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Theorem elfzm1b 9512
Description: An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
elfzm1b  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 1 ... N )  <-> 
( K  -  1 )  e.  ( 0 ... ( N  - 
1 ) ) ) )

Proof of Theorem elfzm1b
StepHypRef Expression
1 1z 8776 . . . 4  |-  1  e.  ZZ
2 fzsubel 9474 . . . . 5  |-  ( ( ( 1  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( K  e.  ( 1 ... N )  <-> 
( K  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
31, 2mpanl1 425 . . . 4  |-  ( ( N  e.  ZZ  /\  ( K  e.  ZZ  /\  1  e.  ZZ ) )  ->  ( K  e.  ( 1 ... N
)  <->  ( K  - 
1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ) )
41, 3mpanr2 429 . . 3  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( K  e.  ( 1 ... N )  <-> 
( K  -  1 )  e.  ( ( 1  -  1 ) ... ( N  - 
1 ) ) ) )
5 1m1e0 8491 . . . . 5  |-  ( 1  -  1 )  =  0
65oveq1i 5662 . . . 4  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
76eleq2i 2154 . . 3  |-  ( ( K  -  1 )  e.  ( ( 1  -  1 ) ... ( N  -  1 ) )  <->  ( K  -  1 )  e.  ( 0 ... ( N  -  1 ) ) )
84, 7syl6bb 194 . 2  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( K  e.  ( 1 ... N )  <-> 
( K  -  1 )  e.  ( 0 ... ( N  - 
1 ) ) ) )
98ancoms 264 1  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 1 ... N )  <-> 
( K  -  1 )  e.  ( 0 ... ( N  - 
1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438  (class class class)co 5652   0cc0 7350   1c1 7351    - cmin 7653   ZZcz 8750   ...cfz 9424
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-fz 9425
This theorem is referenced by:  elfzom1b  9640  bcpasc  10174
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