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

Theorem nnind 9780
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 hypothesis. See nnaddcl 9784 for an example of its use. See nn0ind 10124 for induction on nonnegative integers and uzind 10119, uzind4 10292 for induction on an arbitrary set of upper integers. See indstr 10303 for strong induction. See also nnindALT 9781. (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 9773 . . . . . 6  |-  1  e.  NN
2 nnind.5 . . . . . 6  |-  ps
3 nnind.1 . . . . . . 7  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
43elrab 2936 . . . . . 6  |-  ( 1  e.  { x  e.  NN  |  ph }  <->  ( 1  e.  NN  /\  ps ) )
51, 2, 4mpbir2an 886 . . . . 5  |-  1  e.  { x  e.  NN  |  ph }
6 ssrab2 3271 . . . . . . . 8  |-  { x  e.  NN  |  ph }  C_  NN
76sseli 3189 . . . . . . 7  |-  ( y  e.  { x  e.  NN  |  ph }  ->  y  e.  NN )
8 peano2nn 9774 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
y  +  1 )  e.  NN )
98a1d 22 . . . . . . . . 9  |-  ( y  e.  NN  ->  (
y  e.  NN  ->  ( y  +  1 )  e.  NN ) )
10 nnind.6 . . . . . . . . 9  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
119, 10anim12d 546 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( y  e.  NN  /\ 
ch )  ->  (
( y  +  1 )  e.  NN  /\  th ) ) )
12 nnind.2 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1312elrab 2936 . . . . . . . 8  |-  ( y  e.  { x  e.  NN  |  ph }  <->  ( y  e.  NN  /\  ch ) )
14 nnind.3 . . . . . . . . 9  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
1514elrab 2936 . . . . . . . 8  |-  ( ( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  ( ( y  +  1 )  e.  NN  /\  th ) )
1611, 13, 153imtr4g 261 . . . . . . 7  |-  ( y  e.  NN  ->  (
y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
) )
177, 16mpcom 32 . . . . . 6  |-  ( y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
)
1817rgen 2621 . . . . 5  |-  A. y  e.  { x  e.  NN  |  ph }  ( y  +  1 )  e. 
{ x  e.  NN  |  ph }
19 peano5nni 9765 . . . . 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 } )
205, 18, 19mp2an 653 . . . 4  |-  NN  C_  { x  e.  NN  |  ph }
2120sseli 3189 . . 3  |-  ( A  e.  NN  ->  A  e.  { x  e.  NN  |  ph } )
22 nnind.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
2322elrab 2936 . . 3  |-  ( A  e.  { x  e.  NN  |  ph }  <->  ( A  e.  NN  /\  ta ) )
2421, 23sylib 188 . 2  |-  ( A  e.  NN  ->  ( A  e.  NN  /\  ta ) )
2524simprd 449 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165  (class class class)co 5874   1c1 8754    + caddc 8756   NNcn 9762
This theorem is referenced by:  nnindALT  9781  nn1m1nn  9782  nnaddcl  9784  nnmulcl  9785  nnge1  9788  nnsub  9800  nneo  10111  peano5uzi  10116  uzindOLD  10122  nn0ind-raph  10128  ser1const  11118  expcllem  11130  expeq0  11148  seqcoll  11417  climcndslem2  12325  sqr2irr  12543  gcdmultiple  12745  rplpwr  12751  prmind2  12785  prmdvdsexp  12809  eulerthlem2  12866  pcmpt  12956  prmpwdvds  12967  vdwlem10  13053  mulgnnass  14611  imasdsf1olem  17953  ovolunlem1a  18871  ovolicc2lem3  18894  voliunlem1  18923  volsup  18929  dvexp  19318  plyco  19639  dgrcolem1  19670  vieta1  19708  emcllem6  20310  bposlem5  20543  2sqlem10  20629  dchrisum0flb  20675  iuninc  23174  subfacp1lem6  23731  cvmliftlem10  23840  faclimlem3  24119  faclim  24126  incsequz  26561  bfplem1  26649  2nn0ind  27133  expmordi  27135  fmuldfeq  27816  stoweidlem20  27872  wallispilem4  27920  wallispi2lem1  27923  wallispi2lem2  27924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-1cn 8811
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-nn 9763
  Copyright terms: Public domain W3C validator