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Theorem fzsubel 9231
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 8540 . . 3  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
2 fzaddel 9230 . . 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 397 . 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 8514 . . . 4  |-  ( M  e.  ZZ  ->  M  e.  CC )
5 zcn 8514 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
64, 5anim12i 331 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  CC  /\  N  e.  CC ) )
7 zcn 8514 . . . 4  |-  ( J  e.  ZZ  ->  J  e.  CC )
8 zcn 8514 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  CC )
97, 8anim12i 331 . . 3  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  e.  CC  /\  K  e.  CC ) )
10 negsub 7500 . . . . 5  |-  ( ( J  e.  CC  /\  K  e.  CC )  ->  ( J  +  -u K )  =  ( J  -  K ) )
1110adantl 271 . . . 4  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  ( J  e.  CC  /\  K  e.  CC ) )  -> 
( J  +  -u K )  =  ( J  -  K ) )
12 negsub 7500 . . . . . . 7  |-  ( ( M  e.  CC  /\  K  e.  CC )  ->  ( M  +  -u K )  =  ( M  -  K ) )
13 negsub 7500 . . . . . . 7  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  +  -u K )  =  ( N  -  K ) )
1412, 13oveqan12d 5584 . . . . . 6  |-  ( ( ( M  e.  CC  /\  K  e.  CC )  /\  ( N  e.  CC  /\  K  e.  CC ) )  -> 
( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
1514anandirs 558 . . . . 5  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  K  e.  CC )  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K )
) )
1615adantrl 462 . . . 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 2153 . . 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 283 . 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 186 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 set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434  (class class class)co 5565   CCcc 7118    + caddc 7123    - cmin 7423   -ucneg 7424   ZZcz 8509   ...cfz 9182
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7206  ax-resscn 7207  ax-1cn 7208  ax-1re 7209  ax-icn 7210  ax-addcl 7211  ax-addrcl 7212  ax-mulcl 7213  ax-addcom 7215  ax-addass 7217  ax-distr 7219  ax-i2m1 7220  ax-0lt1 7221  ax-0id 7223  ax-rnegex 7224  ax-cnre 7226  ax-pre-ltirr 7227  ax-pre-ltwlin 7228  ax-pre-lttrn 7229  ax-pre-ltadd 7231
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-br 3807  df-opab 3861  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5521  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-pnf 7294  df-mnf 7295  df-xr 7296  df-ltxr 7297  df-le 7298  df-sub 7425  df-neg 7426  df-inn 8184  df-n0 8433  df-z 8510  df-fz 9183
This theorem is referenced by:  elfzp1b  9267  elfzm1b  9268
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