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Theorem uzind4s 10906
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 hypothesis. (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 3184 . 2  |-  ( j  =  M  ->  ( [ j  /  k ] ph  <->  [. M  /  k ]. ph ) )
2 sbequ 2067 . 2  |-  ( j  =  m  ->  ( [ j  /  k ] ph  <->  [ m  /  k ] ph ) )
3 dfsbcq2 3184 . 2  |-  ( j  =  ( m  + 
1 )  ->  ( [ j  /  k ] ph  <->  [. ( m  + 
1 )  /  k ]. ph ) )
4 dfsbcq2 3184 . 2  |-  ( j  =  N  ->  ( [ j  /  k ] ph  <->  [. N  /  k ]. ph ) )
5 uzind4s.1 . 2  |-  ( M  e.  ZZ  ->  [. M  /  k ]. ph )
6 nfv 1673 . . . 4  |-  F/ k  m  e.  ( ZZ>= `  M )
7 nfs1v 2142 . . . . 5  |-  F/ k [ m  /  k ] ph
8 nfsbc1v 3201 . . . . 5  |-  F/ k
[. ( m  + 
1 )  /  k ]. ph
97, 8nfim 1852 . . . 4  |-  F/ k ( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph )
106, 9nfim 1852 . . 3  |-  F/ k ( m  e.  (
ZZ>= `  M )  -> 
( [ m  / 
k ] ph  ->  [. ( m  +  1 )  /  k ]. ph ) )
11 eleq1 2498 . . . 4  |-  ( k  =  m  ->  (
k  e.  ( ZZ>= `  M )  <->  m  e.  ( ZZ>= `  M )
) )
12 sbequ12 1936 . . . . 5  |-  ( k  =  m  ->  ( ph 
<->  [ m  /  k ] ph ) )
13 oveq1 6093 . . . . . 6  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
14 dfsbcq 3183 . . . . . 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 1957 . 2  |-  ( m  e.  ( ZZ>= `  M
)  ->  ( [
m  /  k ]
ph  ->  [. ( m  + 
1 )  /  k ]. ph ) )
201, 2, 3, 4, 5, 19uzind4 10904 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
k ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369   [wsb 1700    e. wcel 1756   [.wsbc 3181   ` cfv 5413  (class class class)co 6086   1c1 9275    + caddc 9277   ZZcz 10638   ZZ>=cuz 10853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854
This theorem is referenced by: (None)
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