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Theorem fourierdlem52 46113
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 13470 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
62, 5sstrd 4005 . . . . 5 (𝜑𝑇 ⊆ ℝ)
7 fourierdlem52.s . . . . 5 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
8 fourierdlem52.n . . . . 5 𝑁 = ((♯‘𝑇) − 1)
91, 6, 7, 8fourierdlem36 46098 . . . 4 (𝜑𝑆 Isom < , < ((0...𝑁), 𝑇))
10 isof1o 7342 . . . 4 (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto𝑇)
11 f1of 6848 . . . 4 (𝑆:(0...𝑁)–1-1-onto𝑇𝑆:(0...𝑁)⟶𝑇)
129, 10, 113syl 18 . . 3 (𝜑𝑆:(0...𝑁)⟶𝑇)
1312, 2fssd 6753 . 2 (𝜑𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14 f1ofo 6855 . . . . . 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 7126 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐴𝑇) → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
1815, 16, 17syl2anc 584 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
19 elfzle1 13563 . . . . . . . . 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 11302 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
236, 22sstrdi 4007 . . . . . . . . . . 11 (𝜑𝑇 ⊆ ℝ*)
2423adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ*)
25 fzssz 13562 . . . . . . . . . . 11 (0...𝑁) ⊆ ℤ
26 zssre 12617 . . . . . . . . . . . 12 ℤ ⊆ ℝ
2726, 22sstri 4004 . . . . . . . . . . 11 ℤ ⊆ ℝ*
2825, 27sstri 4004 . . . . . . . . . 10 (0...𝑁) ⊆ ℝ*
2924, 28jctil 519 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
30 hashcl 14391 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ0)
311, 30syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑇) ∈ ℕ0)
3216ne0d 4347 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ≠ ∅)
33 hashge1 14424 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (♯‘𝑇))
341, 32, 33syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → 1 ≤ (♯‘𝑇))
35 elnnnn0c 12568 . . . . . . . . . . . . . . 15 ((♯‘𝑇) ∈ ℕ ↔ ((♯‘𝑇) ∈ ℕ0 ∧ 1 ≤ (♯‘𝑇)))
3631, 34, 35sylanbrc 583 . . . . . . . . . . . . . 14 (𝜑 → (♯‘𝑇) ∈ ℕ)
37 nnm1nn0 12564 . . . . . . . . . . . . . 14 ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ ℕ0)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝜑 → ((♯‘𝑇) − 1) ∈ ℕ0)
398, 38eqeltrid 2842 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
40 nn0uz 12917 . . . . . . . . . . . 12 0 = (ℤ‘0)
4139, 40eleqtrdi 2848 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
42 eluzfz1 13567 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
4341, 42syl 17 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑁))
4443anim1i 615 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
45 leisorel 14495 . . . . . . . . 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 232 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆𝑗))
48473adant3 1131 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ (𝑆𝑗))
49 eqcom 2741 . . . . . . . 8 (𝐴 = (𝑆𝑗) ↔ (𝑆𝑗) = 𝐴)
5049biimpi 216 . . . . . . 7 (𝐴 = (𝑆𝑗) → (𝑆𝑗) = 𝐴)
51503ad2ant3 1134 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆𝑗) = 𝐴)
5248, 51breqtrd 5173 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ 𝐴)
5352rexlimdv3a 3156 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗) → (𝑆‘0) ≤ 𝐴))
5418, 53mpd 15 . . 3 (𝜑 → (𝑆‘0) ≤ 𝐴)
553rexrd 11308 . . . 4 (𝜑𝐴 ∈ ℝ*)
564rexrd 11308 . . . 4 (𝜑𝐵 ∈ ℝ*)
5713, 43ffvelcdmd 7104 . . . 4 (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
58 iccgelb 13439 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
5955, 56, 57, 58syl3anc 1370 . . 3 (𝜑𝐴 ≤ (𝑆‘0))
605, 57sseldd 3995 . . . 4 (𝜑 → (𝑆‘0) ∈ ℝ)
6160, 3letri3d 11400 . . 3 (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴𝐴 ≤ (𝑆‘0))))
6254, 59, 61mpbir2and 713 . 2 (𝜑 → (𝑆‘0) = 𝐴)
63 eluzfz2 13568 . . . . . 6 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
6441, 63syl 17 . . . . 5 (𝜑𝑁 ∈ (0...𝑁))
6513, 64ffvelcdmd 7104 . . . 4 (𝜑 → (𝑆𝑁) ∈ (𝐴[,]𝐵))
66 iccleub 13438 . . . 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 7126 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐵𝑇) → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
7015, 68, 69syl2anc 584 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
71 simp3 1137 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 = (𝑆𝑗))
72 elfzle2 13564 . . . . . . . 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 14495 . . . . . . . 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 232 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑆𝑗) ≤ (𝑆𝑁))
8171, 80eqbrtrd 5169 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 ≤ (𝑆𝑁))
8281rexlimdv3a 3156 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗) → 𝐵 ≤ (𝑆𝑁)))
8370, 82mpd 15 . . 3 (𝜑𝐵 ≤ (𝑆𝑁))
845, 65sseldd 3995 . . . 4 (𝜑 → (𝑆𝑁) ∈ ℝ)
8584, 4letri3d 11400 . . 3 (𝜑 → ((𝑆𝑁) = 𝐵 ↔ ((𝑆𝑁) ≤ 𝐵𝐵 ≤ (𝑆𝑁))))
8667, 83, 85mpbir2and 713 . 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wrex 3067  wss 3962  c0 4338   class class class wbr 5147  cio 6513  wf 6558  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562   Isom wiso 6563  (class class class)co 7430  Fincfn 8983  cr 11151  0cc0 11152  1c1 11153  *cxr 11291   < clt 11292  cle 11293  cmin 11489  cn 12263  0cn0 12523  cz 12610  cuz 12875  [,]cicc 13386  ...cfz 13543  chash 14365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-icc 13390  df-fz 13544  df-hash 14366
This theorem is referenced by:  fourierdlem103  46164  fourierdlem104  46165
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