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Theorem fzval2 9425
Description: An alternate 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 9424 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2 zssre 8755 . . . . . . 7  |-  ZZ  C_  RR
3 ressxr 7529 . . . . . . 7  |-  RR  C_  RR*
42, 3sstri 3034 . . . . . 6  |-  ZZ  C_  RR*
54sseli 3021 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  RR* )
64sseli 3021 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  RR* )
7 iccval 9336 . . . . 5  |-  ( ( M  e.  RR*  /\  N  e.  RR* )  ->  ( M [,] N )  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
85, 6, 7syl2an 283 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M [,] N
)  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
98ineq1d 3200 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M [,] N )  i^i  ZZ )  =  ( {
k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ ) )
10 inrab2 3272 . . . 4  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  (
RR*  i^i  ZZ )  |  ( M  <_ 
k  /\  k  <_  N ) }
11 sseqin2 3219 . . . . . 6  |-  ( ZZ  C_  RR*  <->  ( RR*  i^i  ZZ )  =  ZZ )
124, 11mpbi 143 . . . . 5  |-  ( RR*  i^i 
ZZ )  =  ZZ
13 rabeq 2611 . . . . 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 7 . . . 4  |-  { k  e.  ( RR*  i^i  ZZ )  |  ( M  <_  k  /\  k  <_  N ) }  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }
1510, 14eqtri 2108 . . 3  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }
169, 15syl6req 2137 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }  =  ( ( M [,] N
)  i^i  ZZ )
)
171, 16eqtrd 2120 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 102    = wceq 1289    e. wcel 1438   {crab 2363    i^i cin 2998    C_ wss 2999   class class class wbr 3845  (class class class)co 5652   RRcr 7347   RR*cxr 7519    <_ cle 7521   ZZcz 8748   [,]cicc 9307   ...cfz 9422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-neg 7654  df-z 8749  df-icc 9311  df-fz 9423
This theorem is referenced by: (None)
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