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Theorem uzind4s 9337
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 2883 . 2  |-  ( j  =  M  ->  ( [ j  /  k ] ph  <->  [. M  /  k ]. ph ) )
2 sbequ 1794 . 2  |-  ( j  =  m  ->  ( [ j  /  k ] ph  <->  [ m  /  k ] ph ) )
3 dfsbcq2 2883 . 2  |-  ( j  =  ( m  + 
1 )  ->  ( [ j  /  k ] ph  <->  [. ( m  + 
1 )  /  k ]. ph ) )
4 dfsbcq2 2883 . 2  |-  ( j  =  N  ->  ( [ j  /  k ] ph  <->  [. N  /  k ]. ph ) )
5 uzind4s.1 . 2  |-  ( M  e.  ZZ  ->  [. M  /  k ]. ph )
6 nfv 1491 . . . 4  |-  F/ k  m  e.  ( ZZ>= `  M )
7 nfs1v 1890 . . . . 5  |-  F/ k [ m  /  k ] ph
8 nfsbc1v 2898 . . . . 5  |-  F/ k
[. ( m  + 
1 )  /  k ]. ph
97, 8nfim 1534 . . . 4  |-  F/ k ( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph )
106, 9nfim 1534 . . 3  |-  F/ k ( m  e.  (
ZZ>= `  M )  -> 
( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph ) )
11 eleq1 2178 . . . 4  |-  ( k  =  m  ->  (
k  e.  ( ZZ>= `  M )  <->  m  e.  ( ZZ>= `  M )
) )
12 sbequ12 1727 . . . . 5  |-  ( k  =  m  ->  ( ph 
<->  [ m  /  k ] ph ) )
13 oveq1 5747 . . . . . 6  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
1413sbceq1d 2885 . . . . 5  |-  ( k  =  m  ->  ( [. ( k  +  1 )  /  k ]. ph  <->  [. ( m  +  1 )  /  k ]. ph ) )
1512, 14imbi12d 233 . . . 4  |-  ( k  =  m  ->  (
( ph  ->  [. (
k  +  1 )  /  k ]. ph )  <->  ( [ m  /  k ] ph  ->  [. ( m  +  1 )  / 
k ]. ph ) ) )
1611, 15imbi12d 233 . . 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 1713 . 2  |-  ( m  e.  ( ZZ>= `  M
)  ->  ( [
m  /  k ]
ph  ->  [. ( m  + 
1 )  /  k ]. ph ) )
191, 2, 3, 4, 5, 18uzind4 9335 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
k ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463   [wsb 1718   [.wsbc 2880   ` cfv 5091  (class class class)co 5740   1c1 7585    + caddc 7587   ZZcz 9008   ZZ>=cuz 9278
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279
This theorem is referenced by: (None)
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