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Theorem fourierdlem52 42642
Description: d16:d17,d18:jca |- ( ph -> ( ( S 0) ≤ 𝐴𝐴 ≤ (𝑆 0 ) ) ) . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem52.tf (𝜑𝑇 ∈ Fin)
fourierdlem52.n 𝑁 = ((♯‘𝑇) − 1)
fourierdlem52.s 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
fourierdlem52.a (𝜑𝐴 ∈ ℝ)
fourierdlem52.b (𝜑𝐵 ∈ ℝ)
fourierdlem52.t (𝜑𝑇 ⊆ (𝐴[,]𝐵))
fourierdlem52.at (𝜑𝐴𝑇)
fourierdlem52.bt (𝜑𝐵𝑇)
Assertion
Ref Expression
fourierdlem52 (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆𝑁) = 𝐵))
Distinct variable groups:   𝑓,𝑁   𝑆,𝑓   𝑇,𝑓   𝜑,𝑓
Allowed substitution hints:   𝐴(𝑓)   𝐵(𝑓)

Proof of Theorem fourierdlem52
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fourierdlem52.tf . . . . 5 (𝜑𝑇 ∈ Fin)
2 fourierdlem52.t . . . . . 6 (𝜑𝑇 ⊆ (𝐴[,]𝐵))
3 fourierdlem52.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
4 fourierdlem52.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
53, 4iccssred 12810 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
62, 5sstrd 3961 . . . . 5 (𝜑𝑇 ⊆ ℝ)
7 fourierdlem52.s . . . . 5 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
8 fourierdlem52.n . . . . 5 𝑁 = ((♯‘𝑇) − 1)
91, 6, 7, 8fourierdlem36 42627 . . . 4 (𝜑𝑆 Isom < , < ((0...𝑁), 𝑇))
10 isof1o 7058 . . . 4 (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto𝑇)
11 f1of 6596 . . . 4 (𝑆:(0...𝑁)–1-1-onto𝑇𝑆:(0...𝑁)⟶𝑇)
129, 10, 113syl 18 . . 3 (𝜑𝑆:(0...𝑁)⟶𝑇)
1312, 2fssd 6509 . 2 (𝜑𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14 f1ofo 6603 . . . . . 6 (𝑆:(0...𝑁)–1-1-onto𝑇𝑆:(0...𝑁)–onto𝑇)
159, 10, 143syl 18 . . . . 5 (𝜑𝑆:(0...𝑁)–onto𝑇)
16 fourierdlem52.at . . . . 5 (𝜑𝐴𝑇)
17 foelrn 6853 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐴𝑇) → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
1815, 16, 17syl2anc 587 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
19 elfzle1 12903 . . . . . . . . 9 (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
2019adantl 485 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → 0 ≤ 𝑗)
219adantr 484 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
22 ressxr 10670 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
236, 22sstrdi 3963 . . . . . . . . . . 11 (𝜑𝑇 ⊆ ℝ*)
2423adantr 484 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ*)
25 fzssz 12902 . . . . . . . . . . 11 (0...𝑁) ⊆ ℤ
26 zssre 11974 . . . . . . . . . . . 12 ℤ ⊆ ℝ
2726, 22sstri 3960 . . . . . . . . . . 11 ℤ ⊆ ℝ*
2825, 27sstri 3960 . . . . . . . . . 10 (0...𝑁) ⊆ ℝ*
2924, 28jctil 523 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
30 hashcl 13711 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ0)
311, 30syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑇) ∈ ℕ0)
3216ne0d 4282 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ≠ ∅)
33 hashge1 13744 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (♯‘𝑇))
341, 32, 33syl2anc 587 . . . . . . . . . . . . . . 15 (𝜑 → 1 ≤ (♯‘𝑇))
35 elnnnn0c 11928 . . . . . . . . . . . . . . 15 ((♯‘𝑇) ∈ ℕ ↔ ((♯‘𝑇) ∈ ℕ0 ∧ 1 ≤ (♯‘𝑇)))
3631, 34, 35sylanbrc 586 . . . . . . . . . . . . . 14 (𝜑 → (♯‘𝑇) ∈ ℕ)
37 nnm1nn0 11924 . . . . . . . . . . . . . 14 ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ ℕ0)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝜑 → ((♯‘𝑇) − 1) ∈ ℕ0)
398, 38eqeltrid 2920 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
40 nn0uz 12266 . . . . . . . . . . . 12 0 = (ℤ‘0)
4139, 40eleqtrdi 2926 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
42 eluzfz1 12907 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
4341, 42syl 17 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑁))
4443anim1i 617 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
45 leisorel 13812 . . . . . . . . 9 ((𝑆 Isom < , < ((0...𝑁), 𝑇) ∧ ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*) ∧ (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁))) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆𝑗)))
4621, 29, 44, 45syl3anc 1368 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆𝑗)))
4720, 46mpbid 235 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆𝑗))
48473adant3 1129 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ (𝑆𝑗))
49 eqcom 2831 . . . . . . . 8 (𝐴 = (𝑆𝑗) ↔ (𝑆𝑗) = 𝐴)
5049biimpi 219 . . . . . . 7 (𝐴 = (𝑆𝑗) → (𝑆𝑗) = 𝐴)
51503ad2ant3 1132 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆𝑗) = 𝐴)
5248, 51breqtrd 5073 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ 𝐴)
5352rexlimdv3a 3278 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗) → (𝑆‘0) ≤ 𝐴))
5418, 53mpd 15 . . 3 (𝜑 → (𝑆‘0) ≤ 𝐴)
553rexrd 10676 . . . 4 (𝜑𝐴 ∈ ℝ*)
564rexrd 10676 . . . 4 (𝜑𝐵 ∈ ℝ*)
5713, 43ffvelrnd 6833 . . . 4 (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
58 iccgelb 12779 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
5955, 56, 57, 58syl3anc 1368 . . 3 (𝜑𝐴 ≤ (𝑆‘0))
605, 57sseldd 3952 . . . 4 (𝜑 → (𝑆‘0) ∈ ℝ)
6160, 3letri3d 10767 . . 3 (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴𝐴 ≤ (𝑆‘0))))
6254, 59, 61mpbir2and 712 . 2 (𝜑 → (𝑆‘0) = 𝐴)
63 eluzfz2 12908 . . . . . 6 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
6441, 63syl 17 . . . . 5 (𝜑𝑁 ∈ (0...𝑁))
6513, 64ffvelrnd 6833 . . . 4 (𝜑 → (𝑆𝑁) ∈ (𝐴[,]𝐵))
66 iccleub 12778 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆𝑁) ∈ (𝐴[,]𝐵)) → (𝑆𝑁) ≤ 𝐵)
6755, 56, 65, 66syl3anc 1368 . . 3 (𝜑 → (𝑆𝑁) ≤ 𝐵)
68 fourierdlem52.bt . . . . 5 (𝜑𝐵𝑇)
69 foelrn 6853 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐵𝑇) → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
7015, 68, 69syl2anc 587 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
71 simp3 1135 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 = (𝑆𝑗))
72 elfzle2 12904 . . . . . . . 8 (𝑗 ∈ (0...𝑁) → 𝑗𝑁)
73723ad2ant2 1131 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑗𝑁)
7493ad2ant1 1130 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
75293adant3 1129 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
76 simp2 1134 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑗 ∈ (0...𝑁))
77643ad2ant1 1130 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑁 ∈ (0...𝑁))
78 leisorel 13812 . . . . . . . 8 ((𝑆 Isom < , < ((0...𝑁), 𝑇) ∧ ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...𝑁))) → (𝑗𝑁 ↔ (𝑆𝑗) ≤ (𝑆𝑁)))
7974, 75, 76, 77, 78syl112anc 1371 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑗𝑁 ↔ (𝑆𝑗) ≤ (𝑆𝑁)))
8073, 79mpbid 235 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑆𝑗) ≤ (𝑆𝑁))
8171, 80eqbrtrd 5069 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 ≤ (𝑆𝑁))
8281rexlimdv3a 3278 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗) → 𝐵 ≤ (𝑆𝑁)))
8370, 82mpd 15 . . 3 (𝜑𝐵 ≤ (𝑆𝑁))
845, 65sseldd 3952 . . . 4 (𝜑 → (𝑆𝑁) ∈ ℝ)
8584, 4letri3d 10767 . . 3 (𝜑 → ((𝑆𝑁) = 𝐵 ↔ ((𝑆𝑁) ≤ 𝐵𝐵 ≤ (𝑆𝑁))))
8667, 83, 85mpbir2and 712 . 2 (𝜑 → (𝑆𝑁) = 𝐵)
8713, 62, 86jca31 518 1 (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆𝑁) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3013  wrex 3133  wss 3918  c0 4274   class class class wbr 5047  cio 6293  wf 6332  ontowfo 6334  1-1-ontowf1o 6335  cfv 6336   Isom wiso 6337  (class class class)co 7138  Fincfn 8492  cr 10521  0cc0 10522  1c1 10523  *cxr 10659   < clt 10660  cle 10661  cmin 10855  cn 11623  0cn0 11883  cz 11967  cuz 12229  [,]cicc 12727  ...cfz 12883  chash 13684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-inf2 9088  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-se 5496  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-card 9352  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-n0 11884  df-z 11968  df-uz 12230  df-icc 12731  df-fz 12884  df-hash 13685
This theorem is referenced by:  fourierdlem103  42693  fourierdlem104  42694
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