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Theorem fourierdlem52 32187
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 31785 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
62, 5sstrd 3509 . . . . 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 32171 . . . 4  |-  ( ph  ->  S  Isom  <  ,  <  ( ( 0 ... N
) ,  T ) )
10 isof1o 6222 . . . 4  |-  ( S 
Isom  <  ,  <  (
( 0 ... N
) ,  T )  ->  S : ( 0 ... N ) -1-1-onto-> T )
11 f1of 5822 . . . 4  |-  ( S : ( 0 ... N ) -1-1-onto-> T  ->  S :
( 0 ... N
) --> T )
129, 10, 113syl 20 . . 3  |-  ( ph  ->  S : ( 0 ... N ) --> T )
1312, 2fssd 5746 . 2  |-  ( ph  ->  S : ( 0 ... N ) --> ( A [,] B ) )
14 f1ofo 5829 . . . . . 6  |-  ( S : ( 0 ... N ) -1-1-onto-> T  ->  S :
( 0 ... N
) -onto-> T )
159, 10, 143syl 20 . . . . 5  |-  ( ph  ->  S : ( 0 ... N ) -onto-> T )
16 fourierdlem52.at . . . . 5  |-  ( ph  ->  A  e.  T )
17 foelrn 6051 . . . . 5  |-  ( ( S : ( 0 ... N ) -onto-> T  /\  A  e.  T
)  ->  E. j  e.  ( 0 ... N
) A  =  ( S `  j ) )
1815, 16, 17syl2anc 661 . . . 4  |-  ( ph  ->  E. j  e.  ( 0 ... N ) A  =  ( S `
 j ) )
19 elfzle1 11714 . . . . . . . . 9  |-  ( j  e.  ( 0 ... N )  ->  0  <_  j )
2019adantl 466 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  0  <_  j )
219adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  S  Isom  <  ,  <  (
( 0 ... N
) ,  T ) )
22 ressxr 9654 . . . . . . . . . . . 12  |-  RR  C_  RR*
236, 22syl6ss 3511 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  RR* )
2423adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  T  C_ 
RR* )
25 fzssz 31712 . . . . . . . . . . 11  |-  ( 0 ... N )  C_  ZZ
26 zssre 10892 . . . . . . . . . . . 12  |-  ZZ  C_  RR
2726, 22sstri 3508 . . . . . . . . . . 11  |-  ZZ  C_  RR*
2825, 27sstri 3508 . . . . . . . . . 10  |-  ( 0 ... N )  C_  RR*
2924, 28jctil 537 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  (
( 0 ... N
)  C_  RR*  /\  T  C_ 
RR* ) )
30 hashcl 12431 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Fin  ->  ( # `
 T )  e. 
NN0 )
311, 30syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  T
)  e.  NN0 )
32 ne0i 3799 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  T  ->  T  =/=  (/) )
3316, 32syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  T  =/=  (/) )
34 hashge1 12460 . . . . . . . . . . . . . . . 16  |-  ( ( T  e.  Fin  /\  T  =/=  (/) )  ->  1  <_  ( # `  T
) )
351, 33, 34syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  1  <_  ( # `  T
) )
36 elnnnn0c 10862 . . . . . . . . . . . . . . 15  |-  ( (
# `  T )  e.  NN  <->  ( ( # `  T )  e.  NN0  /\  1  <_  ( # `  T
) ) )
3731, 35, 36sylanbrc 664 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  T
)  e.  NN )
38 nnm1nn0 10858 . . . . . . . . . . . . . 14  |-  ( (
# `  T )  e.  NN  ->  ( ( # `
 T )  - 
1 )  e.  NN0 )
3937, 38syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( # `  T
)  -  1 )  e.  NN0 )
408, 39syl5eqel 2549 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  NN0 )
41 nn0uz 11140 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
4240, 41syl6eleq 2555 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
43 eluzfz1 11718 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... N
) )
4442, 43syl 16 . . . . . . . . . 10  |-  ( ph  ->  0  e.  ( 0 ... N ) )
4544anim1i 568 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  (
0  e.  ( 0 ... N )  /\  j  e.  ( 0 ... N ) ) )
46 leisorel 12513 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  (
0  <_  j  <->  ( S `  0 )  <_ 
( S `  j
) ) )
4820, 47mpbid 210 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... N
) )  ->  ( S `  0 )  <_  ( S `  j
) )
49483adant3 1016 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  A  =  ( S `  j ) )  ->  ( S `  0 )  <_ 
( S `  j
) )
50 eqcom 2466 . . . . . . . 8  |-  ( A  =  ( S `  j )  <->  ( S `  j )  =  A )
5150biimpi 194 . . . . . . 7  |-  ( A  =  ( S `  j )  ->  ( S `  j )  =  A )
52513ad2ant3 1019 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  A  =  ( S `  j ) )  ->  ( S `  j )  =  A )
5349, 52breqtrd 4480 . . . . 5  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  A  =  ( S `  j ) )  ->  ( S `  0 )  <_  A )
5453rexlimdv3a 2951 . . . 4  |-  ( ph  ->  ( E. j  e.  ( 0 ... N
) A  =  ( S `  j )  ->  ( S ` 
0 )  <_  A
) )
5518, 54mpd 15 . . 3  |-  ( ph  ->  ( S `  0
)  <_  A )
563rexrd 9660 . . . 4  |-  ( ph  ->  A  e.  RR* )
574rexrd 9660 . . . 4  |-  ( ph  ->  B  e.  RR* )
5813, 44ffvelrnd 6033 . . . 4  |-  ( ph  ->  ( S `  0
)  e.  ( A [,] B ) )
59 iccgelb 11606 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( S `
 0 )  e.  ( A [,] B
) )  ->  A  <_  ( S `  0
) )
6056, 57, 58, 59syl3anc 1228 . . 3  |-  ( ph  ->  A  <_  ( S `  0 ) )
615, 58sseldd 3500 . . . 4  |-  ( ph  ->  ( S `  0
)  e.  RR )
6261, 3letri3d 9744 . . 3  |-  ( ph  ->  ( ( S ` 
0 )  =  A  <-> 
( ( S ` 
0 )  <_  A  /\  A  <_  ( S `
 0 ) ) ) )
6355, 60, 62mpbir2and 922 . 2  |-  ( ph  ->  ( S `  0
)  =  A )
64 eluzfz2 11719 . . . . . 6  |-  ( N  e.  ( ZZ>= `  0
)  ->  N  e.  ( 0 ... N
) )
6542, 64syl 16 . . . . 5  |-  ( ph  ->  N  e.  ( 0 ... N ) )
6613, 65ffvelrnd 6033 . . . 4  |-  ( ph  ->  ( S `  N
)  e.  ( A [,] B ) )
67 iccleub 11605 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( S `
 N )  e.  ( A [,] B
) )  ->  ( S `  N )  <_  B )
6856, 57, 66, 67syl3anc 1228 . . 3  |-  ( ph  ->  ( S `  N
)  <_  B )
69 fourierdlem52.bt . . . . 5  |-  ( ph  ->  B  e.  T )
70 foelrn 6051 . . . . 5  |-  ( ( S : ( 0 ... N ) -onto-> T  /\  B  e.  T
)  ->  E. j  e.  ( 0 ... N
) B  =  ( S `  j ) )
7115, 69, 70syl2anc 661 . . . 4  |-  ( ph  ->  E. j  e.  ( 0 ... N ) B  =  ( S `
 j ) )
72 simp3 998 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  B  =  ( S `  j ) )
73 elfzle2 11715 . . . . . . . 8  |-  ( j  e.  ( 0 ... N )  ->  j  <_  N )
74733ad2ant2 1018 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  j  <_  N )
7593ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  S  Isom  <  ,  <  ( ( 0 ... N ) ,  T ) )
76293adant3 1016 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  ( (
0 ... N )  C_  RR* 
/\  T  C_  RR* )
)
77 simp2 997 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  j  e.  ( 0 ... N
) )
78653ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  N  e.  ( 0 ... N
) )
79 leisorel 12513 . . . . . . . 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 1232 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  ( j  <_  N  <->  ( S `  j )  <_  ( S `  N )
) )
8174, 80mpbid 210 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  ( S `  j )  <_  ( S `  N )
)
8272, 81eqbrtrd 4476 . . . . 5  |-  ( (
ph  /\  j  e.  ( 0 ... N
)  /\  B  =  ( S `  j ) )  ->  B  <_  ( S `  N ) )
8382rexlimdv3a 2951 . . . 4  |-  ( ph  ->  ( E. j  e.  ( 0 ... N
) B  =  ( S `  j )  ->  B  <_  ( S `  N )
) )
8471, 83mpd 15 . . 3  |-  ( ph  ->  B  <_  ( S `  N ) )
855, 66sseldd 3500 . . . 4  |-  ( ph  ->  ( S `  N
)  e.  RR )
8685, 4letri3d 9744 . . 3  |-  ( ph  ->  ( ( S `  N )  =  B  <-> 
( ( S `  N )  <_  B  /\  B  <_  ( S `
 N ) ) ) )
8768, 84, 86mpbir2and 922 . 2  |-  ( ph  ->  ( S `  N
)  =  B )
8813, 63, 87jca31 534 1  |-  ( ph  ->  ( ( S :
( 0 ... N
) --> ( A [,] B )  /\  ( S `  0 )  =  A )  /\  ( S `  N )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    C_ wss 3471   (/)c0 3793   class class class wbr 4456   iotacio 5555   -->wf 5590   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6296   Fincfn 7535   RRcr 9508   0cc0 9509   1c1 9510   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   [,]cicc 11557   ...cfz 11697   #chash 12408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-icc 11561  df-fz 11698  df-hash 12409
This theorem is referenced by:  fourierdlem103  32238  fourierdlem104  32239
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