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Theorem fzval2 10978
Description: An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
fzval2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  ( ( M [,] N )  i^i  ZZ ) )

Proof of Theorem fzval2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fzval 10977 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2 zssre 10221 . . . . . . 7  |-  ZZ  C_  RR
3 ressxr 9062 . . . . . . 7  |-  RR  C_  RR*
42, 3sstri 3300 . . . . . 6  |-  ZZ  C_  RR*
54sseli 3287 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  RR* )
64sseli 3287 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  RR* )
7 iccval 10887 . . . . 5  |-  ( ( M  e.  RR*  /\  N  e.  RR* )  ->  ( M [,] N )  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
85, 6, 7syl2an 464 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M [,] N
)  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
98ineq1d 3484 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M [,] N )  i^i  ZZ )  =  ( {
k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ ) )
10 inrab2 3557 . . . 4  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  (
RR*  i^i  ZZ )  |  ( M  <_ 
k  /\  k  <_  N ) }
11 sseqin2 3503 . . . . . 6  |-  ( ZZ  C_  RR*  <->  ( RR*  i^i  ZZ )  =  ZZ )
124, 11mpbi 200 . . . . 5  |-  ( RR*  i^i 
ZZ )  =  ZZ
13 rabeq 2893 . . . . 5  |-  ( (
RR*  i^i  ZZ )  =  ZZ  ->  { k  e.  ( RR*  i^i  ZZ )  |  ( M  <_  k  /\  k  <_  N ) }  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) } )
1412, 13ax-mp 8 . . . 4  |-  { k  e.  ( RR*  i^i  ZZ )  |  ( M  <_  k  /\  k  <_  N ) }  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }
1510, 14eqtri 2407 . . 3  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }
169, 15syl6req 2436 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }  =  ( ( M [,] N
)  i^i  ZZ )
)
171, 16eqtrd 2419 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  ( ( M [,] N )  i^i  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2653    i^i cin 3262    C_ wss 3263   class class class wbr 4153  (class class class)co 6020   RRcr 8922   RR*cxr 9052    <_ cle 9054   ZZcz 10214   [,]cicc 10851   ...cfz 10975
This theorem is referenced by:  dvfsumle  19772  dvfsumabs  19774  taylplem1  20146  taylplem2  20147  taylpfval  20148  dvtaylp  20153  ppisval  20753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-xr 9057  df-neg 9226  df-z 10215  df-icc 10855  df-fz 10976
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