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Theorem fourierdlem52 45174
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 13416 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
62, 5sstrd 3993 . . . . 5 (𝜑𝑇 ⊆ ℝ)
7 fourierdlem52.s . . . . 5 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
8 fourierdlem52.n . . . . 5 𝑁 = ((♯‘𝑇) − 1)
91, 6, 7, 8fourierdlem36 45159 . . . 4 (𝜑𝑆 Isom < , < ((0...𝑁), 𝑇))
10 isof1o 7323 . . . 4 (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto𝑇)
11 f1of 6834 . . . 4 (𝑆:(0...𝑁)–1-1-onto𝑇𝑆:(0...𝑁)⟶𝑇)
129, 10, 113syl 18 . . 3 (𝜑𝑆:(0...𝑁)⟶𝑇)
1312, 2fssd 6736 . 2 (𝜑𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14 f1ofo 6841 . . . . . 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 7109 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐴𝑇) → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
1815, 16, 17syl2anc 583 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
19 elfzle1 13509 . . . . . . . . 9 (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
2019adantl 481 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → 0 ≤ 𝑗)
219adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
22 ressxr 11263 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
236, 22sstrdi 3995 . . . . . . . . . . 11 (𝜑𝑇 ⊆ ℝ*)
2423adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ*)
25 fzssz 13508 . . . . . . . . . . 11 (0...𝑁) ⊆ ℤ
26 zssre 12570 . . . . . . . . . . . 12 ℤ ⊆ ℝ
2726, 22sstri 3992 . . . . . . . . . . 11 ℤ ⊆ ℝ*
2825, 27sstri 3992 . . . . . . . . . 10 (0...𝑁) ⊆ ℝ*
2924, 28jctil 519 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
30 hashcl 14321 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ0)
311, 30syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑇) ∈ ℕ0)
3216ne0d 4336 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ≠ ∅)
33 hashge1 14354 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (♯‘𝑇))
341, 32, 33syl2anc 583 . . . . . . . . . . . . . . 15 (𝜑 → 1 ≤ (♯‘𝑇))
35 elnnnn0c 12522 . . . . . . . . . . . . . . 15 ((♯‘𝑇) ∈ ℕ ↔ ((♯‘𝑇) ∈ ℕ0 ∧ 1 ≤ (♯‘𝑇)))
3631, 34, 35sylanbrc 582 . . . . . . . . . . . . . 14 (𝜑 → (♯‘𝑇) ∈ ℕ)
37 nnm1nn0 12518 . . . . . . . . . . . . . 14 ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ ℕ0)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝜑 → ((♯‘𝑇) − 1) ∈ ℕ0)
398, 38eqeltrid 2836 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
40 nn0uz 12869 . . . . . . . . . . . 12 0 = (ℤ‘0)
4139, 40eleqtrdi 2842 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
42 eluzfz1 13513 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
4341, 42syl 17 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑁))
4443anim1i 614 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
45 leisorel 14426 . . . . . . . . 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 1370 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆𝑗)))
4720, 46mpbid 231 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆𝑗))
48473adant3 1131 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ (𝑆𝑗))
49 eqcom 2738 . . . . . . . 8 (𝐴 = (𝑆𝑗) ↔ (𝑆𝑗) = 𝐴)
5049biimpi 215 . . . . . . 7 (𝐴 = (𝑆𝑗) → (𝑆𝑗) = 𝐴)
51503ad2ant3 1134 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆𝑗) = 𝐴)
5248, 51breqtrd 5175 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ 𝐴)
5352rexlimdv3a 3158 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗) → (𝑆‘0) ≤ 𝐴))
5418, 53mpd 15 . . 3 (𝜑 → (𝑆‘0) ≤ 𝐴)
553rexrd 11269 . . . 4 (𝜑𝐴 ∈ ℝ*)
564rexrd 11269 . . . 4 (𝜑𝐵 ∈ ℝ*)
5713, 43ffvelcdmd 7088 . . . 4 (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
58 iccgelb 13385 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
5955, 56, 57, 58syl3anc 1370 . . 3 (𝜑𝐴 ≤ (𝑆‘0))
605, 57sseldd 3984 . . . 4 (𝜑 → (𝑆‘0) ∈ ℝ)
6160, 3letri3d 11361 . . 3 (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴𝐴 ≤ (𝑆‘0))))
6254, 59, 61mpbir2and 710 . 2 (𝜑 → (𝑆‘0) = 𝐴)
63 eluzfz2 13514 . . . . . 6 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
6441, 63syl 17 . . . . 5 (𝜑𝑁 ∈ (0...𝑁))
6513, 64ffvelcdmd 7088 . . . 4 (𝜑 → (𝑆𝑁) ∈ (𝐴[,]𝐵))
66 iccleub 13384 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆𝑁) ∈ (𝐴[,]𝐵)) → (𝑆𝑁) ≤ 𝐵)
6755, 56, 65, 66syl3anc 1370 . . 3 (𝜑 → (𝑆𝑁) ≤ 𝐵)
68 fourierdlem52.bt . . . . 5 (𝜑𝐵𝑇)
69 foelrn 7109 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐵𝑇) → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
7015, 68, 69syl2anc 583 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
71 simp3 1137 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 = (𝑆𝑗))
72 elfzle2 13510 . . . . . . . 8 (𝑗 ∈ (0...𝑁) → 𝑗𝑁)
73723ad2ant2 1133 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑗𝑁)
7493ad2ant1 1132 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
75293adant3 1131 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
76 simp2 1136 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑗 ∈ (0...𝑁))
77643ad2ant1 1132 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑁 ∈ (0...𝑁))
78 leisorel 14426 . . . . . . . 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 1373 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑗𝑁 ↔ (𝑆𝑗) ≤ (𝑆𝑁)))
8073, 79mpbid 231 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑆𝑗) ≤ (𝑆𝑁))
8171, 80eqbrtrd 5171 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 ≤ (𝑆𝑁))
8281rexlimdv3a 3158 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗) → 𝐵 ≤ (𝑆𝑁)))
8370, 82mpd 15 . . 3 (𝜑𝐵 ≤ (𝑆𝑁))
845, 65sseldd 3984 . . . 4 (𝜑 → (𝑆𝑁) ∈ ℝ)
8584, 4letri3d 11361 . . 3 (𝜑 → ((𝑆𝑁) = 𝐵 ↔ ((𝑆𝑁) ≤ 𝐵𝐵 ≤ (𝑆𝑁))))
8667, 83, 85mpbir2and 710 . 2 (𝜑 → (𝑆𝑁) = 𝐵)
8713, 62, 86jca31 514 1 (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆𝑁) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wrex 3069  wss 3949  c0 4323   class class class wbr 5149  cio 6494  wf 6540  ontowfo 6542  1-1-ontowf1o 6543  cfv 6544   Isom wiso 6545  (class class class)co 7412  Fincfn 8942  cr 11112  0cc0 11113  1c1 11114  *cxr 11252   < clt 11253  cle 11254  cmin 11449  cn 12217  0cn0 12477  cz 12563  cuz 12827  [,]cicc 13332  ...cfz 13489  chash 14295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-icc 13336  df-fz 13490  df-hash 14296
This theorem is referenced by:  fourierdlem103  45225  fourierdlem104  45226
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