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Theorem fourierdlem54 44475
Description: Given a partition 𝑄 and an arbitrary interval [𝐢, 𝐷], a partition 𝑆 on [𝐢, 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 = (𝐡 βˆ’ 𝐴)
fourierdlem54.p 𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})
fourierdlem54.m (πœ‘ β†’ 𝑀 ∈ β„•)
fourierdlem54.q (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))
fourierdlem54.c (πœ‘ β†’ 𝐢 ∈ ℝ)
fourierdlem54.d (πœ‘ β†’ 𝐷 ∈ ℝ)
fourierdlem54.cd (πœ‘ β†’ 𝐢 < 𝐷)
fourierdlem54.o 𝑂 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐢 ∧ (π‘β€˜π‘š) = 𝐷) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})
fourierdlem54.h 𝐻 = ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄})
fourierdlem54.n 𝑁 = ((β™―β€˜π») βˆ’ 1)
fourierdlem54.s 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
Assertion
Ref Expression
fourierdlem54 (πœ‘ β†’ ((𝑁 ∈ β„• ∧ 𝑆 ∈ (π‘‚β€˜π‘)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
Distinct variable groups:   𝐴,𝑖,π‘š,𝑝   𝐡,𝑖,π‘š,𝑝   𝐢,π‘š,𝑝   π‘₯,𝐢   𝐷,π‘š,𝑝   π‘₯,𝐷   𝑓,𝐻   π‘₯,𝐻   𝑖,𝑀,π‘š,𝑝   𝑓,𝑁   𝑖,𝑁,π‘š,𝑝   π‘₯,𝑁,𝑖   𝑄,𝑖,π‘˜   𝑄,𝑝   π‘₯,𝑄,π‘˜   𝑆,𝑓   𝑆,𝑖,𝑝   π‘₯,𝑆   𝑇,𝑖,π‘˜,π‘₯   πœ‘,𝑓   πœ‘,𝑖,π‘˜
Allowed substitution hints:   πœ‘(π‘₯,π‘š,𝑝)   𝐴(π‘₯,𝑓,π‘˜)   𝐡(π‘₯,𝑓,π‘˜)   𝐢(𝑓,𝑖,π‘˜)   𝐷(𝑓,𝑖,π‘˜)   𝑃(π‘₯,𝑓,𝑖,π‘˜,π‘š,𝑝)   𝑄(𝑓,π‘š)   𝑆(π‘˜,π‘š)   𝑇(𝑓,π‘š,𝑝)   𝐻(𝑖,π‘˜,π‘š,𝑝)   𝑀(π‘₯,𝑓,π‘˜)   𝑁(π‘˜)   𝑂(π‘₯,𝑓,𝑖,π‘˜,π‘š,𝑝)

Proof of Theorem fourierdlem54
Dummy variables 𝑀 β„Ž 𝑦 𝑧 𝑗 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem54.n . . 3 𝑁 = ((β™―β€˜π») βˆ’ 1)
2 2z 12542 . . . . . 6 2 ∈ β„€
32a1i 11 . . . . 5 (πœ‘ β†’ 2 ∈ β„€)
4 fourierdlem54.c . . . . . . . . . 10 (πœ‘ β†’ 𝐢 ∈ ℝ)
5 prid1g 4726 . . . . . . . . . 10 (𝐢 ∈ ℝ β†’ 𝐢 ∈ {𝐢, 𝐷})
6 elun1 4141 . . . . . . . . . 10 (𝐢 ∈ {𝐢, 𝐷} β†’ 𝐢 ∈ ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}))
74, 5, 63syl 18 . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}))
8 fourierdlem54.h . . . . . . . . 9 𝐻 = ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄})
97, 8eleqtrrdi 2849 . . . . . . . 8 (πœ‘ β†’ 𝐢 ∈ 𝐻)
109ne0d 4300 . . . . . . 7 (πœ‘ β†’ 𝐻 β‰  βˆ…)
11 prfi 9273 . . . . . . . . . 10 {𝐢, 𝐷} ∈ Fin
12 fourierdlem54.p . . . . . . . . . . . . 13 𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})
13 fourierdlem54.m . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑀 ∈ β„•)
14 fourierdlem54.q . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))
1512, 13, 14fourierdlem11 44433 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡))
1615simp1d 1143 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ ℝ)
1715simp2d 1144 . . . . . . . . . . 11 (πœ‘ β†’ 𝐡 ∈ ℝ)
1815simp3d 1145 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 < 𝐡)
19 fourierdlem54.t . . . . . . . . . . 11 𝑇 = (𝐡 βˆ’ 𝐴)
2012, 13, 14fourierdlem15 44437 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))
21 frn 6680 . . . . . . . . . . . 12 (𝑄:(0...𝑀)⟢(𝐴[,]𝐡) β†’ ran 𝑄 βŠ† (𝐴[,]𝐡))
2220, 21syl 17 . . . . . . . . . . 11 (πœ‘ β†’ ran 𝑄 βŠ† (𝐴[,]𝐡))
2312fourierdlem2 44424 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ β„• β†’ (𝑄 ∈ (π‘ƒβ€˜π‘€) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))))
2413, 23syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑄 ∈ (π‘ƒβ€˜π‘€) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))))
2514, 24mpbid 231 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)))))
2625simpld 496 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑄 ∈ (ℝ ↑m (0...𝑀)))
27 elmapi 8794 . . . . . . . . . . . . . 14 (𝑄 ∈ (ℝ ↑m (0...𝑀)) β†’ 𝑄:(0...𝑀)βŸΆβ„)
28 ffn 6673 . . . . . . . . . . . . . 14 (𝑄:(0...𝑀)βŸΆβ„ β†’ 𝑄 Fn (0...𝑀))
2926, 27, 283syl 18 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑄 Fn (0...𝑀))
30 fzfid 13885 . . . . . . . . . . . . 13 (πœ‘ β†’ (0...𝑀) ∈ Fin)
31 fnfi 9132 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ (0...𝑀) ∈ Fin) β†’ 𝑄 ∈ Fin)
3229, 30, 31syl2anc 585 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑄 ∈ Fin)
33 rnfi 9286 . . . . . . . . . . . 12 (𝑄 ∈ Fin β†’ ran 𝑄 ∈ Fin)
3432, 33syl 17 . . . . . . . . . . 11 (πœ‘ β†’ ran 𝑄 ∈ Fin)
3525simprd 497 . . . . . . . . . . . . . 14 (πœ‘ β†’ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))
3635simpld 496 . . . . . . . . . . . . 13 (πœ‘ β†’ ((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡))
3736simpld 496 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘„β€˜0) = 𝐴)
3813nnnn0d 12480 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑀 ∈ β„•0)
39 nn0uz 12812 . . . . . . . . . . . . . . 15 β„•0 = (β„€β‰₯β€˜0)
4038, 39eleqtrdi 2848 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜0))
41 eluzfz1 13455 . . . . . . . . . . . . . 14 (𝑀 ∈ (β„€β‰₯β€˜0) β†’ 0 ∈ (0...𝑀))
4240, 41syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 0 ∈ (0...𝑀))
43 fnfvelrn 7036 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ 0 ∈ (0...𝑀)) β†’ (π‘„β€˜0) ∈ ran 𝑄)
4429, 42, 43syl2anc 585 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘„β€˜0) ∈ ran 𝑄)
4537, 44eqeltrrd 2839 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ ran 𝑄)
4636simprd 497 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘„β€˜π‘€) = 𝐡)
47 eluzfz2 13456 . . . . . . . . . . . . . 14 (𝑀 ∈ (β„€β‰₯β€˜0) β†’ 𝑀 ∈ (0...𝑀))
4840, 47syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑀 ∈ (0...𝑀))
49 fnfvelrn 7036 . . . . . . . . . . . . 13 ((𝑄 Fn (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘€) ∈ ran 𝑄)
5029, 48, 49syl2anc 585 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘„β€˜π‘€) ∈ ran 𝑄)
5146, 50eqeltrrd 2839 . . . . . . . . . . 11 (πœ‘ β†’ 𝐡 ∈ ran 𝑄)
52 eqid 2737 . . . . . . . . . . 11 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
53 eqid 2737 . . . . . . . . . . 11 ((ran 𝑄 Γ— ran 𝑄) βˆ– I ) = ((ran 𝑄 Γ— ran 𝑄) βˆ– I )
54 eqid 2737 . . . . . . . . . . 11 ran ((abs ∘ βˆ’ ) β†Ύ ((ran 𝑄 Γ— ran 𝑄) βˆ– I )) = ran ((abs ∘ βˆ’ ) β†Ύ ((ran 𝑄 Γ— ran 𝑄) βˆ– I ))
55 eqid 2737 . . . . . . . . . . 11 inf(ran ((abs ∘ βˆ’ ) β†Ύ ((ran 𝑄 Γ— ran 𝑄) βˆ– I )), ℝ, < ) = inf(ran ((abs ∘ βˆ’ ) β†Ύ ((ran 𝑄 Γ— ran 𝑄) βˆ– I )), ℝ, < )
56 fourierdlem54.d . . . . . . . . . . 11 (πœ‘ β†’ 𝐷 ∈ ℝ)
57 eqid 2737 . . . . . . . . . . 11 (topGenβ€˜ran (,)) = (topGenβ€˜ran (,))
58 eqid 2737 . . . . . . . . . . 11 ((topGenβ€˜ran (,)) β†Ύt (𝐢[,]𝐷)) = ((topGenβ€˜ran (,)) β†Ύt (𝐢[,]𝐷))
59 oveq1 7369 . . . . . . . . . . . . . 14 (π‘₯ = 𝑀 β†’ (π‘₯ + (π‘˜ Β· 𝑇)) = (𝑀 + (π‘˜ Β· 𝑇)))
6059eleq1d 2823 . . . . . . . . . . . . 13 (π‘₯ = 𝑀 β†’ ((π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄 ↔ (𝑀 + (π‘˜ Β· 𝑇)) ∈ ran 𝑄))
6160rexbidv 3176 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ (βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄 ↔ βˆƒπ‘˜ ∈ β„€ (𝑀 + (π‘˜ Β· 𝑇)) ∈ ran 𝑄))
6261cbvrabv 3420 . . . . . . . . . . 11 {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄} = {𝑀 ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (𝑀 + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}
63 oveq1 7369 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 β†’ (𝑖 Β· 𝑇) = (𝑗 Β· 𝑇))
6463oveq2d 7378 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 β†’ (𝑦 + (𝑖 Β· 𝑇)) = (𝑦 + (𝑗 Β· 𝑇)))
6564eleq1d 2823 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 β†’ ((𝑦 + (𝑖 Β· 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 Β· 𝑇)) ∈ ran 𝑄))
6665anbi1d 631 . . . . . . . . . . . . 13 (𝑖 = 𝑗 β†’ (((𝑦 + (𝑖 Β· 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 Β· 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 Β· 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 Β· 𝑇)) ∈ ran 𝑄)))
67 oveq1 7369 . . . . . . . . . . . . . . . 16 (𝑙 = π‘˜ β†’ (𝑙 Β· 𝑇) = (π‘˜ Β· 𝑇))
6867oveq2d 7378 . . . . . . . . . . . . . . 15 (𝑙 = π‘˜ β†’ (𝑧 + (𝑙 Β· 𝑇)) = (𝑧 + (π‘˜ Β· 𝑇)))
6968eleq1d 2823 . . . . . . . . . . . . . 14 (𝑙 = π‘˜ β†’ ((𝑧 + (𝑙 Β· 𝑇)) ∈ ran 𝑄 ↔ (𝑧 + (π‘˜ Β· 𝑇)) ∈ ran 𝑄))
7069anbi2d 630 . . . . . . . . . . . . 13 (𝑙 = π‘˜ β†’ (((𝑦 + (𝑗 Β· 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 Β· 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 Β· 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (π‘˜ Β· 𝑇)) ∈ ran 𝑄)))
7166, 70cbvrex2vw 3231 . . . . . . . . . . . 12 (βˆƒπ‘– ∈ β„€ βˆƒπ‘™ ∈ β„€ ((𝑦 + (𝑖 Β· 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 Β· 𝑇)) ∈ ran 𝑄) ↔ βˆƒπ‘— ∈ β„€ βˆƒπ‘˜ ∈ β„€ ((𝑦 + (𝑗 Β· 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (π‘˜ Β· 𝑇)) ∈ ran 𝑄))
7271anbi2i 624 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑦 < 𝑧)) ∧ βˆƒπ‘– ∈ β„€ βˆƒπ‘™ ∈ β„€ ((𝑦 + (𝑖 Β· 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 Β· 𝑇)) ∈ ran 𝑄)) ↔ ((πœ‘ ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑦 < 𝑧)) ∧ βˆƒπ‘— ∈ β„€ βˆƒπ‘˜ ∈ β„€ ((𝑦 + (𝑗 Β· 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (π‘˜ Β· 𝑇)) ∈ ran 𝑄)))
7316, 17, 18, 19, 22, 34, 45, 51, 52, 53, 54, 55, 4, 56, 57, 58, 62, 72fourierdlem42 44464 . . . . . . . . . 10 (πœ‘ β†’ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄} ∈ Fin)
74 unfi 9123 . . . . . . . . . 10 (({𝐢, 𝐷} ∈ Fin ∧ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄} ∈ Fin) β†’ ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}) ∈ Fin)
7511, 73, 74sylancr 588 . . . . . . . . 9 (πœ‘ β†’ ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}) ∈ Fin)
768, 75eqeltrid 2842 . . . . . . . 8 (πœ‘ β†’ 𝐻 ∈ Fin)
77 hashnncl 14273 . . . . . . . 8 (𝐻 ∈ Fin β†’ ((β™―β€˜π») ∈ β„• ↔ 𝐻 β‰  βˆ…))
7876, 77syl 17 . . . . . . 7 (πœ‘ β†’ ((β™―β€˜π») ∈ β„• ↔ 𝐻 β‰  βˆ…))
7910, 78mpbird 257 . . . . . 6 (πœ‘ β†’ (β™―β€˜π») ∈ β„•)
8079nnzd 12533 . . . . 5 (πœ‘ β†’ (β™―β€˜π») ∈ β„€)
81 fourierdlem54.cd . . . . . . . . 9 (πœ‘ β†’ 𝐢 < 𝐷)
824, 81ltned 11298 . . . . . . . 8 (πœ‘ β†’ 𝐢 β‰  𝐷)
83 hashprg 14302 . . . . . . . . 9 ((𝐢 ∈ ℝ ∧ 𝐷 ∈ ℝ) β†’ (𝐢 β‰  𝐷 ↔ (β™―β€˜{𝐢, 𝐷}) = 2))
844, 56, 83syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (𝐢 β‰  𝐷 ↔ (β™―β€˜{𝐢, 𝐷}) = 2))
8582, 84mpbid 231 . . . . . . 7 (πœ‘ β†’ (β™―β€˜{𝐢, 𝐷}) = 2)
8685eqcomd 2743 . . . . . 6 (πœ‘ β†’ 2 = (β™―β€˜{𝐢, 𝐷}))
87 ssun1 4137 . . . . . . . . 9 {𝐢, 𝐷} βŠ† ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄})
8887a1i 11 . . . . . . . 8 (πœ‘ β†’ {𝐢, 𝐷} βŠ† ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}))
8988, 8sseqtrrdi 4000 . . . . . . 7 (πœ‘ β†’ {𝐢, 𝐷} βŠ† 𝐻)
90 hashssle 43606 . . . . . . 7 ((𝐻 ∈ Fin ∧ {𝐢, 𝐷} βŠ† 𝐻) β†’ (β™―β€˜{𝐢, 𝐷}) ≀ (β™―β€˜π»))
9176, 89, 90syl2anc 585 . . . . . 6 (πœ‘ β†’ (β™―β€˜{𝐢, 𝐷}) ≀ (β™―β€˜π»))
9286, 91eqbrtrd 5132 . . . . 5 (πœ‘ β†’ 2 ≀ (β™―β€˜π»))
93 eluz2 12776 . . . . 5 ((β™―β€˜π») ∈ (β„€β‰₯β€˜2) ↔ (2 ∈ β„€ ∧ (β™―β€˜π») ∈ β„€ ∧ 2 ≀ (β™―β€˜π»)))
943, 80, 92, 93syl3anbrc 1344 . . . 4 (πœ‘ β†’ (β™―β€˜π») ∈ (β„€β‰₯β€˜2))
95 uz2m1nn 12855 . . . 4 ((β™―β€˜π») ∈ (β„€β‰₯β€˜2) β†’ ((β™―β€˜π») βˆ’ 1) ∈ β„•)
9694, 95syl 17 . . 3 (πœ‘ β†’ ((β™―β€˜π») βˆ’ 1) ∈ β„•)
971, 96eqeltrid 2842 . 2 (πœ‘ β†’ 𝑁 ∈ β„•)
98 prssg 4784 . . . . . . . . . . . . 13 ((𝐢 ∈ ℝ ∧ 𝐷 ∈ ℝ) β†’ ((𝐢 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐢, 𝐷} βŠ† ℝ))
994, 56, 98syl2anc 585 . . . . . . . . . . . 12 (πœ‘ β†’ ((𝐢 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐢, 𝐷} βŠ† ℝ))
1004, 56, 99mpbi2and 711 . . . . . . . . . . 11 (πœ‘ β†’ {𝐢, 𝐷} βŠ† ℝ)
101 ssrab2 4042 . . . . . . . . . . . 12 {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄} βŠ† (𝐢[,]𝐷)
1024, 56iccssred 13358 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐢[,]𝐷) βŠ† ℝ)
103101, 102sstrid 3960 . . . . . . . . . . 11 (πœ‘ β†’ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄} βŠ† ℝ)
104100, 103unssd 4151 . . . . . . . . . 10 (πœ‘ β†’ ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}) βŠ† ℝ)
1058, 104eqsstrid 3997 . . . . . . . . 9 (πœ‘ β†’ 𝐻 βŠ† ℝ)
106 fourierdlem54.s . . . . . . . . 9 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
10776, 105, 106, 1fourierdlem36 44458 . . . . . . . 8 (πœ‘ β†’ 𝑆 Isom < , < ((0...𝑁), 𝐻))
108 df-isom 6510 . . . . . . . 8 (𝑆 Isom < , < ((0...𝑁), 𝐻) ↔ (𝑆:(0...𝑁)–1-1-onto→𝐻 ∧ βˆ€π‘₯ ∈ (0...𝑁)βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦))))
109107, 108sylib 217 . . . . . . 7 (πœ‘ β†’ (𝑆:(0...𝑁)–1-1-onto→𝐻 ∧ βˆ€π‘₯ ∈ (0...𝑁)βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦))))
110109simpld 496 . . . . . 6 (πœ‘ β†’ 𝑆:(0...𝑁)–1-1-onto→𝐻)
111 f1of 6789 . . . . . 6 (𝑆:(0...𝑁)–1-1-onto→𝐻 β†’ 𝑆:(0...𝑁)⟢𝐻)
112110, 111syl 17 . . . . 5 (πœ‘ β†’ 𝑆:(0...𝑁)⟢𝐻)
113112, 105fssd 6691 . . . 4 (πœ‘ β†’ 𝑆:(0...𝑁)βŸΆβ„)
114 reex 11149 . . . . 5 ℝ ∈ V
115 ovex 7395 . . . . . 6 (0...𝑁) ∈ V
116115a1i 11 . . . . 5 (πœ‘ β†’ (0...𝑁) ∈ V)
117 elmapg 8785 . . . . 5 ((ℝ ∈ V ∧ (0...𝑁) ∈ V) β†’ (𝑆 ∈ (ℝ ↑m (0...𝑁)) ↔ 𝑆:(0...𝑁)βŸΆβ„))
118114, 116, 117sylancr 588 . . . 4 (πœ‘ β†’ (𝑆 ∈ (ℝ ↑m (0...𝑁)) ↔ 𝑆:(0...𝑁)βŸΆβ„))
119113, 118mpbird 257 . . 3 (πœ‘ β†’ 𝑆 ∈ (ℝ ↑m (0...𝑁)))
120 df-f1o 6508 . . . . . . . . . . 11 (𝑆:(0...𝑁)–1-1-onto→𝐻 ↔ (𝑆:(0...𝑁)–1-1→𝐻 ∧ 𝑆:(0...𝑁)–onto→𝐻))
121110, 120sylib 217 . . . . . . . . . 10 (πœ‘ β†’ (𝑆:(0...𝑁)–1-1→𝐻 ∧ 𝑆:(0...𝑁)–onto→𝐻))
122121simprd 497 . . . . . . . . 9 (πœ‘ β†’ 𝑆:(0...𝑁)–onto→𝐻)
123 dffo3 7057 . . . . . . . . 9 (𝑆:(0...𝑁)–onto→𝐻 ↔ (𝑆:(0...𝑁)⟢𝐻 ∧ βˆ€β„Ž ∈ 𝐻 βˆƒπ‘¦ ∈ (0...𝑁)β„Ž = (π‘†β€˜π‘¦)))
124122, 123sylib 217 . . . . . . . 8 (πœ‘ β†’ (𝑆:(0...𝑁)⟢𝐻 ∧ βˆ€β„Ž ∈ 𝐻 βˆƒπ‘¦ ∈ (0...𝑁)β„Ž = (π‘†β€˜π‘¦)))
125124simprd 497 . . . . . . 7 (πœ‘ β†’ βˆ€β„Ž ∈ 𝐻 βˆƒπ‘¦ ∈ (0...𝑁)β„Ž = (π‘†β€˜π‘¦))
126 eqeq1 2741 . . . . . . . . . 10 (β„Ž = 𝐢 β†’ (β„Ž = (π‘†β€˜π‘¦) ↔ 𝐢 = (π‘†β€˜π‘¦)))
127 eqcom 2744 . . . . . . . . . 10 (𝐢 = (π‘†β€˜π‘¦) ↔ (π‘†β€˜π‘¦) = 𝐢)
128126, 127bitrdi 287 . . . . . . . . 9 (β„Ž = 𝐢 β†’ (β„Ž = (π‘†β€˜π‘¦) ↔ (π‘†β€˜π‘¦) = 𝐢))
129128rexbidv 3176 . . . . . . . 8 (β„Ž = 𝐢 β†’ (βˆƒπ‘¦ ∈ (0...𝑁)β„Ž = (π‘†β€˜π‘¦) ↔ βˆƒπ‘¦ ∈ (0...𝑁)(π‘†β€˜π‘¦) = 𝐢))
130129rspcv 3580 . . . . . . 7 (𝐢 ∈ 𝐻 β†’ (βˆ€β„Ž ∈ 𝐻 βˆƒπ‘¦ ∈ (0...𝑁)β„Ž = (π‘†β€˜π‘¦) β†’ βˆƒπ‘¦ ∈ (0...𝑁)(π‘†β€˜π‘¦) = 𝐢))
1319, 125, 130sylc 65 . . . . . 6 (πœ‘ β†’ βˆƒπ‘¦ ∈ (0...𝑁)(π‘†β€˜π‘¦) = 𝐢)
132 fveq2 6847 . . . . . . . . . . . . . 14 (𝑦 = 0 β†’ (π‘†β€˜π‘¦) = (π‘†β€˜0))
133132eqcomd 2743 . . . . . . . . . . . . 13 (𝑦 = 0 β†’ (π‘†β€˜0) = (π‘†β€˜π‘¦))
134133adantl 483 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ 𝑦 = 0) β†’ (π‘†β€˜0) = (π‘†β€˜π‘¦))
135 simplr 768 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ 𝑦 = 0) β†’ (π‘†β€˜π‘¦) = 𝐢)
136134, 135eqtrd 2777 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ 𝑦 = 0) β†’ (π‘†β€˜0) = 𝐢)
1374ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ 𝑦 = 0) β†’ 𝐢 ∈ ℝ)
138136, 137eqeltrd 2838 . . . . . . . . . 10 (((πœ‘ ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ 𝑦 = 0) β†’ (π‘†β€˜0) ∈ ℝ)
139138, 136eqled 11265 . . . . . . . . 9 (((πœ‘ ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ 𝑦 = 0) β†’ (π‘†β€˜0) ≀ 𝐢)
1401393adantl2 1168 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ 𝑦 = 0) β†’ (π‘†β€˜0) ≀ 𝐢)
1414rexrd 11212 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐢 ∈ ℝ*)
14256rexrd 11212 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐷 ∈ ℝ*)
1434, 56, 81ltled 11310 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐢 ≀ 𝐷)
144 lbicc2 13388 . . . . . . . . . . . . . . . . 17 ((𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐢 ≀ 𝐷) β†’ 𝐢 ∈ (𝐢[,]𝐷))
145141, 142, 143, 144syl3anc 1372 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐢 ∈ (𝐢[,]𝐷))
146 ubicc2 13389 . . . . . . . . . . . . . . . . 17 ((𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐢 ≀ 𝐷) β†’ 𝐷 ∈ (𝐢[,]𝐷))
147141, 142, 143, 146syl3anc 1372 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐷 ∈ (𝐢[,]𝐷))
148 prssg 4784 . . . . . . . . . . . . . . . . 17 ((𝐢 ∈ (𝐢[,]𝐷) ∧ 𝐷 ∈ (𝐢[,]𝐷)) β†’ ((𝐢 ∈ (𝐢[,]𝐷) ∧ 𝐷 ∈ (𝐢[,]𝐷)) ↔ {𝐢, 𝐷} βŠ† (𝐢[,]𝐷)))
149145, 147, 148syl2anc 585 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ ((𝐢 ∈ (𝐢[,]𝐷) ∧ 𝐷 ∈ (𝐢[,]𝐷)) ↔ {𝐢, 𝐷} βŠ† (𝐢[,]𝐷)))
150145, 147, 149mpbi2and 711 . . . . . . . . . . . . . . 15 (πœ‘ β†’ {𝐢, 𝐷} βŠ† (𝐢[,]𝐷))
151101a1i 11 . . . . . . . . . . . . . . 15 (πœ‘ β†’ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄} βŠ† (𝐢[,]𝐷))
152150, 151unssd 4151 . . . . . . . . . . . . . 14 (πœ‘ β†’ ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}) βŠ† (𝐢[,]𝐷))
1538, 152eqsstrid 3997 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐻 βŠ† (𝐢[,]𝐷))
154 nnm1nn0 12461 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜π») ∈ β„• β†’ ((β™―β€˜π») βˆ’ 1) ∈ β„•0)
15579, 154syl 17 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ((β™―β€˜π») βˆ’ 1) ∈ β„•0)
1561, 155eqeltrid 2842 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑁 ∈ β„•0)
157156, 39eleqtrdi 2848 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
158 eluzfz1 13455 . . . . . . . . . . . . . . 15 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ 0 ∈ (0...𝑁))
159157, 158syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ 0 ∈ (0...𝑁))
160112, 159ffvelcdmd 7041 . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘†β€˜0) ∈ 𝐻)
161153, 160sseldd 3950 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘†β€˜0) ∈ (𝐢[,]𝐷))
162102, 161sseldd 3950 . . . . . . . . . . 11 (πœ‘ β†’ (π‘†β€˜0) ∈ ℝ)
163162adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ 𝑦 = 0) β†’ (π‘†β€˜0) ∈ ℝ)
1641633ad2antl1 1186 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ (π‘†β€˜0) ∈ ℝ)
1654adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ 𝑦 = 0) β†’ 𝐢 ∈ ℝ)
1661653ad2antl1 1186 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ 𝐢 ∈ ℝ)
167 elfzelz 13448 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑁) β†’ 𝑦 ∈ β„€)
168167zred 12614 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) β†’ 𝑦 ∈ ℝ)
169168adantr 482 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ Β¬ 𝑦 = 0) β†’ 𝑦 ∈ ℝ)
170 elfzle1 13451 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) β†’ 0 ≀ 𝑦)
171170adantr 482 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ Β¬ 𝑦 = 0) β†’ 0 ≀ 𝑦)
172 neqne 2952 . . . . . . . . . . . . . 14 (Β¬ 𝑦 = 0 β†’ 𝑦 β‰  0)
173172adantl 483 . . . . . . . . . . . . 13 ((𝑦 ∈ (0...𝑁) ∧ Β¬ 𝑦 = 0) β†’ 𝑦 β‰  0)
174169, 171, 173ne0gt0d 11299 . . . . . . . . . . . 12 ((𝑦 ∈ (0...𝑁) ∧ Β¬ 𝑦 = 0) β†’ 0 < 𝑦)
1751743ad2antl2 1187 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ 0 < 𝑦)
176 simpl1 1192 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ πœ‘)
177 simpl2 1193 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ 𝑦 ∈ (0...𝑁))
178109simprd 497 . . . . . . . . . . . . . 14 (πœ‘ β†’ βˆ€π‘₯ ∈ (0...𝑁)βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)))
179 breq1 5113 . . . . . . . . . . . . . . . . 17 (π‘₯ = 0 β†’ (π‘₯ < 𝑦 ↔ 0 < 𝑦))
180 fveq2 6847 . . . . . . . . . . . . . . . . . 18 (π‘₯ = 0 β†’ (π‘†β€˜π‘₯) = (π‘†β€˜0))
181180breq1d 5120 . . . . . . . . . . . . . . . . 17 (π‘₯ = 0 β†’ ((π‘†β€˜π‘₯) < (π‘†β€˜π‘¦) ↔ (π‘†β€˜0) < (π‘†β€˜π‘¦)))
182179, 181bibi12d 346 . . . . . . . . . . . . . . . 16 (π‘₯ = 0 β†’ ((π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) ↔ (0 < 𝑦 ↔ (π‘†β€˜0) < (π‘†β€˜π‘¦))))
183182ralbidv 3175 . . . . . . . . . . . . . . 15 (π‘₯ = 0 β†’ (βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ (0...𝑁)(0 < 𝑦 ↔ (π‘†β€˜0) < (π‘†β€˜π‘¦))))
184183rspcv 3580 . . . . . . . . . . . . . 14 (0 ∈ (0...𝑁) β†’ (βˆ€π‘₯ ∈ (0...𝑁)βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) β†’ βˆ€π‘¦ ∈ (0...𝑁)(0 < 𝑦 ↔ (π‘†β€˜0) < (π‘†β€˜π‘¦))))
185159, 178, 184sylc 65 . . . . . . . . . . . . 13 (πœ‘ β†’ βˆ€π‘¦ ∈ (0...𝑁)(0 < 𝑦 ↔ (π‘†β€˜0) < (π‘†β€˜π‘¦)))
186185r19.21bi 3237 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ (0 < 𝑦 ↔ (π‘†β€˜0) < (π‘†β€˜π‘¦)))
187176, 177, 186syl2anc 585 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ (0 < 𝑦 ↔ (π‘†β€˜0) < (π‘†β€˜π‘¦)))
188175, 187mpbid 231 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ (π‘†β€˜0) < (π‘†β€˜π‘¦))
189 simpl3 1194 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ (π‘†β€˜π‘¦) = 𝐢)
190188, 189breqtrd 5136 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ (π‘†β€˜0) < 𝐢)
191164, 166, 190ltled 11310 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) ∧ Β¬ 𝑦 = 0) β†’ (π‘†β€˜0) ≀ 𝐢)
192140, 191pm2.61dan 812 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐢) β†’ (π‘†β€˜0) ≀ 𝐢)
193192rexlimdv3a 3157 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘¦ ∈ (0...𝑁)(π‘†β€˜π‘¦) = 𝐢 β†’ (π‘†β€˜0) ≀ 𝐢))
194131, 193mpd 15 . . . . 5 (πœ‘ β†’ (π‘†β€˜0) ≀ 𝐢)
195 elicc2 13336 . . . . . . . 8 ((𝐢 ∈ ℝ ∧ 𝐷 ∈ ℝ) β†’ ((π‘†β€˜0) ∈ (𝐢[,]𝐷) ↔ ((π‘†β€˜0) ∈ ℝ ∧ 𝐢 ≀ (π‘†β€˜0) ∧ (π‘†β€˜0) ≀ 𝐷)))
1964, 56, 195syl2anc 585 . . . . . . 7 (πœ‘ β†’ ((π‘†β€˜0) ∈ (𝐢[,]𝐷) ↔ ((π‘†β€˜0) ∈ ℝ ∧ 𝐢 ≀ (π‘†β€˜0) ∧ (π‘†β€˜0) ≀ 𝐷)))
197161, 196mpbid 231 . . . . . 6 (πœ‘ β†’ ((π‘†β€˜0) ∈ ℝ ∧ 𝐢 ≀ (π‘†β€˜0) ∧ (π‘†β€˜0) ≀ 𝐷))
198197simp2d 1144 . . . . 5 (πœ‘ β†’ 𝐢 ≀ (π‘†β€˜0))
199162, 4letri3d 11304 . . . . 5 (πœ‘ β†’ ((π‘†β€˜0) = 𝐢 ↔ ((π‘†β€˜0) ≀ 𝐢 ∧ 𝐢 ≀ (π‘†β€˜0))))
200194, 198, 199mpbir2and 712 . . . 4 (πœ‘ β†’ (π‘†β€˜0) = 𝐢)
201 eluzfz2 13456 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ 𝑁 ∈ (0...𝑁))
202157, 201syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝑁 ∈ (0...𝑁))
203112, 202ffvelcdmd 7041 . . . . . . . 8 (πœ‘ β†’ (π‘†β€˜π‘) ∈ 𝐻)
204153, 203sseldd 3950 . . . . . . 7 (πœ‘ β†’ (π‘†β€˜π‘) ∈ (𝐢[,]𝐷))
205 elicc2 13336 . . . . . . . 8 ((𝐢 ∈ ℝ ∧ 𝐷 ∈ ℝ) β†’ ((π‘†β€˜π‘) ∈ (𝐢[,]𝐷) ↔ ((π‘†β€˜π‘) ∈ ℝ ∧ 𝐢 ≀ (π‘†β€˜π‘) ∧ (π‘†β€˜π‘) ≀ 𝐷)))
2064, 56, 205syl2anc 585 . . . . . . 7 (πœ‘ β†’ ((π‘†β€˜π‘) ∈ (𝐢[,]𝐷) ↔ ((π‘†β€˜π‘) ∈ ℝ ∧ 𝐢 ≀ (π‘†β€˜π‘) ∧ (π‘†β€˜π‘) ≀ 𝐷)))
207204, 206mpbid 231 . . . . . 6 (πœ‘ β†’ ((π‘†β€˜π‘) ∈ ℝ ∧ 𝐢 ≀ (π‘†β€˜π‘) ∧ (π‘†β€˜π‘) ≀ 𝐷))
208207simp3d 1145 . . . . 5 (πœ‘ β†’ (π‘†β€˜π‘) ≀ 𝐷)
209 prid2g 4727 . . . . . . . . 9 (𝐷 ∈ ℝ β†’ 𝐷 ∈ {𝐢, 𝐷})
210 elun1 4141 . . . . . . . . 9 (𝐷 ∈ {𝐢, 𝐷} β†’ 𝐷 ∈ ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}))
21156, 209, 2103syl 18 . . . . . . . 8 (πœ‘ β†’ 𝐷 ∈ ({𝐢, 𝐷} βˆͺ {π‘₯ ∈ (𝐢[,]𝐷) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ ran 𝑄}))
212211, 8eleqtrrdi 2849 . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ 𝐻)
213 eqeq1 2741 . . . . . . . . . 10 (β„Ž = 𝐷 β†’ (β„Ž = (π‘†β€˜π‘¦) ↔ 𝐷 = (π‘†β€˜π‘¦)))
214 eqcom 2744 . . . . . . . . . 10 (𝐷 = (π‘†β€˜π‘¦) ↔ (π‘†β€˜π‘¦) = 𝐷)
215213, 214bitrdi 287 . . . . . . . . 9 (β„Ž = 𝐷 β†’ (β„Ž = (π‘†β€˜π‘¦) ↔ (π‘†β€˜π‘¦) = 𝐷))
216215rexbidv 3176 . . . . . . . 8 (β„Ž = 𝐷 β†’ (βˆƒπ‘¦ ∈ (0...𝑁)β„Ž = (π‘†β€˜π‘¦) ↔ βˆƒπ‘¦ ∈ (0...𝑁)(π‘†β€˜π‘¦) = 𝐷))
217216rspcv 3580 . . . . . . 7 (𝐷 ∈ 𝐻 β†’ (βˆ€β„Ž ∈ 𝐻 βˆƒπ‘¦ ∈ (0...𝑁)β„Ž = (π‘†β€˜π‘¦) β†’ βˆƒπ‘¦ ∈ (0...𝑁)(π‘†β€˜π‘¦) = 𝐷))
218212, 125, 217sylc 65 . . . . . 6 (πœ‘ β†’ βˆƒπ‘¦ ∈ (0...𝑁)(π‘†β€˜π‘¦) = 𝐷)
219214biimpri 227 . . . . . . . . 9 ((π‘†β€˜π‘¦) = 𝐷 β†’ 𝐷 = (π‘†β€˜π‘¦))
2202193ad2ant3 1136 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐷) β†’ 𝐷 = (π‘†β€˜π‘¦))
221113ffvelcdmda 7040 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ (π‘†β€˜π‘¦) ∈ ℝ)
222102, 204sseldd 3950 . . . . . . . . . . 11 (πœ‘ β†’ (π‘†β€˜π‘) ∈ ℝ)
223222adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ (π‘†β€˜π‘) ∈ ℝ)
224168adantl 483 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ 𝑦 ∈ ℝ)
225 elfzel2 13446 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...𝑁) β†’ 𝑁 ∈ β„€)
226225zred 12614 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑁) β†’ 𝑁 ∈ ℝ)
227226adantl 483 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ 𝑁 ∈ ℝ)
228 elfzle2 13452 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑁) β†’ 𝑦 ≀ 𝑁)
229228adantl 483 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ 𝑦 ≀ 𝑁)
230224, 227, 229lensymd 11313 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ Β¬ 𝑁 < 𝑦)
231 breq1 5113 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑁 β†’ (π‘₯ < 𝑦 ↔ 𝑁 < 𝑦))
232 fveq2 6847 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑁 β†’ (π‘†β€˜π‘₯) = (π‘†β€˜π‘))
233232breq1d 5120 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑁 β†’ ((π‘†β€˜π‘₯) < (π‘†β€˜π‘¦) ↔ (π‘†β€˜π‘) < (π‘†β€˜π‘¦)))
234231, 233bibi12d 346 . . . . . . . . . . . . . . 15 (π‘₯ = 𝑁 β†’ ((π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) ↔ (𝑁 < 𝑦 ↔ (π‘†β€˜π‘) < (π‘†β€˜π‘¦))))
235234ralbidv 3175 . . . . . . . . . . . . . 14 (π‘₯ = 𝑁 β†’ (βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (π‘†β€˜π‘) < (π‘†β€˜π‘¦))))
236235rspcv 3580 . . . . . . . . . . . . 13 (𝑁 ∈ (0...𝑁) β†’ (βˆ€π‘₯ ∈ (0...𝑁)βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) β†’ βˆ€π‘¦ ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (π‘†β€˜π‘) < (π‘†β€˜π‘¦))))
237202, 178, 236sylc 65 . . . . . . . . . . . 12 (πœ‘ β†’ βˆ€π‘¦ ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (π‘†β€˜π‘) < (π‘†β€˜π‘¦)))
238237r19.21bi 3237 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ (𝑁 < 𝑦 ↔ (π‘†β€˜π‘) < (π‘†β€˜π‘¦)))
239230, 238mtbid 324 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ Β¬ (π‘†β€˜π‘) < (π‘†β€˜π‘¦))
240221, 223, 239nltled 11312 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁)) β†’ (π‘†β€˜π‘¦) ≀ (π‘†β€˜π‘))
2412403adant3 1133 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐷) β†’ (π‘†β€˜π‘¦) ≀ (π‘†β€˜π‘))
242220, 241eqbrtrd 5132 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (0...𝑁) ∧ (π‘†β€˜π‘¦) = 𝐷) β†’ 𝐷 ≀ (π‘†β€˜π‘))
243242rexlimdv3a 3157 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘¦ ∈ (0...𝑁)(π‘†β€˜π‘¦) = 𝐷 β†’ 𝐷 ≀ (π‘†β€˜π‘)))
244218, 243mpd 15 . . . . 5 (πœ‘ β†’ 𝐷 ≀ (π‘†β€˜π‘))
245222, 56letri3d 11304 . . . . 5 (πœ‘ β†’ ((π‘†β€˜π‘) = 𝐷 ↔ ((π‘†β€˜π‘) ≀ 𝐷 ∧ 𝐷 ≀ (π‘†β€˜π‘))))
246208, 244, 245mpbir2and 712 . . . 4 (πœ‘ β†’ (π‘†β€˜π‘) = 𝐷)
247 elfzoelz 13579 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) β†’ 𝑖 ∈ β„€)
248247zred 12614 . . . . . . . 8 (𝑖 ∈ (0..^𝑁) β†’ 𝑖 ∈ ℝ)
249248ltp1d 12092 . . . . . . 7 (𝑖 ∈ (0..^𝑁) β†’ 𝑖 < (𝑖 + 1))
250249adantl 483 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ 𝑖 < (𝑖 + 1))
251178adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ βˆ€π‘₯ ∈ (0...𝑁)βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)))
252 elfzofz 13595 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) β†’ 𝑖 ∈ (0...𝑁))
253252adantl 483 . . . . . . . 8 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ 𝑖 ∈ (0...𝑁))
254 fzofzp1 13676 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) β†’ (𝑖 + 1) ∈ (0...𝑁))
255254adantl 483 . . . . . . . 8 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ (𝑖 + 1) ∈ (0...𝑁))
256 breq1 5113 . . . . . . . . . 10 (π‘₯ = 𝑖 β†’ (π‘₯ < 𝑦 ↔ 𝑖 < 𝑦))
257 fveq2 6847 . . . . . . . . . . 11 (π‘₯ = 𝑖 β†’ (π‘†β€˜π‘₯) = (π‘†β€˜π‘–))
258257breq1d 5120 . . . . . . . . . 10 (π‘₯ = 𝑖 β†’ ((π‘†β€˜π‘₯) < (π‘†β€˜π‘¦) ↔ (π‘†β€˜π‘–) < (π‘†β€˜π‘¦)))
259256, 258bibi12d 346 . . . . . . . . 9 (π‘₯ = 𝑖 β†’ ((π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) ↔ (𝑖 < 𝑦 ↔ (π‘†β€˜π‘–) < (π‘†β€˜π‘¦))))
260 breq2 5114 . . . . . . . . . 10 (𝑦 = (𝑖 + 1) β†’ (𝑖 < 𝑦 ↔ 𝑖 < (𝑖 + 1)))
261 fveq2 6847 . . . . . . . . . . 11 (𝑦 = (𝑖 + 1) β†’ (π‘†β€˜π‘¦) = (π‘†β€˜(𝑖 + 1)))
262261breq2d 5122 . . . . . . . . . 10 (𝑦 = (𝑖 + 1) β†’ ((π‘†β€˜π‘–) < (π‘†β€˜π‘¦) ↔ (π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1))))
263260, 262bibi12d 346 . . . . . . . . 9 (𝑦 = (𝑖 + 1) β†’ ((𝑖 < 𝑦 ↔ (π‘†β€˜π‘–) < (π‘†β€˜π‘¦)) ↔ (𝑖 < (𝑖 + 1) ↔ (π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1)))))
264259, 263rspc2v 3593 . . . . . . . 8 ((𝑖 ∈ (0...𝑁) ∧ (𝑖 + 1) ∈ (0...𝑁)) β†’ (βˆ€π‘₯ ∈ (0...𝑁)βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) β†’ (𝑖 < (𝑖 + 1) ↔ (π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1)))))
265253, 255, 264syl2anc 585 . . . . . . 7 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ (βˆ€π‘₯ ∈ (0...𝑁)βˆ€π‘¦ ∈ (0...𝑁)(π‘₯ < 𝑦 ↔ (π‘†β€˜π‘₯) < (π‘†β€˜π‘¦)) β†’ (𝑖 < (𝑖 + 1) ↔ (π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1)))))
266251, 265mpd 15 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ (𝑖 < (𝑖 + 1) ↔ (π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1))))
267250, 266mpbid 231 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ (π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1)))
268267ralrimiva 3144 . . . 4 (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑁)(π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1)))
269200, 246, 268jca31 516 . . 3 (πœ‘ β†’ (((π‘†β€˜0) = 𝐢 ∧ (π‘†β€˜π‘) = 𝐷) ∧ βˆ€π‘– ∈ (0..^𝑁)(π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1))))
270 fourierdlem54.o . . . . 5 𝑂 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐢 ∧ (π‘β€˜π‘š) = 𝐷) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})
271270fourierdlem2 44424 . . . 4 (𝑁 ∈ β„• β†’ (𝑆 ∈ (π‘‚β€˜π‘) ↔ (𝑆 ∈ (ℝ ↑m (0...𝑁)) ∧ (((π‘†β€˜0) = 𝐢 ∧ (π‘†β€˜π‘) = 𝐷) ∧ βˆ€π‘– ∈ (0..^𝑁)(π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1))))))
27297, 271syl 17 . . 3 (πœ‘ β†’ (𝑆 ∈ (π‘‚β€˜π‘) ↔ (𝑆 ∈ (ℝ ↑m (0...𝑁)) ∧ (((π‘†β€˜0) = 𝐢 ∧ (π‘†β€˜π‘) = 𝐷) ∧ βˆ€π‘– ∈ (0..^𝑁)(π‘†β€˜π‘–) < (π‘†β€˜(𝑖 + 1))))))
273119, 269, 272mpbir2and 712 . 2 (πœ‘ β†’ 𝑆 ∈ (π‘‚β€˜π‘))
27497, 273, 107jca31 516 1 (πœ‘ β†’ ((𝑁 ∈ β„• ∧ 𝑆 ∈ (π‘‚β€˜π‘)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βˆ– cdif 3912   βˆͺ cun 3913   βŠ† wss 3915  βˆ…c0 4287  {cpr 4593   class class class wbr 5110   ↦ cmpt 5193   I cid 5535   Γ— cxp 5636  ran crn 5639   β†Ύ cres 5640   ∘ ccom 5642  β„©cio 6451   Fn wfn 6496  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501   Isom wiso 6502  (class class class)co 7362   ↑m cmap 8772  Fincfn 8890  infcinf 9384  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   Β· cmul 11063  β„*cxr 11195   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392  β„•cn 12160  2c2 12215  β„•0cn0 12420  β„€cz 12506  β„€β‰₯cuz 12770  (,)cioo 13271  [,]cicc 13274  ...cfz 13431  ..^cfzo 13574  β™―chash 14237  abscabs 15126   β†Ύt crest 17309  topGenctg 17326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fi 9354  df-sup 9385  df-inf 9386  df-oi 9453  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-ioo 13275  df-icc 13278  df-fz 13432  df-fzo 13575  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-rest 17311  df-topgen 17332  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-top 22259  df-topon 22276  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-lp 22503  df-cmp 22754
This theorem is referenced by:  fourierdlem63  44484  fourierdlem64  44485  fourierdlem65  44486  fourierdlem79  44500  fourierdlem89  44510  fourierdlem90  44511  fourierdlem91  44512  fourierdlem100  44521  fourierdlem107  44528  fourierdlem109  44530  fourierdlem112  44533
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