MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0indALT Unicode version

Theorem nn0indALT 10109
Description: Principle of Mathematical Induction (inference schema) on nonnegative integers. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction hypothesis. Either nn0ind 10108 or nn0indALT 10109 may be used; see comment for nnind 9764. (Contributed by NM, 28-Nov-2005.)
Hypotheses
Ref Expression
nn0indALT.6  |-  ( y  e.  NN0  ->  ( ch 
->  th ) )
nn0indALT.5  |-  ps
nn0indALT.1  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
nn0indALT.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
nn0indALT.3  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
nn0indALT.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
Assertion
Ref Expression
nn0indALT  |-  ( A  e.  NN0  ->  ta )
Distinct variable groups:    x, y    x, A    ps, x    ch, x    th, x    ta, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)    ta( y)    A( y)

Proof of Theorem nn0indALT
StepHypRef Expression
1 nn0indALT.1 . 2  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2 nn0indALT.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
3 nn0indALT.3 . 2  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
4 nn0indALT.4 . 2  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
5 nn0indALT.5 . 2  |-  ps
6 nn0indALT.6 . 2  |-  ( y  e.  NN0  ->  ( ch 
->  th ) )
71, 2, 3, 4, 5, 6nn0ind 10108 1  |-  ( A  e.  NN0  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740   NN0cn0 9965
This theorem is referenced by:  uzaddcl  10275  faclbnd4lem4  11309  ipasslem1  21409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025
  Copyright terms: Public domain W3C validator