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Theorem uzind4s2 8796
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 8795 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 2827 . 2  |-  ( m  =  M  ->  ( [. m  /  j ]. ph  <->  [. M  /  j ]. ph ) )
2 dfsbcq 2827 . 2  |-  ( m  =  n  ->  ( [. m  /  j ]. ph  <->  [. n  /  j ]. ph ) )
3 dfsbcq 2827 . 2  |-  ( m  =  ( n  + 
1 )  ->  ( [. m  /  j ]. ph  <->  [. ( n  + 
1 )  /  j ]. ph ) )
4 dfsbcq 2827 . 2  |-  ( m  =  N  ->  ( [. m  /  j ]. ph  <->  [. N  /  j ]. ph ) )
5 uzind4s2.1 . 2  |-  ( M  e.  ZZ  ->  [. M  /  j ]. ph )
6 dfsbcq 2827 . . . 4  |-  ( k  =  n  ->  ( [. k  /  j ]. ph  <->  [. n  /  j ]. ph ) )
7 oveq1 5571 . . . . 5  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
87sbceq1d 2830 . . . 4  |-  ( k  =  n  ->  ( [. ( k  +  1 )  /  j ]. ph  <->  [. ( n  +  1 )  /  j ]. ph ) )
96, 8imbi12d 232 . . 3  |-  ( k  =  n  ->  (
( [. k  /  j ]. ph  ->  [. ( k  +  1 )  / 
j ]. ph )  <->  ( [. n  /  j ]. ph  ->  [. ( n  +  1 )  /  j ]. ph ) ) )
10 uzind4s2.2 . . 3  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( [. k  /  j ]. ph  ->  [. ( k  +  1 )  /  j ]. ph ) )
119, 10vtoclga 2673 . 2  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( [. n  /  j ]. ph  ->  [. ( n  +  1 )  /  j ]. ph ) )
121, 2, 3, 4, 5, 11uzind4 8793 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  [. N  / 
j ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   [.wsbc 2825   ` cfv 4953  (class class class)co 5564   1c1 7080    + caddc 7082   ZZcz 8468   ZZ>=cuz 8736
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 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-addcom 7174  ax-addass 7176  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-0id 7182  ax-rnegex 7183  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-ltadd 7190
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 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-inn 8143  df-n0 8392  df-z 8469  df-uz 8737
This theorem is referenced by: (None)
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