ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uzind4s Unicode version

Theorem uzind4s 9068
Description: Induction on the upper set of integers that starts at an integer  M, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)
Hypotheses
Ref Expression
uzind4s.1  |-  ( M  e.  ZZ  ->  [. M  /  k ]. ph )
uzind4s.2  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ph  ->  [. ( k  +  1 )  /  k ]. ph ) )
Assertion
Ref Expression
uzind4s  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
k ]. ph )
Distinct variable group:    k, M
Allowed substitution hints:    ph( k)    N( k)

Proof of Theorem uzind4s
Dummy variables  m  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2843 . 2  |-  ( j  =  M  ->  ( [ j  /  k ] ph  <->  [. M  /  k ]. ph ) )
2 sbequ 1768 . 2  |-  ( j  =  m  ->  ( [ j  /  k ] ph  <->  [ m  /  k ] ph ) )
3 dfsbcq2 2843 . 2  |-  ( j  =  ( m  + 
1 )  ->  ( [ j  /  k ] ph  <->  [. ( m  + 
1 )  /  k ]. ph ) )
4 dfsbcq2 2843 . 2  |-  ( j  =  N  ->  ( [ j  /  k ] ph  <->  [. N  /  k ]. ph ) )
5 uzind4s.1 . 2  |-  ( M  e.  ZZ  ->  [. M  /  k ]. ph )
6 nfv 1466 . . . 4  |-  F/ k  m  e.  ( ZZ>= `  M )
7 nfs1v 1863 . . . . 5  |-  F/ k [ m  /  k ] ph
8 nfsbc1v 2858 . . . . 5  |-  F/ k
[. ( m  + 
1 )  /  k ]. ph
97, 8nfim 1509 . . . 4  |-  F/ k ( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph )
106, 9nfim 1509 . . 3  |-  F/ k ( m  e.  (
ZZ>= `  M )  -> 
( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph ) )
11 eleq1 2150 . . . 4  |-  ( k  =  m  ->  (
k  e.  ( ZZ>= `  M )  <->  m  e.  ( ZZ>= `  M )
) )
12 sbequ12 1701 . . . . 5  |-  ( k  =  m  ->  ( ph 
<->  [ m  /  k ] ph ) )
13 oveq1 5651 . . . . . 6  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
1413sbceq1d 2845 . . . . 5  |-  ( k  =  m  ->  ( [. ( k  +  1 )  /  k ]. ph  <->  [. ( m  +  1 )  /  k ]. ph ) )
1512, 14imbi12d 232 . . . 4  |-  ( k  =  m  ->  (
( ph  ->  [. (
k  +  1 )  /  k ]. ph )  <->  ( [ m  /  k ] ph  ->  [. ( m  +  1 )  / 
k ]. ph ) ) )
1611, 15imbi12d 232 . . 3  |-  ( k  =  m  ->  (
( k  e.  (
ZZ>= `  M )  -> 
( ph  ->  [. (
k  +  1 )  /  k ]. ph )
)  <->  ( m  e.  ( ZZ>= `  M )  ->  ( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph ) ) ) )
17 uzind4s.2 . . 3  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ph  ->  [. ( k  +  1 )  /  k ]. ph ) )
1810, 16, 17chvar 1687 . 2  |-  ( m  e.  ( ZZ>= `  M
)  ->  ( [
m  /  k ]
ph  ->  [. ( m  + 
1 )  /  k ]. ph ) )
191, 2, 3, 4, 5, 18uzind4 9066 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
k ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   [wsb 1692   [.wsbc 2840   ` cfv 5010  (class class class)co 5644   1c1 7341    + caddc 7343   ZZcz 8740   ZZ>=cuz 9009
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 3955  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-addcom 7435  ax-addass 7437  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-ltadd 7451
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 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-inn 8413  df-n0 8664  df-z 8741  df-uz 9010
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator