 7 (𝜑 → (𝑆:(0...𝑁)–1-1-onto𝐻 ∧ ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦))))
110109simpld 488 . . . . . 6 (𝜑𝑆:(0...𝑁)–1-1-onto𝐻)
111 f1of 6320 . . . . . 6 (𝑆:(0...𝑁)–1-1-onto𝐻𝑆:(0...𝑁)⟶𝐻)
112110, 111syl 17 . . . . 5 (𝜑𝑆:(0...𝑁)⟶𝐻)
113112, 105fssd 6237 . . . 4 (𝜑𝑆:(0...𝑁)⟶ℝ)
114 reex 10280 . . . . 5 ℝ ∈ V
115 ovex 6874 . . . . . 6 (0...𝑁) ∈ V
116115a1i 11 . . . . 5 (𝜑 → (0...𝑁) ∈ V)
117 elmapg 8073 . . . . 5 ((ℝ ∈ V ∧ (0...𝑁) ∈ V) → (𝑆 ∈ (ℝ ↑𝑚 (0...𝑁)) ↔ 𝑆:(0...𝑁)⟶ℝ))
118114, 116, 117sylancr 581 . . . 4 (𝜑 → (𝑆 ∈ (ℝ ↑𝑚 (0...𝑁)) ↔ 𝑆:(0...𝑁)⟶ℝ))
119113, 118mpbird 248 . . 3 (𝜑𝑆 ∈ (ℝ ↑𝑚 (0...𝑁)))
120 df-f1o 6075 . . . . . . . . . . 11 (𝑆:(0...𝑁)–1-1-onto𝐻 ↔ (𝑆:(0...𝑁)–1-1𝐻𝑆:(0...𝑁)–onto𝐻))
121110, 120sylib 209 . . . . . . . . . 10 (𝜑 → (𝑆:(0...𝑁)–1-1𝐻𝑆:(0...𝑁)–onto𝐻))
122121simprd 489 . . . . . . . . 9 (𝜑𝑆:(0...𝑁)–onto𝐻)
123 dffo3 6564 . . . . . . . . 9 (𝑆:(0...𝑁)–onto𝐻 ↔ (𝑆:(0...𝑁)⟶𝐻 ∧ ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦)))
124122, 123sylib 209 . . . . . . . 8 (𝜑 → (𝑆:(0...𝑁)⟶𝐻 ∧ ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦)))
125124simprd 489 . . . . . . 7 (𝜑 → ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦))
126 eqeq1 2769 . . . . . . . . . 10 ( = 𝐶 → ( = (𝑆𝑦) ↔ 𝐶 = (𝑆𝑦)))
127 eqcom 2772 . . . . . . . . . 10 (𝐶 = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐶)
128126, 127syl6bb 278 . . . . . . . . 9 ( = 𝐶 → ( = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐶))
129128rexbidv 3199 . . . . . . . 8 ( = 𝐶 → (∃𝑦 ∈ (0...𝑁) = (𝑆𝑦) ↔ ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶))
130129rspcv 3457 . . . . . . 7 (𝐶𝐻 → (∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦) → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶))
1319, 125, 130sylc 65 . . . . . 6 (𝜑 → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶)
132 fveq2 6375 . . . . . . . . . . . . . 14 (𝑦 = 0 → (𝑆𝑦) = (𝑆‘0))
133132eqcomd 2771 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑆‘0) = (𝑆𝑦))
134133adantl 473 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) = (𝑆𝑦))
135 simplr 785 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆𝑦) = 𝐶)
136134, 135eqtrd 2799 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) = 𝐶)
1374ad2antrr 717 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → 𝐶 ∈ ℝ)
138136, 137eqeltrd 2844 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
139138, 136eqled 10394 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
1401393adantl2 1208 . . . . . . . 8 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
1414rexrd 10343 . . . . . . . . . . . . . . . . 17 (𝜑𝐶 ∈ ℝ*)
14256rexrd 10343 . . . . . . . . . . . . . . . . 17 (𝜑𝐷 ∈ ℝ*)
1434, 56, 81ltled 10439 . . . . . . . . . . . . . . . . 17 (𝜑𝐶𝐷)
144 lbicc2 12492 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*𝐶𝐷) → 𝐶 ∈ (𝐶[,]𝐷))
145141, 142, 143, 144syl3anc 1490 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ (𝐶[,]𝐷))
146 ubicc2 12493 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*𝐶𝐷) → 𝐷 ∈ (𝐶[,]𝐷))
147141, 142, 143, 146syl3anc 1490 . . . . . . . . . . . . . . . 16 (𝜑𝐷 ∈ (𝐶[,]𝐷))
148 prssg 4504 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) → ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) ↔ {𝐶, 𝐷} ⊆ (𝐶[,]𝐷)))
149145, 147, 148syl2anc 579 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) ↔ {𝐶, 𝐷} ⊆ (𝐶[,]𝐷)))
150145, 147, 149mpbi2and 703 . . . . . . . . . . . . . . 15 (𝜑 → {𝐶, 𝐷} ⊆ (𝐶[,]𝐷))
151101a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ (𝐶[,]𝐷))
152150, 151unssd 3951 . . . . . . . . . . . . . 14 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ⊆ (𝐶[,]𝐷))
1538, 152syl5eqss 3809 . . . . . . . . . . . . 13 (𝜑𝐻 ⊆ (𝐶[,]𝐷))
154 nnm1nn0 11581 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐻) ∈ ℕ → ((♯‘𝐻) − 1) ∈ ℕ0)
15579, 154syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((♯‘𝐻) − 1) ∈ ℕ0)
1561, 155syl5eqel 2848 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ0)
157156, 39syl6eleq 2854 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘0))
158 eluzfz1 12555 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
159157, 158syl 17 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ (0...𝑁))
160112, 159ffvelrnd 6550 . . . . . . . . . . . . 13 (𝜑 → (𝑆‘0) ∈ 𝐻)
161153, 160sseldd 3762 . . . . . . . . . . . 12 (𝜑 → (𝑆‘0) ∈ (𝐶[,]𝐷))
162102, 161sseldd 3762 . . . . . . . . . . 11 (𝜑 → (𝑆‘0) ∈ ℝ)
163162adantr 472 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
1641633ad2antl1 1236 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
1654adantr 472 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑦 = 0) → 𝐶 ∈ ℝ)
1661653ad2antl1 1236 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝐶 ∈ ℝ)
167 elfzelz 12549 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑁) → 𝑦 ∈ ℤ)
168167zred 11729 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 𝑦 ∈ ℝ)
169168adantr 472 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 𝑦 ∈ ℝ)
170 elfzle1 12551 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 0 ≤ 𝑦)
171170adantr 472 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 0 ≤ 𝑦)
172 neqne 2945 . . . . . . . . . . . . . 14 𝑦 = 0 → 𝑦 ≠ 0)
173172adantl 473 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 𝑦 ≠ 0)
174169, 171, 173ne0gt0d 10428 . . . . . . . . . . . 12 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 0 < 𝑦)
1751743ad2antl2 1237 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 0 < 𝑦)
176 simpl1 1242 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝜑)
177 simpl2 1244 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝑦 ∈ (0...𝑁))
178109simprd 489 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)))
179 breq1 4812 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
180 fveq2 6375 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → (𝑆𝑥) = (𝑆‘0))
181180breq1d 4819 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆‘0) < (𝑆𝑦)))
182179, 181bibi12d 336 . . . . . . . . . . . . . . . 16 (𝑥 = 0 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦))))
183182ralbidv 3133 . . . . . . . . . . . . . . 15 (𝑥 = 0 → (∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦))))
184183rspcv 3457 . . . . . . . . . . . . . 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 3079 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦)))
187176, 177, 186syl2anc 579 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦)))
188175, 187mpbid 223 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) < (𝑆𝑦))
189 simpl3 1246 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆𝑦) = 𝐶)
190188, 189breqtrd 4835 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) < 𝐶)
191164, 166, 190ltled 10439 . . . . . . . 8 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
192140, 191pm2.61dan 847 . . . . . . 7 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) → (𝑆‘0) ≤ 𝐶)
193192rexlimdv3a 3180 . . . . . 6 (𝜑 → (∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶 → (𝑆‘0) ≤ 𝐶))
194131, 193mpd 15 . . . . 5 (𝜑 → (𝑆‘0) ≤ 𝐶)
195 elicc2 12440 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘0) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷)))
1964, 56, 195syl2anc 579 . . . . . . 7 (𝜑 → ((𝑆‘0) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷)))
197161, 196mpbid 223 . . . . . 6 (𝜑 → ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷))
198197simp2d 1173 . . . . 5 (𝜑𝐶 ≤ (𝑆‘0))
199162, 4letri3d 10433 . . . . 5 (𝜑 → ((𝑆‘0) = 𝐶 ↔ ((𝑆‘0) ≤ 𝐶𝐶 ≤ (𝑆‘0))))
200194, 198, 199mpbir2and 704 . . . 4 (𝜑 → (𝑆‘0) = 𝐶)
201 eluzfz2 12556 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
202157, 201syl 17 . . . . . . . . 9 (𝜑𝑁 ∈ (0...𝑁))
203112, 202ffvelrnd 6550 . . . . . . . 8 (𝜑 → (𝑆𝑁) ∈ 𝐻)
204153, 203sseldd 3762 . . . . . . 7 (𝜑 → (𝑆𝑁) ∈ (𝐶[,]𝐷))
205 elicc2 12440 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆𝑁) ∈ (𝐶[,]𝐷) ↔ ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷)))
2064, 56, 205syl2anc 579 . . . . . . 7 (𝜑 → ((𝑆𝑁) ∈ (𝐶[,]𝐷) ↔ ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷)))
207204, 206mpbid 223 . . . . . 6 (𝜑 → ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷))
208207simp3d 1174 . . . . 5 (𝜑 → (𝑆𝑁) ≤ 𝐷)
209 prid2g 4451 . . . . . . . . 9 (𝐷 ∈ ℝ → 𝐷 ∈ {𝐶, 𝐷})
210 elun1 3942 . . . . . . . . 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, 8syl6eleqr 2855 . . . . . . 7 (𝜑𝐷𝐻)
213 eqeq1 2769 . . . . . . . . . 10 ( = 𝐷 → ( = (𝑆𝑦) ↔ 𝐷 = (𝑆𝑦)))
214 eqcom 2772 . . . . . . . . . 10 (𝐷 = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐷)
215213, 214syl6bb 278 . . . . . . . . 9 ( = 𝐷 → ( = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐷))
216215rexbidv 3199 . . . . . . . 8 ( = 𝐷 → (∃𝑦 ∈ (0...𝑁) = (𝑆𝑦) ↔ ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷))
217216rspcv 3457 . . . . . . 7 (𝐷𝐻 → (∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦) → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷))
218212, 125, 217sylc 65 . . . . . 6 (𝜑 → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷)
219214biimpri 219 . . . . . . . . 9 ((𝑆𝑦) = 𝐷𝐷 = (𝑆𝑦))
2202193ad2ant3 1165 . . . . . . . 8 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → 𝐷 = (𝑆𝑦))
221113ffvelrnda 6549 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑦) ∈ ℝ)
222102, 204sseldd 3762 . . . . . . . . . . 11 (𝜑 → (𝑆𝑁) ∈ ℝ)
223222adantr 472 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑁) ∈ ℝ)
224168adantl 473 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑦 ∈ ℝ)
225 elfzel2 12547 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
226225zred 11729 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
227226adantl 473 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
228 elfzle2 12552 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑁) → 𝑦𝑁)
229228adantl 473 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑦𝑁)
230224, 227, 229lensymd 10442 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...𝑁)) → ¬ 𝑁 < 𝑦)
231 breq1 4812 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑁 → (𝑥 < 𝑦𝑁 < 𝑦))
232 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑁 → (𝑆𝑥) = (𝑆𝑁))
233232breq1d 4819 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑁 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆𝑁) < (𝑆𝑦)))
234231, 233bibi12d 336 . . . . . . . . . . . . . . 15 (𝑥 = 𝑁 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦))))
235234ralbidv 3133 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦))))
236235rspcv 3457 . . . . . . . . . . . . 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 3079 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦)))
239230, 238mtbid 315 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → ¬ (𝑆𝑁) < (𝑆𝑦))
240221, 223, 239nltled 10441 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑦) ≤ (𝑆𝑁))
2412403adant3 1162 . . . . . . . 8 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → (𝑆𝑦) ≤ (𝑆𝑁))
242220, 241eqbrtrd 4831 . . . . . . 7 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → 𝐷 ≤ (𝑆𝑁))
243242rexlimdv3a 3180 . . . . . 6 (𝜑 → (∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷𝐷 ≤ (𝑆𝑁)))
244218, 243mpd 15 . . . . 5 (𝜑𝐷 ≤ (𝑆𝑁))
245222, 56letri3d 10433 . . . . 5 (𝜑 → ((𝑆𝑁) = 𝐷 ↔ ((𝑆𝑁) ≤ 𝐷𝐷 ≤ (𝑆𝑁))))
246208, 244, 245mpbir2and 704 . . . 4 (𝜑 → (𝑆𝑁) = 𝐷)
247 elfzoelz 12678 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
248247zred 11729 . . . . . . . 8 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
249248ltp1d 11208 . . . . . . 7 (𝑖 ∈ (0..^𝑁) → 𝑖 < (𝑖 + 1))
250249adantl 473 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 < (𝑖 + 1))
251178adantr 472 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑁)) → ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)))
252 elfzofz 12693 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0...𝑁))
253252adantl 473 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0...𝑁))
254 fzofzp1 12773 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
255254adantl 473 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
256 breq1 4812 . . . . . . . . . 10 (𝑥 = 𝑖 → (𝑥 < 𝑦𝑖 < 𝑦))
257 fveq2 6375 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑆𝑥) = (𝑆𝑖))
258257breq1d 4819 . . . . . . . . . 10 (𝑥 = 𝑖 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆𝑖) < (𝑆𝑦)))
259256, 258bibi12d 336 . . . . . . . . 9 (𝑥 = 𝑖 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (𝑖 < 𝑦 ↔ (𝑆𝑖) < (𝑆𝑦))))
260 breq2 4813 . . . . . . . . . 10 (𝑦 = (𝑖 + 1) → (𝑖 < 𝑦𝑖 < (𝑖 + 1)))
261 fveq2 6375 . . . . . . . . . . 11 (𝑦 = (𝑖 + 1) → (𝑆𝑦) = (𝑆‘(𝑖 + 1)))
262261breq2d 4821 . . . . . . . . . 10 (𝑦 = (𝑖 + 1) → ((𝑆𝑖) < (𝑆𝑦) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1))))
263260, 262bibi12d 336 . . . . . . . . 9 (𝑦 = (𝑖 + 1) → ((𝑖 < 𝑦 ↔ (𝑆𝑖) < (𝑆𝑦)) ↔ (𝑖 < (𝑖 + 1) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1)))))
264259, 263rspc2v 3474 . . . . . . . 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 579 . . . . . . 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 223 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑆𝑖) < (𝑆‘(𝑖 + 1)))
268267ralrimiva 3113 . . . 4 (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1)))
269200, 246, 268jca31 510 . . 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 40895 . . . 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 704 . 2 (𝜑𝑆 ∈ (𝑂𝑁))
27497, 273, 107jca31 510 1 (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  cun 3730  wss 3732  c0 4079  {cpr 4336   class class class wbr 4809  cmpt 4888   I cid 5184   × cxp 5275  ran crn 5278  cres 5279  ccom 5281  cio 6029   Fn wfn 6063  wf 6064  1-1wf1 6065  ontowfo 6066  1-1-ontowf1o 6067  cfv 6068   Isom wiso 6069  (class class class)co 6842  𝑚 cmap 8060  Fincfn 8160  infcinf 8554  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194  *cxr 10327   < clt 10328  cle 10329  cmin 10520  cn 11274  2c2 11327  0cn0 11538  cz 11624  cuz 11886  (,)cioo 12377  [,]cicc 12380  ...cfz 12533  ..^cfzo 12673  chash 13321  abscabs 14261  t crest 16349  topGenctg 16366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-icc 12384  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-rest 16351  df-topgen 16372  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-nei 21182  df-lp 21220  df-cmp 21470
This theorem is referenced by:  fourierdlem63  40955  fourierdlem64  40956  fourierdlem65  40957  fourierdlem79  40971  fourierdlem89  40981  fourierdlem90  40982  fourierdlem91  40983  fourierdlem100  40992  fourierdlem107  40999  fourierdlem109  41001  fourierdlem112  41004
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