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Theorem fourierdlem36 31879
Description:  F is an isomorphism. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem36.a  |-  ( ph  ->  A  e.  Fin )
fourierdlem36.assr  |-  ( ph  ->  A  C_  RR )
fourierdlem36.f  |-  F  =  ( iota f f 
Isom  <  ,  <  (
( 0 ... N
) ,  A ) )
fourierdlem36.n  |-  N  =  ( ( # `  A
)  -  1 )
Assertion
Ref Expression
fourierdlem36  |-  ( ph  ->  F  Isom  <  ,  <  ( ( 0 ... N
) ,  A ) )
Distinct variable groups:    A, f    f, F    f, N    ph, f

Proof of Theorem fourierdlem36
StepHypRef Expression
1 fourierdlem36.f . . 3  |-  F  =  ( iota f f 
Isom  <  ,  <  (
( 0 ... N
) ,  A ) )
2 fourierdlem36.a . . . . . 6  |-  ( ph  ->  A  e.  Fin )
3 fourierdlem36.assr . . . . . . 7  |-  ( ph  ->  A  C_  RR )
4 ltso 9668 . . . . . . 7  |-  <  Or  RR
5 soss 4808 . . . . . . 7  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
63, 4, 5mpisyl 18 . . . . . 6  |-  ( ph  ->  <  Or  A )
7 0zd 10883 . . . . . 6  |-  ( ph  ->  0  e.  ZZ )
8 eqid 2443 . . . . . 6  |-  ( (
# `  A )  +  ( 0  -  1 ) )  =  ( ( # `  A
)  +  ( 0  -  1 ) )
92, 6, 7, 8fzisoeu 31454 . . . . 5  |-  ( ph  ->  E! f  f  Isom  <  ,  <  ( ( 0 ... ( ( # `  A )  +  ( 0  -  1 ) ) ) ,  A
) )
10 hashcl 12410 . . . . . . . . . . . 12  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
112, 10syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
1211nn0cnd 10861 . . . . . . . . . 10  |-  ( ph  ->  ( # `  A
)  e.  CC )
13 1cnd 9615 . . . . . . . . . 10  |-  ( ph  ->  1  e.  CC )
1412, 13negsubd 9942 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  A
)  +  -u 1
)  =  ( (
# `  A )  -  1 ) )
15 df-neg 9813 . . . . . . . . . . 11  |-  -u 1  =  ( 0  -  1 )
1615eqcomi 2456 . . . . . . . . . 10  |-  ( 0  -  1 )  = 
-u 1
1716oveq2i 6292 . . . . . . . . 9  |-  ( (
# `  A )  +  ( 0  -  1 ) )  =  ( ( # `  A
)  +  -u 1
)
18 fourierdlem36.n . . . . . . . . 9  |-  N  =  ( ( # `  A
)  -  1 )
1914, 17, 183eqtr4g 2509 . . . . . . . 8  |-  ( ph  ->  ( ( # `  A
)  +  ( 0  -  1 ) )  =  N )
2019oveq2d 6297 . . . . . . 7  |-  ( ph  ->  ( 0 ... (
( # `  A )  +  ( 0  -  1 ) ) )  =  ( 0 ... N ) )
21 isoeq4 6203 . . . . . . 7  |-  ( ( 0 ... ( (
# `  A )  +  ( 0  -  1 ) ) )  =  ( 0 ... N )  ->  (
f  Isom  <  ,  <  ( ( 0 ... (
( # `  A )  +  ( 0  -  1 ) ) ) ,  A )  <->  f  Isom  <  ,  <  ( ( 0 ... N ) ,  A ) ) )
2220, 21syl 16 . . . . . 6  |-  ( ph  ->  ( f  Isom  <  ,  <  ( ( 0 ... ( ( # `  A )  +  ( 0  -  1 ) ) ) ,  A
)  <->  f  Isom  <  ,  <  ( ( 0 ... N ) ,  A ) ) )
2322eubidv 2290 . . . . 5  |-  ( ph  ->  ( E! f  f 
Isom  <  ,  <  (
( 0 ... (
( # `  A )  +  ( 0  -  1 ) ) ) ,  A )  <->  E! f 
f  Isom  <  ,  <  ( ( 0 ... N
) ,  A ) ) )
249, 23mpbid 210 . . . 4  |-  ( ph  ->  E! f  f  Isom  <  ,  <  ( ( 0 ... N ) ,  A ) )
25 iotacl 5564 . . . 4  |-  ( E! f  f  Isom  <  ,  <  ( ( 0 ... N ) ,  A )  ->  ( iota f f  Isom  <  ,  <  ( ( 0 ... N ) ,  A ) )  e. 
{ f  |  f 
Isom  <  ,  <  (
( 0 ... N
) ,  A ) } )
2624, 25syl 16 . . 3  |-  ( ph  ->  ( iota f f 
Isom  <  ,  <  (
( 0 ... N
) ,  A ) )  e.  { f  |  f  Isom  <  ,  <  ( ( 0 ... N ) ,  A ) } )
271, 26syl5eqel 2535 . 2  |-  ( ph  ->  F  e.  { f  |  f  Isom  <  ,  <  ( ( 0 ... N ) ,  A ) } )
28 iotaex 5558 . . . 4  |-  ( iota f f  Isom  <  ,  <  ( ( 0 ... N ) ,  A ) )  e. 
_V
291, 28eqeltri 2527 . . 3  |-  F  e. 
_V
30 isoeq1 6200 . . 3  |-  ( f  =  F  ->  (
f  Isom  <  ,  <  ( ( 0 ... N
) ,  A )  <-> 
F  Isom  <  ,  <  ( ( 0 ... N
) ,  A ) ) )
3129, 30elab 3232 . 2  |-  ( F  e.  { f  |  f  Isom  <  ,  <  ( ( 0 ... N
) ,  A ) }  <->  F  Isom  <  ,  <  ( ( 0 ... N ) ,  A
) )
3227, 31sylib 196 1  |-  ( ph  ->  F  Isom  <  ,  <  ( ( 0 ... N
) ,  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1383    e. wcel 1804   E!weu 2268   {cab 2428   _Vcvv 3095    C_ wss 3461    Or wor 4789   iotacio 5539   ` cfv 5578    Isom wiso 5579  (class class class)co 6281   Fincfn 7518   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    < clt 9631    - cmin 9810   -ucneg 9811   NN0cn0 10802   ...cfz 11683   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388
This theorem is referenced by:  fourierdlem50  31893  fourierdlem51  31894  fourierdlem52  31895  fourierdlem54  31897  fourierdlem76  31919  fourierdlem102  31945  fourierdlem103  31946  fourierdlem104  31947  fourierdlem114  31957
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