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Theorem fzval2 10785
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 10784 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2 zssre 10031 . . . . . . 7  |-  ZZ  C_  RR
3 ressxr 8876 . . . . . . 7  |-  RR  C_  RR*
42, 3sstri 3188 . . . . . 6  |-  ZZ  C_  RR*
54sseli 3176 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  RR* )
64sseli 3176 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  RR* )
7 iccval 10695 . . . . 5  |-  ( ( M  e.  RR*  /\  N  e.  RR* )  ->  ( M [,] N )  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
85, 6, 7syl2an 463 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M [,] N
)  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
98ineq1d 3369 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M [,] N )  i^i  ZZ )  =  ( {
k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ ) )
10 inrab2 3441 . . . 4  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  (
RR*  i^i  ZZ )  |  ( M  <_ 
k  /\  k  <_  N ) }
11 sseqin2 3388 . . . . . 6  |-  ( ZZ  C_  RR*  <->  ( RR*  i^i  ZZ )  =  ZZ )
124, 11mpbi 199 . . . . 5  |-  ( RR*  i^i 
ZZ )  =  ZZ
13 rabeq 2782 . . . . 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 2303 . . 3  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }
169, 15syl6req 2332 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }  =  ( ( M [,] N
)  i^i  ZZ )
)
171, 16eqtrd 2315 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 358    = wceq 1623    e. wcel 1684   {crab 2547    i^i cin 3151    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   RRcr 8736   RR*cxr 8866    <_ cle 8868   ZZcz 10024   [,]cicc 10659   ...cfz 10782
This theorem is referenced by:  dvfsumle  19368  dvfsumabs  19370  taylplem1  19742  taylplem2  19743  taylpfval  19744  dvtaylp  19749  ppisval  20341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-xr 8871  df-neg 9040  df-z 10025  df-icc 10663  df-fz 10783
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