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Theorem fzval2 9968
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 9967 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2 zssre 9219 . . . . . . 7  |-  ZZ  C_  RR
3 ressxr 7963 . . . . . . 7  |-  RR  C_  RR*
42, 3sstri 3156 . . . . . 6  |-  ZZ  C_  RR*
54sseli 3143 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  RR* )
64sseli 3143 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  RR* )
7 iccval 9877 . . . . 5  |-  ( ( M  e.  RR*  /\  N  e.  RR* )  ->  ( M [,] N )  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
85, 6, 7syl2an 287 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M [,] N
)  =  { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) } )
98ineq1d 3327 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M [,] N )  i^i  ZZ )  =  ( {
k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ ) )
10 inrab2 3400 . . . 4  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  (
RR*  i^i  ZZ )  |  ( M  <_ 
k  /\  k  <_  N ) }
11 sseqin2 3346 . . . . . 6  |-  ( ZZ  C_  RR*  <->  ( RR*  i^i  ZZ )  =  ZZ )
124, 11mpbi 144 . . . . 5  |-  ( RR*  i^i 
ZZ )  =  ZZ
13 rabeq 2722 . . . . 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 2191 . . 3  |-  ( { k  e.  RR*  |  ( M  <_  k  /\  k  <_  N ) }  i^i  ZZ )  =  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }
169, 15eqtr2di 2220 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }  =  ( ( M [,] N
)  i^i  ZZ )
)
171, 16eqtrd 2203 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 103    = wceq 1348    e. wcel 2141   {crab 2452    i^i cin 3120    C_ wss 3121   class class class wbr 3989  (class class class)co 5853   RRcr 7773   RR*cxr 7953    <_ cle 7955   ZZcz 9212   [,]cicc 9848   ...cfz 9965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-neg 8093  df-z 9213  df-icc 9852  df-fz 9966
This theorem is referenced by: (None)
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