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Theorem uzind4s 8829
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 2827 . 2  |-  ( j  =  M  ->  ( [ j  /  k ] ph  <->  [. M  /  k ]. ph ) )
2 sbequ 1763 . 2  |-  ( j  =  m  ->  ( [ j  /  k ] ph  <->  [ m  /  k ] ph ) )
3 dfsbcq2 2827 . 2  |-  ( j  =  ( m  + 
1 )  ->  ( [ j  /  k ] ph  <->  [. ( m  + 
1 )  /  k ]. ph ) )
4 dfsbcq2 2827 . 2  |-  ( j  =  N  ->  ( [ j  /  k ] ph  <->  [. N  /  k ]. ph ) )
5 uzind4s.1 . 2  |-  ( M  e.  ZZ  ->  [. M  /  k ]. ph )
6 nfv 1462 . . . 4  |-  F/ k  m  e.  ( ZZ>= `  M )
7 nfs1v 1858 . . . . 5  |-  F/ k [ m  /  k ] ph
8 nfsbc1v 2842 . . . . 5  |-  F/ k
[. ( m  + 
1 )  /  k ]. ph
97, 8nfim 1505 . . . 4  |-  F/ k ( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph )
106, 9nfim 1505 . . 3  |-  F/ k ( m  e.  (
ZZ>= `  M )  -> 
( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph ) )
11 eleq1 2145 . . . 4  |-  ( k  =  m  ->  (
k  e.  ( ZZ>= `  M )  <->  m  e.  ( ZZ>= `  M )
) )
12 sbequ12 1696 . . . . 5  |-  ( k  =  m  ->  ( ph 
<->  [ m  /  k ] ph ) )
13 oveq1 5571 . . . . . 6  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
1413sbceq1d 2829 . . . . 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 1682 . 2  |-  ( m  e.  ( ZZ>= `  M
)  ->  ( [
m  /  k ]
ph  ->  [. ( m  + 
1 )  /  k ]. ph ) )
191, 2, 3, 4, 5, 18uzind4 8827 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
k ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   [wsb 1687   [.wsbc 2824   ` cfv 4952  (class class class)co 5564   1c1 7114    + caddc 7116   ZZcz 8502   ZZ>=cuz 8770
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 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-addcom 7208  ax-addass 7210  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-0id 7216  ax-rnegex 7217  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-ltadd 7224
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-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771
This theorem is referenced by: (None)
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