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Theorem fzsubel 11595
Description: Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
Assertion
Ref Expression
fzsubel  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( J  e.  ( M ... N )  <-> 
( J  -  K
)  e.  ( ( M  -  K ) ... ( N  -  K ) ) ) )

Proof of Theorem fzsubel
StepHypRef Expression
1 znegcl 10781 . . 3  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
2 fzaddel 11594 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  -u K  e.  ZZ ) )  -> 
( J  e.  ( M ... N )  <-> 
( J  +  -u K )  e.  ( ( M  +  -u K ) ... ( N  +  -u K ) ) ) )
31, 2sylanr2 653 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( J  e.  ( M ... N )  <-> 
( J  +  -u K )  e.  ( ( M  +  -u K ) ... ( N  +  -u K ) ) ) )
4 zcn 10752 . . . 4  |-  ( M  e.  ZZ  ->  M  e.  CC )
5 zcn 10752 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
64, 5anim12i 566 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  CC  /\  N  e.  CC ) )
7 zcn 10752 . . . 4  |-  ( J  e.  ZZ  ->  J  e.  CC )
8 zcn 10752 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  CC )
97, 8anim12i 566 . . 3  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  e.  CC  /\  K  e.  CC ) )
10 negsub 9758 . . . . 5  |-  ( ( J  e.  CC  /\  K  e.  CC )  ->  ( J  +  -u K )  =  ( J  -  K ) )
1110adantl 466 . . . 4  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  ( J  e.  CC  /\  K  e.  CC ) )  -> 
( J  +  -u K )  =  ( J  -  K ) )
12 negsub 9758 . . . . . . 7  |-  ( ( M  e.  CC  /\  K  e.  CC )  ->  ( M  +  -u K )  =  ( M  -  K ) )
13 negsub 9758 . . . . . . 7  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  +  -u K )  =  ( N  -  K ) )
1412, 13oveqan12d 6209 . . . . . 6  |-  ( ( ( M  e.  CC  /\  K  e.  CC )  /\  ( N  e.  CC  /\  K  e.  CC ) )  -> 
( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
1514anandirs 827 . . . . 5  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  K  e.  CC )  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K )
) )
1615adantrl 715 . . . 4  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  ( J  e.  CC  /\  K  e.  CC ) )  -> 
( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
1711, 16eleq12d 2533 . . 3  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  ( J  e.  CC  /\  K  e.  CC ) )  -> 
( ( J  +  -u K )  e.  ( ( M  +  -u K ) ... ( N  +  -u K ) )  <->  ( J  -  K )  e.  ( ( M  -  K
) ... ( N  -  K ) ) ) )
186, 9, 17syl2an 477 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( ( J  +  -u K )  e.  ( ( M  +  -u K ) ... ( N  +  -u K ) )  <->  ( J  -  K )  e.  ( ( M  -  K
) ... ( N  -  K ) ) ) )
193, 18bitrd 253 1  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( J  e.  ( M ... N )  <-> 
( J  -  K
)  e.  ( ( M  -  K ) ... ( N  -  K ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6190   CCcc 9381    + caddc 9386    - cmin 9696   -ucneg 9697   ZZcz 10747   ...cfz 11538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-fz 11539
This theorem is referenced by:  elfzp1b  11638  elfzm1b  11639  fsum0diag2  13352  vdwapun  14137  sylow1lem1  16201  ballotlemfrceq  27045  fprodser  27596  fdc  28779  stoweidlem11  29944  stoweidlem34  29967
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