ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elfzp1b Unicode version

Theorem elfzp1b 9908
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 9114 . . . 4  |-  ( K  e.  ZZ  ->  ( K  +  1 )  e.  ZZ )
2 1z 9104 . . . . 5  |-  1  e.  ZZ
3 fzsubel 9871 . . . . . 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 431 . . . . 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 435 . . . 4  |-  ( ( N  e.  ZZ  /\  ( K  +  1
)  e.  ZZ )  ->  ( ( K  +  1 )  e.  ( 1 ... N
)  <->  ( ( K  +  1 )  - 
1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ) )
61, 5sylan2 284 . . 3  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( K  + 
1 )  e.  ( 1 ... N )  <-> 
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
76ancoms 266 . 2  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  + 
1 )  e.  ( 1 ... N )  <-> 
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
8 zcn 9083 . . . . 5  |-  ( K  e.  ZZ  ->  K  e.  CC )
9 ax-1cn 7737 . . . . 5  |-  1  e.  CC
10 pncan 7992 . . . . 5  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
118, 9, 10sylancl 410 . . . 4  |-  ( K  e.  ZZ  ->  (
( K  +  1 )  -  1 )  =  K )
12 1m1e0 8813 . . . . . 6  |-  ( 1  -  1 )  =  0
1312oveq1i 5792 . . . . 5  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
1413a1i 9 . . . 4  |-  ( K  e.  ZZ  ->  (
( 1  -  1 ) ... ( N  -  1 ) )  =  ( 0 ... ( N  -  1 ) ) )
1511, 14eleq12d 2211 . . 3  |-  ( K  e.  ZZ  ->  (
( ( K  + 
1 )  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) )  <->  K  e.  ( 0 ... ( N  -  1 ) ) ) )
1615adantr 274 . 2  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( K  +  1 )  - 
1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) )  <-> 
K  e.  ( 0 ... ( N  - 
1 ) ) ) )
177, 16bitr2d 188 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 103    <-> wb 104    = wceq 1332    e. wcel 1481  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645    + caddc 7647    - cmin 7957   ZZcz 9078   ...cfz 9821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-fz 9822
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator