MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uzind4s2 Structured version   Unicode version

Theorem uzind4s2 10934
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. Use this instead of uzind4s 10933 when  j and  k must be distinct in  [. ( k  +  1 )  /  j ]. ph. (Contributed by NM, 16-Nov-2005.)
Hypotheses
Ref Expression
uzind4s2.1  |-  ( M  e.  ZZ  ->  [. M  /  j ]. ph )
uzind4s2.2  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( [. k  /  j ]. ph  ->  [. ( k  +  1 )  /  j ]. ph ) )
Assertion
Ref Expression
uzind4s2  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
j ]. ph )
Distinct variable groups:    k, M    ph, k    j, k
Allowed substitution hints:    ph( j)    M( j)    N( j, k)

Proof of Theorem uzind4s2
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3207 . 2  |-  ( m  =  M  ->  ( [. m  /  j ]. ph  <->  [. M  /  j ]. ph ) )
2 dfsbcq 3207 . 2  |-  ( m  =  n  ->  ( [. m  /  j ]. ph  <->  [. n  /  j ]. ph ) )
3 dfsbcq 3207 . 2  |-  ( m  =  ( n  + 
1 )  ->  ( [. m  /  j ]. ph  <->  [. ( n  + 
1 )  /  j ]. ph ) )
4 dfsbcq 3207 . 2  |-  ( m  =  N  ->  ( [. m  /  j ]. ph  <->  [. N  /  j ]. ph ) )
5 uzind4s2.1 . 2  |-  ( M  e.  ZZ  ->  [. M  /  j ]. ph )
6 dfsbcq 3207 . . . 4  |-  ( k  =  n  ->  ( [. k  /  j ]. ph  <->  [. n  /  j ]. ph ) )
7 oveq1 6117 . . . . 5  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
8 dfsbcq 3207 . . . . 5  |-  ( ( k  +  1 )  =  ( n  + 
1 )  ->  ( [. ( k  +  1 )  /  j ]. ph  <->  [. ( n  +  1 )  /  j ]. ph ) )
97, 8syl 16 . . . 4  |-  ( k  =  n  ->  ( [. ( k  +  1 )  /  j ]. ph  <->  [. ( n  +  1 )  /  j ]. ph ) )
106, 9imbi12d 320 . . 3  |-  ( k  =  n  ->  (
( [. k  /  j ]. ph  ->  [. ( k  +  1 )  / 
j ]. ph )  <->  ( [. n  /  j ]. ph  ->  [. ( n  +  1 )  /  j ]. ph ) ) )
11 uzind4s2.2 . . 3  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( [. k  /  j ]. ph  ->  [. ( k  +  1 )  /  j ]. ph ) )
1210, 11vtoclga 3055 . 2  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( [. n  /  j ]. ph  ->  [. ( n  +  1 )  /  j ]. ph ) )
131, 2, 3, 4, 5, 12uzind4 10931 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
j ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   [.wsbc 3205   ` cfv 5437  (class class class)co 6110   1c1 9302    + caddc 9304   ZZcz 10665   ZZ>=cuz 10880
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 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-recs 6851  df-rdg 6885  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-n0 10599  df-z 10666  df-uz 10881
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator