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Theorem uzind4s 11153
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 3339 . 2  |-  ( j  =  M  ->  ( [ j  /  k ] ph  <->  [. M  /  k ]. ph ) )
2 sbequ 2090 . 2  |-  ( j  =  m  ->  ( [ j  /  k ] ph  <->  [ m  /  k ] ph ) )
3 dfsbcq2 3339 . 2  |-  ( j  =  ( m  + 
1 )  ->  ( [ j  /  k ] ph  <->  [. ( m  + 
1 )  /  k ]. ph ) )
4 dfsbcq2 3339 . 2  |-  ( j  =  N  ->  ( [ j  /  k ] ph  <->  [. N  /  k ]. ph ) )
5 uzind4s.1 . 2  |-  ( M  e.  ZZ  ->  [. M  /  k ]. ph )
6 nfv 1683 . . . 4  |-  F/ k  m  e.  ( ZZ>= `  M )
7 nfs1v 2164 . . . . 5  |-  F/ k [ m  /  k ] ph
8 nfsbc1v 3356 . . . . 5  |-  F/ k
[. ( m  + 
1 )  /  k ]. ph
97, 8nfim 1867 . . . 4  |-  F/ k ( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph )
106, 9nfim 1867 . . 3  |-  F/ k ( m  e.  (
ZZ>= `  M )  -> 
( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph ) )
11 eleq1 2539 . . . 4  |-  ( k  =  m  ->  (
k  e.  ( ZZ>= `  M )  <->  m  e.  ( ZZ>= `  M )
) )
12 sbequ12 1961 . . . . 5  |-  ( k  =  m  ->  ( ph 
<->  [ m  /  k ] ph ) )
13 oveq1 6302 . . . . . 6  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
14 dfsbcq 3338 . . . . . 6  |-  ( ( k  +  1 )  =  ( m  + 
1 )  ->  ( [. ( k  +  1 )  /  k ]. ph  <->  [. ( m  +  1 )  /  k ]. ph ) )
1513, 14syl 16 . . . . 5  |-  ( k  =  m  ->  ( [. ( k  +  1 )  /  k ]. ph  <->  [. ( m  +  1 )  /  k ]. ph ) )
1612, 15imbi12d 320 . . . 4  |-  ( k  =  m  ->  (
( ph  ->  [. (
k  +  1 )  /  k ]. ph )  <->  ( [ m  /  k ] ph  ->  [. ( m  +  1 )  / 
k ]. ph ) ) )
1711, 16imbi12d 320 . . 3  |-  ( k  =  m  ->  (
( k  e.  (
ZZ>= `  M )  -> 
( ph  ->  [. (
k  +  1 )  /  k ]. ph )
)  <->  ( m  e.  ( ZZ>= `  M )  ->  ( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph ) ) ) )
18 uzind4s.2 . . 3  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ph  ->  [. ( k  +  1 )  /  k ]. ph ) )
1910, 17, 18chvar 1982 . 2  |-  ( m  e.  ( ZZ>= `  M
)  ->  ( [
m  /  k ]
ph  ->  [. ( m  + 
1 )  /  k ]. ph ) )
201, 2, 3, 4, 5, 19uzind4 11151 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
k ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   [wsb 1711    e. wcel 1767   [.wsbc 3336   ` cfv 5594  (class class class)co 6295   1c1 9505    + caddc 9507   ZZcz 10876   ZZ>=cuz 11094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095
This theorem is referenced by: (None)
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