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Theorem elfzp1b 9242
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 8520 . . . 4  |-  ( K  e.  ZZ  ->  ( K  +  1 )  e.  ZZ )
2 1z 8510 . . . . 5  |-  1  e.  ZZ
3 fzsubel 9206 . . . . . 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 425 . . . . 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 429 . . . 4  |-  ( ( N  e.  ZZ  /\  ( K  +  1
)  e.  ZZ )  ->  ( ( K  +  1 )  e.  ( 1 ... N
)  <->  ( ( K  +  1 )  - 
1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ) )
61, 5sylan2 280 . . 3  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( K  + 
1 )  e.  ( 1 ... N )  <-> 
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
76ancoms 264 . 2  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  + 
1 )  e.  ( 1 ... N )  <-> 
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
8 zcn 8489 . . . . 5  |-  ( K  e.  ZZ  ->  K  e.  CC )
9 ax-1cn 7183 . . . . 5  |-  1  e.  CC
10 pncan 7433 . . . . 5  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
118, 9, 10sylancl 404 . . . 4  |-  ( K  e.  ZZ  ->  (
( K  +  1 )  -  1 )  =  K )
12 1m1e0 8227 . . . . . 6  |-  ( 1  -  1 )  =  0
1312oveq1i 5573 . . . . 5  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
1413a1i 9 . . . 4  |-  ( K  e.  ZZ  ->  (
( 1  -  1 ) ... ( N  -  1 ) )  =  ( 0 ... ( N  -  1 ) ) )
1511, 14eleq12d 2153 . . 3  |-  ( K  e.  ZZ  ->  (
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) )  <->  K  e.  ( 0 ... ( N  -  1 ) ) ) )
1615adantr 270 . 2  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( K  +  1 )  - 
1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) )  <-> 
K  e.  ( 0 ... ( N  - 
1 ) ) ) )
177, 16bitr2d 187 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 102    <-> wb 103    = wceq 1285    e. wcel 1434  (class class class)co 5563   CCcc 7093   0cc0 7095   1c1 7096    + caddc 7098    - cmin 7398   ZZcz 8484   ...cfz 9157
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 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
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 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-fz 9158
This theorem is referenced by: (None)
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