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Theorem uzind4i 8813
Description: Induction on the upper integers that start at  M. The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 4-Sep-2005.)
Hypotheses
Ref Expression
uzind4i.1  |-  M  e.  ZZ
uzind4i.2  |-  ( j  =  M  ->  ( ph 
<->  ps ) )
uzind4i.3  |-  ( j  =  k  ->  ( ph 
<->  ch ) )
uzind4i.4  |-  ( j  =  ( k  +  1 )  ->  ( ph 
<->  th ) )
uzind4i.5  |-  ( j  =  N  ->  ( ph 
<->  ta ) )
uzind4i.6  |-  ps
uzind4i.7  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ch  ->  th ) )
Assertion
Ref Expression
uzind4i  |-  ( N  e.  ( ZZ>= `  M
)  ->  ta )
Distinct variable groups:    j, N    ps, j    ch, j    th, j    ta, j    ph, k    j, k, M
Allowed substitution hints:    ph( j)    ps( k)    ch( k)    th( k)    ta( k)    N( k)

Proof of Theorem uzind4i
StepHypRef Expression
1 uzind4i.2 . 2  |-  ( j  =  M  ->  ( ph 
<->  ps ) )
2 uzind4i.3 . 2  |-  ( j  =  k  ->  ( ph 
<->  ch ) )
3 uzind4i.4 . 2  |-  ( j  =  ( k  +  1 )  ->  ( ph 
<->  th ) )
4 uzind4i.5 . 2  |-  ( j  =  N  ->  ( ph 
<->  ta ) )
5 uzind4i.6 . . 3  |-  ps
65a1i 9 . 2  |-  ( M  e.  ZZ  ->  ps )
7 uzind4i.7 . 2  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ch  ->  th ) )
81, 2, 3, 4, 6, 7uzind4 8809 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   ` cfv 4952  (class class class)co 5563   1c1 7096    + caddc 7098   ZZcz 8484   ZZ>=cuz 8752
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 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
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 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753
This theorem is referenced by:  rebtwn2zlemshrink  9392
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