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Theorem fourierdlem52 37848
Description: d16:d17,d18:jca |- ( ph -> ( ( S  0 )  <_  A  /\  A  <_  ( S 0 ) ) ) (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem52.tf  |-  ( ph  ->  T  e.  Fin )
fourierdlem52.n  |-  N  =  ( ( # `  T
)  -  1 )
fourierdlem52.s  |-  S  =  ( iota f f 
Isom  <  ,  <  (
( 0 ... N
) ,  T ) )
fourierdlem52.a  |-  ( ph  ->  A  e.  RR )
fourierdlem52.b  |-  ( ph  ->  B  e.  RR )
fourierdlem52.t  |-  ( ph  ->  T  C_  ( A [,] B ) )
fourierdlem52.at  |-  ( ph  ->  A  e.  T )
fourierdlem52.bt  |-  ( ph  ->  B  e.  T )
Assertion
Ref Expression
fourierdlem52  |-  ( ph  ->  ( ( S :
( 0 ... N
) --> ( A [,] B )  /\  ( S `  0 )  =  A )  /\  ( S `  N )  =  B ) )
Distinct variable groups:    f, N    S, f    T, f    ph, 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  |-  ( ph  ->  T  e.  Fin )
2 fourierdlem52.t . . . . . 6  |-  ( ph  ->  T  C_  ( A [,] B ) )
3 fourierdlem52.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
4 fourierdlem52.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
53, 4iccssred 37439 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
62, 5sstrd 3475 . . . . 5  |-  ( ph  ->  T  C_  RR )
7 fourierdlem52.s . . . . 5  |-  S  =  ( iota f f 
Isom  <  ,  <  (
( 0 ... N
) ,  T ) )
8 fourierdlem52.n . . . . 5  |-  N  =  ( ( # `  T
)  -  1 )
91, 6, 7, 8fourierdlem36 37832 . . . 4  |-  ( ph  ->  S  Isom  <  ,  <  ( ( 0 ... N
) ,  T ) )
10 isof1o 6229 . . . 4  |-  ( S 
Isom  <  ,  <  (
( 0 ... N
) ,  T )  ->  S : ( 0 ... N ) -1-1-onto-> T )
11 f1of 5829 . . . 4  |-  ( S : ( 0 ... N ) -1-1-onto-> T  ->  S :
( 0 ... N
) --> T )
129, 10, 113syl 18 . . 3  |-  ( ph  ->  S : ( 0 ... N ) --> T )
1312, 2fssd 5753 . 2  |-  ( ph  ->  S : ( 0 ... N ) --> ( A [,] B ) )
14 f1ofo 5836 . . . . . 6  |-  ( S : ( 0 ... N ) -1-1-onto-> T  ->  S :
( 0 ... N
) -onto-> T )
159, 10, 143syl 18 . . . . 5  |-  ( ph  ->  S : ( 0 ... N ) -onto-> T )
16 fourierdlem52.at . . . . 5  |-  ( ph  ->  A  e.  T )
17 foelrn 6054 . . . . 5  |-  ( ( S : ( 0 ... N ) -onto-> T  /\  A  e.  T
)  ->  E. j  e.  ( 0 ... N
) A  =  ( S `  j ) )
1815, 16, 17syl2anc 666 . . . 4  |-  ( ph  ->  E. j  e.  ( 0 ... N ) A  =  ( S `
 j ) )
19 elfzle1 11804 . . . . . . . . 9  |-  ( j  e.  ( 0 ... N )  ->  0  <_  j )
2019adantl 468 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  0  <_  j )
219adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  S  Isom  <  ,  <  (
( 0 ... N
) ,  T ) )
22 ressxr 9686 . . . . . . . . . . . 12  |-  RR  C_  RR*
236, 22syl6ss 3477 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  RR* )
2423adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  T  C_ 
RR* )
25 fzssz 11803 . . . . . . . . . . 11  |-  ( 0 ... N )  C_  ZZ
26 zssre 10946 . . . . . . . . . . . 12  |-  ZZ  C_  RR
2726, 22sstri 3474 . . . . . . . . . . 11  |-  ZZ  C_  RR*
2825, 27sstri 3474 . . . . . . . . . 10  |-  ( 0 ... N )  C_  RR*
2924, 28jctil 540 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  (
( 0 ... N
)  C_  RR*  /\  T  C_ 
RR* ) )
30 hashcl 12539 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Fin  ->  ( # `
 T )  e. 
NN0 )
311, 30syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  T
)  e.  NN0 )
32 ne0i 3768 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  T  ->  T  =/=  (/) )
3316, 32syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  T  =/=  (/) )
34 hashge1 12569 . . . . . . . . . . . . . . . 16  |-  ( ( T  e.  Fin  /\  T  =/=  (/) )  ->  1  <_  ( # `  T
) )
351, 33, 34syl2anc 666 . . . . . . . . . . . . . . 15  |-  ( ph  ->  1  <_  ( # `  T
) )
36 elnnnn0c 10917 . . . . . . . . . . . . . . 15  |-  ( (
# `  T )  e.  NN  <->  ( ( # `  T )  e.  NN0  /\  1  <_  ( # `  T
) ) )
3731, 35, 36sylanbrc 669 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  T
)  e.  NN )
38 nnm1nn0 10913 . . . . . . . . . . . . . 14  |-  ( (
# `  T )  e.  NN  ->  ( ( # `
 T )  - 
1 )  e.  NN0 )
3937, 38syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( # `  T
)  -  1 )  e.  NN0 )
408, 39syl5eqel 2515 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  NN0 )
41 nn0uz 11195 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
4240, 41syl6eleq 2521 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
43 eluzfz1 11808 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... N
) )
4442, 43syl 17 . . . . . . . . . 10  |-  ( ph  ->  0  e.  ( 0 ... N ) )
4544anim1i 571 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  (
0  e.  ( 0 ... N )  /\  j  e.  ( 0 ... N ) ) )
46 leisorel 12622 . . . . . . . . 9  |-  ( ( S  Isom  <  ,  <  ( ( 0 ... N
) ,  T )  /\  ( ( 0 ... N )  C_  RR* 
/\  T  C_  RR* )  /\  ( 0  e.  ( 0 ... N )  /\  j  e.  ( 0 ... N ) ) )  ->  (
0  <_  j  <->  ( S `  0 )  <_ 
( S `  j
) ) )
4721, 29, 45, 46syl3anc 1265 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  (
0  <_  j  <->  ( S `  0 )  <_ 
( S `  j
) ) )
4820, 47mpbid 214 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  ( S `  0 )  <_  ( S `  j
) )
49483adant3 1026 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  A  =  ( S `  j ) )  ->  ( S `  0 )  <_ 
( S `  j
) )
50 eqcom 2432 . . . . . . . 8  |-  ( A  =  ( S `  j )  <->  ( S `  j )  =  A )
5150biimpi 198 . . . . . . 7  |-  ( A  =  ( S `  j )  ->  ( S `  j )  =  A )
52513ad2ant3 1029 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  A  =  ( S `  j ) )  ->  ( S `  j )  =  A )
5349, 52breqtrd 4446 . . . . 5  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  A  =  ( S `  j ) )  ->  ( S `  0 )  <_  A )
5453rexlimdv3a 2920 . . . 4  |-  ( ph  ->  ( E. j  e.  ( 0 ... N
) A  =  ( S `  j )  ->  ( S ` 
0 )  <_  A
) )
5518, 54mpd 15 . . 3  |-  ( ph  ->  ( S `  0
)  <_  A )
563rexrd 9692 . . . 4  |-  ( ph  ->  A  e.  RR* )
574rexrd 9692 . . . 4  |-  ( ph  ->  B  e.  RR* )
5813, 44ffvelrnd 6036 . . . 4  |-  ( ph  ->  ( S `  0
)  e.  ( A [,] B ) )
59 iccgelb 11693 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( S `
 0 )  e.  ( A [,] B
) )  ->  A  <_  ( S `  0
) )
6056, 57, 58, 59syl3anc 1265 . . 3  |-  ( ph  ->  A  <_  ( S `  0 ) )
615, 58sseldd 3466 . . . 4  |-  ( ph  ->  ( S `  0
)  e.  RR )
6261, 3letri3d 9779 . . 3  |-  ( ph  ->  ( ( S ` 
0 )  =  A  <-> 
( ( S ` 
0 )  <_  A  /\  A  <_  ( S `
 0 ) ) ) )
6355, 60, 62mpbir2and 931 . 2  |-  ( ph  ->  ( S `  0
)  =  A )
64 eluzfz2 11809 . . . . . 6  |-  ( N  e.  ( ZZ>= `  0
)  ->  N  e.  ( 0 ... N
) )
6542, 64syl 17 . . . . 5  |-  ( ph  ->  N  e.  ( 0 ... N ) )
6613, 65ffvelrnd 6036 . . . 4  |-  ( ph  ->  ( S `  N
)  e.  ( A [,] B ) )
67 iccleub 11692 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( S `
 N )  e.  ( A [,] B
) )  ->  ( S `  N )  <_  B )
6856, 57, 66, 67syl3anc 1265 . . 3  |-  ( ph  ->  ( S `  N
)  <_  B )
69 fourierdlem52.bt . . . . 5  |-  ( ph  ->  B  e.  T )
70 foelrn 6054 . . . . 5  |-  ( ( S : ( 0 ... N ) -onto-> T  /\  B  e.  T
)  ->  E. j  e.  ( 0 ... N
) B  =  ( S `  j ) )
7115, 69, 70syl2anc 666 . . . 4  |-  ( ph  ->  E. j  e.  ( 0 ... N ) B  =  ( S `
 j ) )
72 simp3 1008 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  B  =  ( S `  j ) )
73 elfzle2 11805 . . . . . . . 8  |-  ( j  e.  ( 0 ... N )  ->  j  <_  N )
74733ad2ant2 1028 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  j  <_  N )
7593ad2ant1 1027 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  S  Isom  <  ,  <  ( ( 0 ... N ) ,  T ) )
76293adant3 1026 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  ( (
0 ... N )  C_  RR* 
/\  T  C_  RR* )
)
77 simp2 1007 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  j  e.  ( 0 ... N
) )
78653ad2ant1 1027 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  N  e.  ( 0 ... N
) )
79 leisorel 12622 . . . . . . . 8  |-  ( ( S  Isom  <  ,  <  ( ( 0 ... N
) ,  T )  /\  ( ( 0 ... N )  C_  RR* 
/\  T  C_  RR* )  /\  ( j  e.  ( 0 ... N )  /\  N  e.  ( 0 ... N ) ) )  ->  (
j  <_  N  <->  ( S `  j )  <_  ( S `  N )
) )
8075, 76, 77, 78, 79syl112anc 1269 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  ( j  <_  N  <->  ( S `  j )  <_  ( S `  N )
) )
8174, 80mpbid 214 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  ( S `  j )  <_  ( S `  N )
)
8272, 81eqbrtrd 4442 . . . . 5  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  B  <_  ( S `  N ) )
8382rexlimdv3a 2920 . . . 4  |-  ( ph  ->  ( E. j  e.  ( 0 ... N
) B  =  ( S `  j )  ->  B  <_  ( S `  N )
) )
8471, 83mpd 15 . . 3  |-  ( ph  ->  B  <_  ( S `  N ) )
855, 66sseldd 3466 . . . 4  |-  ( ph  ->  ( S `  N
)  e.  RR )
8685, 4letri3d 9779 . . 3  |-  ( ph  ->  ( ( S `  N )  =  B  <-> 
( ( S `  N )  <_  B  /\  B  <_  ( S `
 N ) ) ) )
8768, 84, 86mpbir2and 931 . 2  |-  ( ph  ->  ( S `  N
)  =  B )
8813, 63, 87jca31 537 1  |-  ( ph  ->  ( ( S :
( 0 ... N
) --> ( A [,] B )  /\  ( S `  0 )  =  A )  /\  ( S `  N )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   E.wrex 2777    C_ wss 3437   (/)c0 3762   class class class wbr 4421   iotacio 5561   -->wf 5595   -onto->wfo 5597   -1-1-onto->wf1o 5598   ` cfv 5599    Isom wiso 5600  (class class class)co 6303   Fincfn 7575   RRcr 9540   0cc0 9541   1c1 9542   RR*cxr 9676    < clt 9677    <_ cle 9678    - cmin 9862   NNcn 10611   NN0cn0 10871   ZZcz 10939   ZZ>=cuz 11161   [,]cicc 11640   ...cfz 11786   #chash 12516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-oi 8029  df-card 8376  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-n0 10872  df-z 10940  df-uz 11162  df-icc 11644  df-fz 11787  df-hash 12517
This theorem is referenced by:  fourierdlem103  37899  fourierdlem104  37900
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