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Theorem nnind 8122
Description: Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 8126 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
Hypotheses
Ref Expression
nnind.1  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
nnind.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
nnind.3  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
nnind.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
nnind.5  |-  ps
nnind.6  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
Assertion
Ref Expression
nnind  |-  ( A  e.  NN  ->  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 nnind
StepHypRef Expression
1 1nn 8117 . . . . . 6  |-  1  e.  NN
2 nnind.5 . . . . . 6  |-  ps
3 nnind.1 . . . . . . 7  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
43elrab 2750 . . . . . 6  |-  ( 1  e.  { x  e.  NN  |  ph }  <->  ( 1  e.  NN  /\  ps ) )
51, 2, 4mpbir2an 884 . . . . 5  |-  1  e.  { x  e.  NN  |  ph }
6 elrabi 2747 . . . . . . 7  |-  ( y  e.  { x  e.  NN  |  ph }  ->  y  e.  NN )
7 peano2nn 8118 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
y  +  1 )  e.  NN )
87a1d 22 . . . . . . . . 9  |-  ( y  e.  NN  ->  (
y  e.  NN  ->  ( y  +  1 )  e.  NN ) )
9 nnind.6 . . . . . . . . 9  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
108, 9anim12d 328 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( y  e.  NN  /\ 
ch )  ->  (
( y  +  1 )  e.  NN  /\  th ) ) )
11 nnind.2 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1211elrab 2750 . . . . . . . 8  |-  ( y  e.  { x  e.  NN  |  ph }  <->  ( y  e.  NN  /\  ch ) )
13 nnind.3 . . . . . . . . 9  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
1413elrab 2750 . . . . . . . 8  |-  ( ( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  ( ( y  +  1 )  e.  NN  /\  th ) )
1510, 12, 143imtr4g 203 . . . . . . 7  |-  ( y  e.  NN  ->  (
y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
) )
166, 15mpcom 36 . . . . . 6  |-  ( y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
)
1716rgen 2417 . . . . 5  |-  A. y  e.  { x  e.  NN  |  ph }  ( y  +  1 )  e. 
{ x  e.  NN  |  ph }
18 peano5nni 8109 . . . . 5  |-  ( ( 1  e.  { x  e.  NN  |  ph }  /\  A. y  e.  {
x  e.  NN  |  ph }  ( y  +  1 )  e.  {
x  e.  NN  |  ph } )  ->  NN  C_ 
{ x  e.  NN  |  ph } )
195, 17, 18mp2an 417 . . . 4  |-  NN  C_  { x  e.  NN  |  ph }
2019sseli 2996 . . 3  |-  ( A  e.  NN  ->  A  e.  { x  e.  NN  |  ph } )
21 nnind.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
2221elrab 2750 . . 3  |-  ( A  e.  { x  e.  NN  |  ph }  <->  ( A  e.  NN  /\  ta ) )
2320, 22sylib 120 . 2  |-  ( A  e.  NN  ->  ( A  e.  NN  /\  ta ) )
2423simprd 112 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   {crab 2353    C_ wss 2974  (class class class)co 5543   1c1 7044    + caddc 7046   NNcn 8106
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-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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-cnex 7129  ax-resscn 7130  ax-1re 7132  ax-addrcl 7135
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546  df-inn 8107
This theorem is referenced by:  nnindALT  8123  nn1m1nn  8124  nnaddcl  8126  nnmulcl  8127  nnge1  8129  nn1gt1  8139  nnsub  8144  zaddcllempos  8469  zaddcllemneg  8471  nneoor  8530  peano5uzti  8536  nn0ind-raph  8545  indstr  8762  exbtwnzlemshrink  9335  expivallem  9574  expcllem  9584  expap0  9603  resqrexlemover  10034  resqrexlemlo  10037  resqrexlemcalc3  10040  gcdmultiple  10553  rplpwr  10560  prmind2  10646  prmdvdsexp  10671  sqrt2irr  10685  pw2dvdslemn  10687
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