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Theorem fzsubel 11044
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 10269 . . 3  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
2 fzaddel 11043 . . 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 635 . 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 10243 . . . 4  |-  ( M  e.  ZZ  ->  M  e.  CC )
5 zcn 10243 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
64, 5anim12i 550 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  CC  /\  N  e.  CC ) )
7 zcn 10243 . . . 4  |-  ( J  e.  ZZ  ->  J  e.  CC )
8 zcn 10243 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  CC )
97, 8anim12i 550 . . 3  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  e.  CC  /\  K  e.  CC ) )
10 negsub 9305 . . . . 5  |-  ( ( J  e.  CC  /\  K  e.  CC )  ->  ( J  +  -u K )  =  ( J  -  K ) )
1110adantl 453 . . . 4  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  ( J  e.  CC  /\  K  e.  CC ) )  -> 
( J  +  -u K )  =  ( J  -  K ) )
12 negsub 9305 . . . . . . 7  |-  ( ( M  e.  CC  /\  K  e.  CC )  ->  ( M  +  -u K )  =  ( M  -  K ) )
13 negsub 9305 . . . . . . 7  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  +  -u K )  =  ( N  -  K ) )
1412, 13oveqan12d 6059 . . . . . 6  |-  ( ( ( M  e.  CC  /\  K  e.  CC )  /\  ( N  e.  CC  /\  K  e.  CC ) )  -> 
( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
1514anandirs 805 . . . . 5  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  K  e.  CC )  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K )
) )
1615adantrl 697 . . . 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 2472 . . 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 464 . 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 245 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    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721  (class class class)co 6040   CCcc 8944    + caddc 8949    - cmin 9247   -ucneg 9248   ZZcz 10238   ...cfz 10999
This theorem is referenced by:  elfzm1b  11080  fsum0diag2  12521  vdwapun  13297  sylow1lem1  15187  ballotlemfrceq  24739  elfzp1b  25069  fprodser  25228  fdc  26339  stoweidlem11  27627  stoweidlem34  27650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-fz 11000
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