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

Theorem nn0indALT 10368
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 10367 or nn0indALT 10368 may be used; see comment for nnind 10019. (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 10367 1  |-  ( A  e.  NN0  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726  (class class class)co 6082   0cc0 8991   1c1 8992    + caddc 8994   NN0cn0 10222
This theorem is referenced by:  uzaddcl  10534  faclbnd4lem4  11588  ipasslem1  22333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284
  Copyright terms: Public domain W3C validator