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Theorem fzval2 11678
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 11677 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2 zssre 10867 . . . . . . 7  |-  ZZ  C_  RR
3 ressxr 9626 . . . . . . 7  |-  RR  C_  RR*
42, 3sstri 3498 . . . . . 6  |-  ZZ  C_  RR*
54sseli 3485 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  RR* )
64sseli 3485 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  RR* )
7 iccval 11571 . . . . 5  |-  ( ( M  e.  RR*  /\  N  e.  RR* )  ->  ( M [,] N )  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
85, 6, 7syl2an 475 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M [,] N
)  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
98ineq1d 3685 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M [,] N )  i^i  ZZ )  =  ( {
k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ ) )
10 inrab2 3768 . . . 4  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  (
RR*  i^i  ZZ )  |  ( M  <_ 
k  /\  k  <_  N ) }
11 sseqin2 3703 . . . . . 6  |-  ( ZZ  C_  RR*  <->  ( RR*  i^i  ZZ )  =  ZZ )
124, 11mpbi 208 . . . . 5  |-  ( RR*  i^i 
ZZ )  =  ZZ
13 rabeq 3100 . . . . 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 5 . . . 4  |-  { k  e.  ( RR*  i^i  ZZ )  |  ( M  <_  k  /\  k  <_  N ) }  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }
1510, 14eqtri 2483 . . 3  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }
169, 15syl6req 2512 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }  =  ( ( M [,] N
)  i^i  ZZ )
)
171, 16eqtrd 2495 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  ( ( M [,] N )  i^i  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808    i^i cin 3460    C_ wss 3461   class class class wbr 4439  (class class class)co 6270   RRcr 9480   RR*cxr 9616    <_ cle 9618   ZZcz 10860   [,]cicc 11535   ...cfz 11675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-xr 9621  df-neg 9799  df-z 10861  df-icc 11539  df-fz 11676
This theorem is referenced by:  dvfsumle  22591  dvfsumabs  22593  taylplem1  22927  taylplem2  22928  taylpfval  22929  dvtaylp  22934  ppisval  23578
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