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Theorem fourierdlem54 40695
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 11447 . . . . . 6 2 ∈ ℤ
32a1i 11 . . . . 5 (𝜑 → 2 ∈ ℤ)
4 fourierdlem54.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ)
5 prid1g 4327 . . . . . . . . . 10 (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶, 𝐷})
6 elun1 3813 . . . . . . . . . 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 2741 . . . . . . . 8 (𝜑𝐶𝐻)
10 ne0i 3954 . . . . . . . 8 (𝐶𝐻𝐻 ≠ ∅)
119, 10syl 17 . . . . . . 7 (𝜑𝐻 ≠ ∅)
12 prfi 8276 . . . . . . . . . 10 {𝐶, 𝐷} ∈ Fin
13 fourierdlem54.p . . . . . . . . . . . . 13 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
14 fourierdlem54.m . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℕ)
15 fourierdlem54.q . . . . . . . . . . . . 13 (𝜑𝑄 ∈ (𝑃𝑀))
1613, 14, 15fourierdlem11 40653 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
1716simp1d 1093 . . . . . . . . . . 11 (𝜑𝐴 ∈ ℝ)
1816simp2d 1094 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℝ)
1916simp3d 1095 . . . . . . . . . . 11 (𝜑𝐴 < 𝐵)
20 fourierdlem54.t . . . . . . . . . . 11 𝑇 = (𝐵𝐴)
2113, 14, 15fourierdlem15 40657 . . . . . . . . . . . 12 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
22 frn 6091 . . . . . . . . . . . 12 (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) → ran 𝑄 ⊆ (𝐴[,]𝐵))
2321, 22syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
2413fourierdlem2 40644 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
2514, 24syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
2615, 25mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
2726simpld 474 . . . . . . . . . . . . . 14 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
28 elmapi 7921 . . . . . . . . . . . . . 14 (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
29 ffn 6083 . . . . . . . . . . . . . 14 (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
3027, 28, 293syl 18 . . . . . . . . . . . . 13 (𝜑𝑄 Fn (0...𝑀))
31 fzfid 12812 . . . . . . . . . . . . 13 (𝜑 → (0...𝑀) ∈ Fin)
32 fnfi 8279 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ (0...𝑀) ∈ Fin) → 𝑄 ∈ Fin)
3330, 31, 32syl2anc 694 . . . . . . . . . . . 12 (𝜑𝑄 ∈ Fin)
34 rnfi 8290 . . . . . . . . . . . 12 (𝑄 ∈ Fin → ran 𝑄 ∈ Fin)
3533, 34syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑄 ∈ Fin)
3626simprd 478 . . . . . . . . . . . . . 14 (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
3736simpld 474 . . . . . . . . . . . . 13 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
3837simpld 474 . . . . . . . . . . . 12 (𝜑 → (𝑄‘0) = 𝐴)
3914nnnn0d 11389 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℕ0)
40 nn0uz 11760 . . . . . . . . . . . . . . 15 0 = (ℤ‘0)
4139, 40syl6eleq 2740 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ‘0))
42 eluzfz1 12386 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
4341, 42syl 17 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ (0...𝑀))
44 fnfvelrn 6396 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ 0 ∈ (0...𝑀)) → (𝑄‘0) ∈ ran 𝑄)
4530, 43, 44syl2anc 694 . . . . . . . . . . . 12 (𝜑 → (𝑄‘0) ∈ ran 𝑄)
4638, 45eqeltrrd 2731 . . . . . . . . . . 11 (𝜑𝐴 ∈ ran 𝑄)
4737simprd 478 . . . . . . . . . . . 12 (𝜑 → (𝑄𝑀) = 𝐵)
48 eluzfz2 12387 . . . . . . . . . . . . . 14 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
4941, 48syl 17 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (0...𝑀))
50 fnfvelrn 6396 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)) → (𝑄𝑀) ∈ ran 𝑄)
5130, 49, 50syl2anc 694 . . . . . . . . . . . 12 (𝜑 → (𝑄𝑀) ∈ ran 𝑄)
5247, 51eqeltrrd 2731 . . . . . . . . . . 11 (𝜑𝐵 ∈ ran 𝑄)
53 eqid 2651 . . . . . . . . . . 11 (abs ∘ − ) = (abs ∘ − )
54 eqid 2651 . . . . . . . . . . 11 ((ran 𝑄 × ran 𝑄) ∖ I ) = ((ran 𝑄 × ran 𝑄) ∖ I )
55 eqid 2651 . . . . . . . . . . 11 ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )) = ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I ))
56 eqid 2651 . . . . . . . . . . 11 inf(ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )), ℝ, < ) = inf(ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )), ℝ, < )
57 fourierdlem54.d . . . . . . . . . . 11 (𝜑𝐷 ∈ ℝ)
58 eqid 2651 . . . . . . . . . . 11 (topGen‘ran (,)) = (topGen‘ran (,))
59 eqid 2651 . . . . . . . . . . 11 ((topGen‘ran (,)) ↾t (𝐶[,]𝐷)) = ((topGen‘ran (,)) ↾t (𝐶[,]𝐷))
60 oveq1 6697 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥 + (𝑘 · 𝑇)) = (𝑤 + (𝑘 · 𝑇)))
6160eleq1d 2715 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6261rexbidv 3081 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6362cbvrabv 3230 . . . . . . . . . . 11 {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑤 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}
64 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑖 · 𝑇) = (𝑗 · 𝑇))
6564oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑦 + (𝑖 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
6665eleq1d 2715 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
6766anbi1d 741 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄)))
68 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑘 → (𝑙 · 𝑇) = (𝑘 · 𝑇))
6968oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑙 = 𝑘 → (𝑧 + (𝑙 · 𝑇)) = (𝑧 + (𝑘 · 𝑇)))
7069eleq1d 2715 . . . . . . . . . . . . . 14 (𝑙 = 𝑘 → ((𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7170anbi2d 740 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → (((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄)))
7267, 71cbvrex2v 3210 . . . . . . . . . . . 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))
7372anbi2i 730 . . . . . . . . . . 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)))
7417, 18, 19, 20, 23, 35, 46, 52, 53, 54, 55, 56, 4, 57, 58, 59, 63, 73fourierdlem42 40684 . . . . . . . . . 10 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ∈ Fin)
75 unfi 8268 . . . . . . . . . 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)
7612, 74, 75sylancr 696 . . . . . . . . 9 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ∈ Fin)
778, 76syl5eqel 2734 . . . . . . . 8 (𝜑𝐻 ∈ Fin)
78 hashnncl 13195 . . . . . . . 8 (𝐻 ∈ Fin → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
7977, 78syl 17 . . . . . . 7 (𝜑 → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
8011, 79mpbird 247 . . . . . 6 (𝜑 → (#‘𝐻) ∈ ℕ)
8180nnzd 11519 . . . . 5 (𝜑 → (#‘𝐻) ∈ ℤ)
82 fourierdlem54.cd . . . . . . . . 9 (𝜑𝐶 < 𝐷)
834, 82ltned 10211 . . . . . . . 8 (𝜑𝐶𝐷)
84 hashprg 13220 . . . . . . . . 9 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶𝐷 ↔ (#‘{𝐶, 𝐷}) = 2))
854, 57, 84syl2anc 694 . . . . . . . 8 (𝜑 → (𝐶𝐷 ↔ (#‘{𝐶, 𝐷}) = 2))
8683, 85mpbid 222 . . . . . . 7 (𝜑 → (#‘{𝐶, 𝐷}) = 2)
8786eqcomd 2657 . . . . . 6 (𝜑 → 2 = (#‘{𝐶, 𝐷}))
88 ssun1 3809 . . . . . . . . 9 {𝐶, 𝐷} ⊆ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
8988a1i 11 . . . . . . . 8 (𝜑 → {𝐶, 𝐷} ⊆ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
9089, 8syl6sseqr 3685 . . . . . . 7 (𝜑 → {𝐶, 𝐷} ⊆ 𝐻)
91 hashssle 39826 . . . . . . 7 ((𝐻 ∈ Fin ∧ {𝐶, 𝐷} ⊆ 𝐻) → (#‘{𝐶, 𝐷}) ≤ (#‘𝐻))
9277, 90, 91syl2anc 694 . . . . . 6 (𝜑 → (#‘{𝐶, 𝐷}) ≤ (#‘𝐻))
9387, 92eqbrtrd 4707 . . . . 5 (𝜑 → 2 ≤ (#‘𝐻))
94 eluz2 11731 . . . . 5 ((#‘𝐻) ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ (#‘𝐻) ∈ ℤ ∧ 2 ≤ (#‘𝐻)))
953, 81, 93, 94syl3anbrc 1265 . . . 4 (𝜑 → (#‘𝐻) ∈ (ℤ‘2))
96 uz2m1nn 11801 . . . 4 ((#‘𝐻) ∈ (ℤ‘2) → ((#‘𝐻) − 1) ∈ ℕ)
9795, 96syl 17 . . 3 (𝜑 → ((#‘𝐻) − 1) ∈ ℕ)
981, 97syl5eqel 2734 . 2 (𝜑𝑁 ∈ ℕ)
99 prssg 4382 . . . . . . . . . . . . 13 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐶, 𝐷} ⊆ ℝ))
1004, 57, 99syl2anc 694 . . . . . . . . . . . 12 (𝜑 → ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐶, 𝐷} ⊆ ℝ))
1014, 57, 100mpbi2and 976 . . . . . . . . . . 11 (𝜑 → {𝐶, 𝐷} ⊆ ℝ)
102 ssrab2 3720 . . . . . . . . . . . 12 {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ (𝐶[,]𝐷)
1034, 57iccssred 40045 . . . . . . . . . . . 12 (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
104102, 103syl5ss 3647 . . . . . . . . . . 11 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ ℝ)
105101, 104unssd 3822 . . . . . . . . . 10 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ⊆ ℝ)
1068, 105syl5eqss 3682 . . . . . . . . 9 (𝜑𝐻 ⊆ ℝ)
107 fourierdlem54.s . . . . . . . . 9 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
10877, 106, 107, 1fourierdlem36 40678 . . . . . . . 8 (𝜑𝑆 Isom < , < ((0...𝑁), 𝐻))
109 df-isom 5935 . . . . . . . 8 (𝑆 Isom < , < ((0...𝑁), 𝐻) ↔ (𝑆:(0...𝑁)–1-1-onto𝐻 ∧ ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦))))
110108, 109sylib 208 . . . . . . 7 (𝜑 → (𝑆:(0...𝑁)–1-1-onto𝐻 ∧ ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦))))
111110simpld 474 . . . . . 6 (𝜑𝑆:(0...𝑁)–1-1-onto𝐻)
112 f1of 6175 . . . . . 6 (𝑆:(0...𝑁)–1-1-onto𝐻𝑆:(0...𝑁)⟶𝐻)
113111, 112syl 17 . . . . 5 (𝜑𝑆:(0...𝑁)⟶𝐻)
114113, 106fssd 6095 . . . 4 (𝜑𝑆:(0...𝑁)⟶ℝ)
115 reex 10065 . . . . 5 ℝ ∈ V
116 ovex 6718 . . . . . 6 (0...𝑁) ∈ V
117116a1i 11 . . . . 5 (𝜑 → (0...𝑁) ∈ V)
118 elmapg 7912 . . . . 5 ((ℝ ∈ V ∧ (0...𝑁) ∈ V) → (𝑆 ∈ (ℝ ↑𝑚 (0...𝑁)) ↔ 𝑆:(0...𝑁)⟶ℝ))
119115, 117, 118sylancr 696 . . . 4 (𝜑 → (𝑆 ∈ (ℝ ↑𝑚 (0...𝑁)) ↔ 𝑆:(0...𝑁)⟶ℝ))
120114, 119mpbird 247 . . 3 (𝜑𝑆 ∈ (ℝ ↑𝑚 (0...𝑁)))
121 df-f1o 5933 . . . . . . . . . . 11 (𝑆:(0...𝑁)–1-1-onto𝐻 ↔ (𝑆:(0...𝑁)–1-1𝐻𝑆:(0...𝑁)–onto𝐻))
122111, 121sylib 208 . . . . . . . . . 10 (𝜑 → (𝑆:(0...𝑁)–1-1𝐻𝑆:(0...𝑁)–onto𝐻))
123122simprd 478 . . . . . . . . 9 (𝜑𝑆:(0...𝑁)–onto𝐻)
124 dffo3 6414 . . . . . . . . 9 (𝑆:(0...𝑁)–onto𝐻 ↔ (𝑆:(0...𝑁)⟶𝐻 ∧ ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦)))
125123, 124sylib 208 . . . . . . . 8 (𝜑 → (𝑆:(0...𝑁)⟶𝐻 ∧ ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦)))
126125simprd 478 . . . . . . 7 (𝜑 → ∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦))
127 eqeq1 2655 . . . . . . . . . 10 ( = 𝐶 → ( = (𝑆𝑦) ↔ 𝐶 = (𝑆𝑦)))
128 eqcom 2658 . . . . . . . . . 10 (𝐶 = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐶)
129127, 128syl6bb 276 . . . . . . . . 9 ( = 𝐶 → ( = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐶))
130129rexbidv 3081 . . . . . . . 8 ( = 𝐶 → (∃𝑦 ∈ (0...𝑁) = (𝑆𝑦) ↔ ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶))
131130rspcv 3336 . . . . . . 7 (𝐶𝐻 → (∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦) → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶))
1329, 126, 131sylc 65 . . . . . 6 (𝜑 → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶)
133 fveq2 6229 . . . . . . . . . . . . . 14 (𝑦 = 0 → (𝑆𝑦) = (𝑆‘0))
134133eqcomd 2657 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑆‘0) = (𝑆𝑦))
135134adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) = (𝑆𝑦))
136 simplr 807 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆𝑦) = 𝐶)
137135, 136eqtrd 2685 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) = 𝐶)
1384ad2antrr 762 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → 𝐶 ∈ ℝ)
139137, 138eqeltrd 2730 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
140139, 137eqled 10178 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
1411403adantl2 1238 . . . . . . . 8 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
1424rexrd 10127 . . . . . . . . . . . . . . . . 17 (𝜑𝐶 ∈ ℝ*)
14357rexrd 10127 . . . . . . . . . . . . . . . . 17 (𝜑𝐷 ∈ ℝ*)
1444, 57, 82ltled 10223 . . . . . . . . . . . . . . . . 17 (𝜑𝐶𝐷)
145 lbicc2 12326 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*𝐶𝐷) → 𝐶 ∈ (𝐶[,]𝐷))
146142, 143, 144, 145syl3anc 1366 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ (𝐶[,]𝐷))
147 ubicc2 12327 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*𝐶𝐷) → 𝐷 ∈ (𝐶[,]𝐷))
148142, 143, 144, 147syl3anc 1366 . . . . . . . . . . . . . . . 16 (𝜑𝐷 ∈ (𝐶[,]𝐷))
149 prssg 4382 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) → ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) ↔ {𝐶, 𝐷} ⊆ (𝐶[,]𝐷)))
150146, 148, 149syl2anc 694 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) ↔ {𝐶, 𝐷} ⊆ (𝐶[,]𝐷)))
151146, 148, 150mpbi2and 976 . . . . . . . . . . . . . . 15 (𝜑 → {𝐶, 𝐷} ⊆ (𝐶[,]𝐷))
152102a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ (𝐶[,]𝐷))
153151, 152unssd 3822 . . . . . . . . . . . . . 14 (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ⊆ (𝐶[,]𝐷))
1548, 153syl5eqss 3682 . . . . . . . . . . . . 13 (𝜑𝐻 ⊆ (𝐶[,]𝐷))
155 nnm1nn0 11372 . . . . . . . . . . . . . . . . . 18 ((#‘𝐻) ∈ ℕ → ((#‘𝐻) − 1) ∈ ℕ0)
15680, 155syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((#‘𝐻) − 1) ∈ ℕ0)
1571, 156syl5eqel 2734 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ0)
158157, 40syl6eleq 2740 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘0))
159 eluzfz1 12386 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
160158, 159syl 17 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ (0...𝑁))
161113, 160ffvelrnd 6400 . . . . . . . . . . . . 13 (𝜑 → (𝑆‘0) ∈ 𝐻)
162154, 161sseldd 3637 . . . . . . . . . . . 12 (𝜑 → (𝑆‘0) ∈ (𝐶[,]𝐷))
163103, 162sseldd 3637 . . . . . . . . . . 11 (𝜑 → (𝑆‘0) ∈ ℝ)
164163adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
1651643ad2antl1 1243 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
1664adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑦 = 0) → 𝐶 ∈ ℝ)
1671663ad2antl1 1243 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝐶 ∈ ℝ)
168 elfzelz 12380 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑁) → 𝑦 ∈ ℤ)
169168zred 11520 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 𝑦 ∈ ℝ)
170169adantr 480 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 𝑦 ∈ ℝ)
171 elfzle1 12382 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 0 ≤ 𝑦)
172171adantr 480 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 0 ≤ 𝑦)
173 neqne 2831 . . . . . . . . . . . . . 14 𝑦 = 0 → 𝑦 ≠ 0)
174173adantl 481 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 𝑦 ≠ 0)
175170, 172, 174ne0gt0d 10212 . . . . . . . . . . . 12 ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 0 < 𝑦)
1761753ad2antl2 1244 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 0 < 𝑦)
177 simpl1 1084 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝜑)
178 simpl2 1085 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝑦 ∈ (0...𝑁))
179110simprd 478 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)))
180 breq1 4688 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
181 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → (𝑆𝑥) = (𝑆‘0))
182181breq1d 4695 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆‘0) < (𝑆𝑦)))
183180, 182bibi12d 334 . . . . . . . . . . . . . . . 16 (𝑥 = 0 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦))))
184183ralbidv 3015 . . . . . . . . . . . . . . 15 (𝑥 = 0 → (∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦))))
185184rspcv 3336 . . . . . . . . . . . . . 14 (0 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) → ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦))))
186160, 179, 185sylc 65 . . . . . . . . . . . . 13 (𝜑 → ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦)))
187186r19.21bi 2961 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦)))
188177, 178, 187syl2anc 694 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (0 < 𝑦 ↔ (𝑆‘0) < (𝑆𝑦)))
189176, 188mpbid 222 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) < (𝑆𝑦))
190 simpl3 1086 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆𝑦) = 𝐶)
191189, 190breqtrd 4711 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) < 𝐶)
192165, 167, 191ltled 10223 . . . . . . . 8 (((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
193141, 192pm2.61dan 849 . . . . . . 7 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐶) → (𝑆‘0) ≤ 𝐶)
194193rexlimdv3a 3062 . . . . . 6 (𝜑 → (∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐶 → (𝑆‘0) ≤ 𝐶))
195132, 194mpd 15 . . . . 5 (𝜑 → (𝑆‘0) ≤ 𝐶)
196 elicc2 12276 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘0) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷)))
1974, 57, 196syl2anc 694 . . . . . . 7 (𝜑 → ((𝑆‘0) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷)))
198162, 197mpbid 222 . . . . . 6 (𝜑 → ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷))
199198simp2d 1094 . . . . 5 (𝜑𝐶 ≤ (𝑆‘0))
200163, 4letri3d 10217 . . . . 5 (𝜑 → ((𝑆‘0) = 𝐶 ↔ ((𝑆‘0) ≤ 𝐶𝐶 ≤ (𝑆‘0))))
201195, 199, 200mpbir2and 977 . . . 4 (𝜑 → (𝑆‘0) = 𝐶)
202 eluzfz2 12387 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
203158, 202syl 17 . . . . . . . . 9 (𝜑𝑁 ∈ (0...𝑁))
204113, 203ffvelrnd 6400 . . . . . . . 8 (𝜑 → (𝑆𝑁) ∈ 𝐻)
205154, 204sseldd 3637 . . . . . . 7 (𝜑 → (𝑆𝑁) ∈ (𝐶[,]𝐷))
206 elicc2 12276 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆𝑁) ∈ (𝐶[,]𝐷) ↔ ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷)))
2074, 57, 206syl2anc 694 . . . . . . 7 (𝜑 → ((𝑆𝑁) ∈ (𝐶[,]𝐷) ↔ ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷)))
208205, 207mpbid 222 . . . . . 6 (𝜑 → ((𝑆𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆𝑁) ∧ (𝑆𝑁) ≤ 𝐷))
209208simp3d 1095 . . . . 5 (𝜑 → (𝑆𝑁) ≤ 𝐷)
210 prid2g 4328 . . . . . . . . 9 (𝐷 ∈ ℝ → 𝐷 ∈ {𝐶, 𝐷})
211 elun1 3813 . . . . . . . . 9 (𝐷 ∈ {𝐶, 𝐷} → 𝐷 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
21257, 210, 2113syl 18 . . . . . . . 8 (𝜑𝐷 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
213212, 8syl6eleqr 2741 . . . . . . 7 (𝜑𝐷𝐻)
214 eqeq1 2655 . . . . . . . . . 10 ( = 𝐷 → ( = (𝑆𝑦) ↔ 𝐷 = (𝑆𝑦)))
215 eqcom 2658 . . . . . . . . . 10 (𝐷 = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐷)
216214, 215syl6bb 276 . . . . . . . . 9 ( = 𝐷 → ( = (𝑆𝑦) ↔ (𝑆𝑦) = 𝐷))
217216rexbidv 3081 . . . . . . . 8 ( = 𝐷 → (∃𝑦 ∈ (0...𝑁) = (𝑆𝑦) ↔ ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷))
218217rspcv 3336 . . . . . . 7 (𝐷𝐻 → (∀𝐻𝑦 ∈ (0...𝑁) = (𝑆𝑦) → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷))
219213, 126, 218sylc 65 . . . . . 6 (𝜑 → ∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷)
220215biimpri 218 . . . . . . . . 9 ((𝑆𝑦) = 𝐷𝐷 = (𝑆𝑦))
2212203ad2ant3 1104 . . . . . . . 8 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → 𝐷 = (𝑆𝑦))
222114ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑦) ∈ ℝ)
223103, 205sseldd 3637 . . . . . . . . . . 11 (𝜑 → (𝑆𝑁) ∈ ℝ)
224223adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑁) ∈ ℝ)
225169adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑦 ∈ ℝ)
226 elfzel2 12378 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
227226zred 11520 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
228227adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
229 elfzle2 12383 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑁) → 𝑦𝑁)
230229adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...𝑁)) → 𝑦𝑁)
231225, 228, 230lensymd 10226 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...𝑁)) → ¬ 𝑁 < 𝑦)
232 breq1 4688 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑁 → (𝑥 < 𝑦𝑁 < 𝑦))
233 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑁 → (𝑆𝑥) = (𝑆𝑁))
234233breq1d 4695 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑁 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆𝑁) < (𝑆𝑦)))
235232, 234bibi12d 334 . . . . . . . . . . . . . . 15 (𝑥 = 𝑁 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦))))
236235ralbidv 3015 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦))))
237236rspcv 3336 . . . . . . . . . . . . 13 (𝑁 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) → ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦))))
238203, 179, 237sylc 65 . . . . . . . . . . . 12 (𝜑 → ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦)))
239238r19.21bi 2961 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑁 < 𝑦 ↔ (𝑆𝑁) < (𝑆𝑦)))
240231, 239mtbid 313 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...𝑁)) → ¬ (𝑆𝑁) < (𝑆𝑦))
241222, 224, 240nltled 10225 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...𝑁)) → (𝑆𝑦) ≤ (𝑆𝑁))
2422413adant3 1101 . . . . . . . 8 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → (𝑆𝑦) ≤ (𝑆𝑁))
243221, 242eqbrtrd 4707 . . . . . . 7 ((𝜑𝑦 ∈ (0...𝑁) ∧ (𝑆𝑦) = 𝐷) → 𝐷 ≤ (𝑆𝑁))
244243rexlimdv3a 3062 . . . . . 6 (𝜑 → (∃𝑦 ∈ (0...𝑁)(𝑆𝑦) = 𝐷𝐷 ≤ (𝑆𝑁)))
245219, 244mpd 15 . . . . 5 (𝜑𝐷 ≤ (𝑆𝑁))
246223, 57letri3d 10217 . . . . 5 (𝜑 → ((𝑆𝑁) = 𝐷 ↔ ((𝑆𝑁) ≤ 𝐷𝐷 ≤ (𝑆𝑁))))
247209, 245, 246mpbir2and 977 . . . 4 (𝜑 → (𝑆𝑁) = 𝐷)
248 elfzoelz 12509 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
249248zred 11520 . . . . . . . 8 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
250249ltp1d 10992 . . . . . . 7 (𝑖 ∈ (0..^𝑁) → 𝑖 < (𝑖 + 1))
251250adantl 481 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 < (𝑖 + 1))
252179adantr 480 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑁)) → ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)))
253 elfzofz 12524 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0...𝑁))
254253adantl 481 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0...𝑁))
255 fzofzp1 12605 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
256255adantl 481 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
257 breq1 4688 . . . . . . . . . 10 (𝑥 = 𝑖 → (𝑥 < 𝑦𝑖 < 𝑦))
258 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑆𝑥) = (𝑆𝑖))
259258breq1d 4695 . . . . . . . . . 10 (𝑥 = 𝑖 → ((𝑆𝑥) < (𝑆𝑦) ↔ (𝑆𝑖) < (𝑆𝑦)))
260257, 259bibi12d 334 . . . . . . . . 9 (𝑥 = 𝑖 → ((𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) ↔ (𝑖 < 𝑦 ↔ (𝑆𝑖) < (𝑆𝑦))))
261 breq2 4689 . . . . . . . . . 10 (𝑦 = (𝑖 + 1) → (𝑖 < 𝑦𝑖 < (𝑖 + 1)))
262 fveq2 6229 . . . . . . . . . . 11 (𝑦 = (𝑖 + 1) → (𝑆𝑦) = (𝑆‘(𝑖 + 1)))
263262breq2d 4697 . . . . . . . . . 10 (𝑦 = (𝑖 + 1) → ((𝑆𝑖) < (𝑆𝑦) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1))))
264261, 263bibi12d 334 . . . . . . . . 9 (𝑦 = (𝑖 + 1) → ((𝑖 < 𝑦 ↔ (𝑆𝑖) < (𝑆𝑦)) ↔ (𝑖 < (𝑖 + 1) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1)))))
265260, 264rspc2v 3353 . . . . . . . 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)))))
266254, 256, 265syl2anc 694 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆𝑥) < (𝑆𝑦)) → (𝑖 < (𝑖 + 1) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1)))))
267252, 266mpd 15 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 < (𝑖 + 1) ↔ (𝑆𝑖) < (𝑆‘(𝑖 + 1))))
268251, 267mpbid 222 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑆𝑖) < (𝑆‘(𝑖 + 1)))
269268ralrimiva 2995 . . . 4 (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1)))
270201, 247, 269jca31 556 . . 3 (𝜑 → (((𝑆‘0) = 𝐶 ∧ (𝑆𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1))))
271 fourierdlem54.o . . . . 5 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
272271fourierdlem2 40644 . . . 4 (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂𝑁) ↔ (𝑆 ∈ (ℝ ↑𝑚 (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1))))))
27398, 272syl 17 . . 3 (𝜑 → (𝑆 ∈ (𝑂𝑁) ↔ (𝑆 ∈ (ℝ ↑𝑚 (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆𝑖) < (𝑆‘(𝑖 + 1))))))
274120, 270, 273mpbir2and 977 . 2 (𝜑𝑆 ∈ (𝑂𝑁))
27598, 274, 108jca31 556 1 (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  {cpr 4212   class class class wbr 4685  cmpt 4762   I cid 5052   × cxp 5141  ran crn 5144  cres 5145  ccom 5147  cio 5887   Fn wfn 5921  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  infcinf 8388  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  (,)cioo 12213  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504  #chash 13157  abscabs 14018  t crest 16128  topGenctg 16145
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-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  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  ax-pre-sup 10052
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-rmo 2949  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-int 4508  df-iun 4554  df-iin 4555  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-se 5103  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-isom 5935  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-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-cmp 21238
This theorem is referenced by:  fourierdlem63  40704  fourierdlem64  40705  fourierdlem65  40706  fourierdlem79  40720  fourierdlem89  40730  fourierdlem90  40731  fourierdlem91  40732  fourierdlem100  40741  fourierdlem107  40748  fourierdlem109  40750  fourierdlem112  40753
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