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Theorem elfzp1b 10094
Description: An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
elfzp1b  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 0 ... ( N  -  1 ) )  <-> 
( K  +  1 )  e.  ( 1 ... N ) ) )

Proof of Theorem elfzp1b
StepHypRef Expression
1 peano2z 9287 . . . 4  |-  ( K  e.  ZZ  ->  ( K  +  1 )  e.  ZZ )
2 1z 9277 . . . . 5  |-  1  e.  ZZ
3 fzsubel 10057 . . . . . 6  |-  ( ( ( 1  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  +  1 )  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( ( K  + 
1 )  e.  ( 1 ... N )  <-> 
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
42, 3mpanl1 434 . . . . 5  |-  ( ( N  e.  ZZ  /\  ( ( K  + 
1 )  e.  ZZ  /\  1  e.  ZZ ) )  ->  ( ( K  +  1 )  e.  ( 1 ... N )  <->  ( ( K  +  1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ) )
52, 4mpanr2 438 . . . 4  |-  ( ( N  e.  ZZ  /\  ( K  +  1
)  e.  ZZ )  ->  ( ( K  +  1 )  e.  ( 1 ... N
)  <->  ( ( K  +  1 )  - 
1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ) )
61, 5sylan2 286 . . 3  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( K  + 
1 )  e.  ( 1 ... N )  <-> 
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
76ancoms 268 . 2  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  + 
1 )  e.  ( 1 ... N )  <-> 
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
8 zcn 9256 . . . . 5  |-  ( K  e.  ZZ  ->  K  e.  CC )
9 ax-1cn 7903 . . . . 5  |-  1  e.  CC
10 pncan 8161 . . . . 5  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
118, 9, 10sylancl 413 . . . 4  |-  ( K  e.  ZZ  ->  (
( K  +  1 )  -  1 )  =  K )
12 1m1e0 8986 . . . . . 6  |-  ( 1  -  1 )  =  0
1312oveq1i 5884 . . . . 5  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
1413a1i 9 . . . 4  |-  ( K  e.  ZZ  ->  (
( 1  -  1 ) ... ( N  -  1 ) )  =  ( 0 ... ( N  -  1 ) ) )
1511, 14eleq12d 2248 . . 3  |-  ( K  e.  ZZ  ->  (
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) )  <->  K  e.  ( 0 ... ( N  -  1 ) ) ) )
1615adantr 276 . 2  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( K  +  1 )  - 
1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) )  <-> 
K  e.  ( 0 ... ( N  - 
1 ) ) ) )
177, 16bitr2d 189 1  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 0 ... ( N  -  1 ) )  <-> 
( K  +  1 )  e.  ( 1 ... N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148  (class class class)co 5874   CCcc 7808   0cc0 7810   1c1 7811    + caddc 7813    - cmin 8126   ZZcz 9251   ...cfz 10006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-fz 10007
This theorem is referenced by: (None)
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