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Theorem elfz2 11679
Description: Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show  M  e.  ZZ and  N  e.  ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
elfz2  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) ) )

Proof of Theorem elfz2
StepHypRef Expression
1 anass 649 . 2  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) )
2 df-3an 975 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ ) )
32anbi1i 695 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N
) ) )
4 elfz1 11677 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N ) ) )
5 3anass 977 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N )  <->  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) )
6 ibar 504 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
75, 6syl5bb 257 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N
)  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
84, 7bitrd 253 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
9 fzf 11676 . . . . . . 7  |-  ... :
( ZZ  X.  ZZ )
--> ~P ZZ
109fdmi 5736 . . . . . 6  |-  dom  ...  =  ( ZZ  X.  ZZ )
1110ndmov 6443 . . . . 5  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  (/) )
1211eleq2d 2537 . . . 4  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N
)  <->  K  e.  (/) ) )
13 noel 3789 . . . . . 6  |-  -.  K  e.  (/)
1413pm2.21i 131 . . . . 5  |-  ( K  e.  (/)  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
15 simpl 457 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
1614, 15pm5.21ni 352 . . . 4  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  (/) 
<->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
1712, 16bitrd 253 . . 3  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N
)  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
188, 17pm2.61i 164 . 2  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) )
191, 3, 183bitr4ri 278 1  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447    X. cxp 4997  (class class class)co 6284    <_ cle 9629   ZZcz 10864   ...cfz 11672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-neg 9808  df-z 10865  df-fz 11673
This theorem is referenced by:  elfz4  11681  elfzuzb  11682  uzsubsubfz  11707  fzmmmeqm  11717  fzpreddisj  11729  elfz1b  11748  elfz0ubfz0  11776  elfz0fzfz0  11777  fz0fzelfz0  11778  fz0fzdiffz0  11781  elfzmlbm  11782  elfzmlbp  11783  fzind2  11892  swrdswrdlem  12647  swrdswrd  12648  swrdccatin12lem2a  12673  swrdccatin12lem2b  12674  swrdccatin2  12675  swrdccatin12lem2  12677  swrdccat3  12680  2cshwcshw  12756  cshwcsh2id  12759  chfacfscmulgsum  19156  chfacfpmmulgsum  19160  wwlkextproplem1  24445  wwlkextproplem2  24446  clwlkfclwwlk  24548  fzp1nel  28621  fprodntriv  28679  fprodeq0  28710  preduz  28885  monoords  31101  fmul01lt1lem1  31162  fmul01lt1lem2  31163  sumnnodd  31200  itgspltprt  31325  stoweidlem3  31331  stoweidlem34  31362  stoweidlem51  31379  fourierdlem12  31447  fourierdlem14  31449  fourierdlem41  31476  fourierdlem48  31483  fourierdlem49  31484  fourierdlem50  31485  fourierdlem79  31514  fourierdlem92  31527  fourierdlem93  31528  elfzelfzlble  31832
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