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Theorem fzsubel 10916
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 10144 . . 3  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
2 fzaddel 10915 . . 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 634 . 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 10118 . . . 4  |-  ( M  e.  ZZ  ->  M  e.  CC )
5 zcn 10118 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
64, 5anim12i 549 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  CC  /\  N  e.  CC ) )
7 zcn 10118 . . . 4  |-  ( J  e.  ZZ  ->  J  e.  CC )
8 zcn 10118 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  CC )
97, 8anim12i 549 . . 3  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  e.  CC  /\  K  e.  CC ) )
10 negsub 9182 . . . . 5  |-  ( ( J  e.  CC  /\  K  e.  CC )  ->  ( J  +  -u K )  =  ( J  -  K ) )
1110adantl 452 . . . 4  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  ( J  e.  CC  /\  K  e.  CC ) )  -> 
( J  +  -u K )  =  ( J  -  K ) )
12 negsub 9182 . . . . . . 7  |-  ( ( M  e.  CC  /\  K  e.  CC )  ->  ( M  +  -u K )  =  ( M  -  K ) )
13 negsub 9182 . . . . . . 7  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  +  -u K )  =  ( N  -  K ) )
1412, 13oveqan12d 5961 . . . . . 6  |-  ( ( ( M  e.  CC  /\  K  e.  CC )  /\  ( N  e.  CC  /\  K  e.  CC ) )  -> 
( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
1514anandirs 804 . . . . 5  |-  ( ( ( M  e.  CC  /\  N  e.  CC )  /\  K  e.  CC )  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K )
) )
1615adantrl 696 . . . 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 2426 . . 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 463 . 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 244 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 176    /\ wa 358    = wceq 1642    e. wcel 1710  (class class class)co 5942   CCcc 8822    + caddc 8827    - cmin 9124   -ucneg 9125   ZZcz 10113   ...cfz 10871
This theorem is referenced by:  elfzm1b  10949  fsum0diag2  12336  vdwapun  13112  sylow1lem1  15002  ballotlemfrceq  24035  elfzp1b  24416  fprodser  24576  fdc  25779  stoweidlem11  27083  stoweidlem34  27106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-n0 10055  df-z 10114  df-fz 10872
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