MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0ind-raph Structured version   Unicode version

Theorem nn0ind-raph 10981
Description: Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
nn0ind-raph.1  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
nn0ind-raph.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
nn0ind-raph.3  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
nn0ind-raph.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
nn0ind-raph.5  |-  ps
nn0ind-raph.6  |-  ( y  e.  NN0  ->  ( ch 
->  th ) )
Assertion
Ref Expression
nn0ind-raph  |-  ( 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 nn0ind-raph
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elnn0 10817 . 2  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 dfsbcq2 3240 . . . 4  |-  ( z  =  1  ->  ( [ z  /  x ] ph  <->  [. 1  /  x ]. ph ) )
3 nfv 1755 . . . . 5  |-  F/ x ch
4 nn0ind-raph.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
53, 4sbhypf 3065 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ch ) )
6 nfv 1755 . . . . 5  |-  F/ x th
7 nn0ind-raph.3 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
86, 7sbhypf 3065 . . . 4  |-  ( z  =  ( y  +  1 )  ->  ( [ z  /  x ] ph  <->  th ) )
9 nfv 1755 . . . . 5  |-  F/ x ta
10 nn0ind-raph.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
119, 10sbhypf 3065 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  ta ) )
12 nfsbc1v 3257 . . . . 5  |-  F/ x [. 1  /  x ]. ph
13 1ex 9584 . . . . 5  |-  1  e.  _V
14 c0ex 9583 . . . . . . 7  |-  0  e.  _V
15 0nn0 10830 . . . . . . . . . . . 12  |-  0  e.  NN0
16 eleq1a 2496 . . . . . . . . . . . 12  |-  ( 0  e.  NN0  ->  ( y  =  0  ->  y  e.  NN0 ) )
1715, 16ax-mp 5 . . . . . . . . . . 11  |-  ( y  =  0  ->  y  e.  NN0 )
18 nn0ind-raph.5 . . . . . . . . . . . . . . 15  |-  ps
19 nn0ind-raph.1 . . . . . . . . . . . . . . 15  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2018, 19mpbiri 236 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  ph )
21 eqeq2 2434 . . . . . . . . . . . . . . . 16  |-  ( y  =  0  ->  (
x  =  y  <->  x  = 
0 ) )
2221, 4syl6bir 232 . . . . . . . . . . . . . . 15  |-  ( y  =  0  ->  (
x  =  0  -> 
( ph  <->  ch ) ) )
2322pm5.74d 250 . . . . . . . . . . . . . 14  |-  ( y  =  0  ->  (
( x  =  0  ->  ph )  <->  ( x  =  0  ->  ch ) ) )
2420, 23mpbii 214 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
x  =  0  ->  ch ) )
2524com12 32 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
y  =  0  ->  ch ) )
2614, 25vtocle 3093 . . . . . . . . . . 11  |-  ( y  =  0  ->  ch )
27 nn0ind-raph.6 . . . . . . . . . . 11  |-  ( y  e.  NN0  ->  ( ch 
->  th ) )
2817, 26, 27sylc 62 . . . . . . . . . 10  |-  ( y  =  0  ->  th )
2928adantr 466 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  th )
30 oveq1 6251 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
y  +  1 )  =  ( 0  +  1 ) )
31 0p1e1 10667 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
3230, 31syl6eq 2473 . . . . . . . . . . . 12  |-  ( y  =  0  ->  (
y  +  1 )  =  1 )
3332eqeq2d 2433 . . . . . . . . . . 11  |-  ( y  =  0  ->  (
x  =  ( y  +  1 )  <->  x  = 
1 ) )
3433, 7syl6bir 232 . . . . . . . . . 10  |-  ( y  =  0  ->  (
x  =  1  -> 
( ph  <->  th ) ) )
3534imp 430 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  ( ph  <->  th )
)
3629, 35mpbird 235 . . . . . . . 8  |-  ( ( y  =  0  /\  x  =  1 )  ->  ph )
3736ex 435 . . . . . . 7  |-  ( y  =  0  ->  (
x  =  1  ->  ph ) )
3814, 37vtocle 3093 . . . . . 6  |-  ( x  =  1  ->  ph )
39 sbceq1a 3248 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<-> 
[. 1  /  x ]. ph ) )
4038, 39mpbid 213 . . . . 5  |-  ( x  =  1  ->  [. 1  /  x ]. ph )
4112, 13, 40vtoclef 3092 . . . 4  |-  [. 1  /  x ]. ph
42 nnnn0 10822 . . . . 5  |-  ( y  e.  NN  ->  y  e.  NN0 )
4342, 27syl 17 . . . 4  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
442, 5, 8, 11, 41, 43nnind 10573 . . 3  |-  ( A  e.  NN  ->  ta )
45 nfv 1755 . . . . 5  |-  F/ x
( 0  =  A  ->  ta )
46 eqeq1 2427 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  <->  0  =  A ) )
4719bicomd 204 . . . . . . . . 9  |-  ( x  =  0  ->  ( ps 
<-> 
ph ) )
4847, 10sylan9bb 704 . . . . . . . 8  |-  ( ( x  =  0  /\  x  =  A )  ->  ( ps  <->  ta )
)
4918, 48mpbii 214 . . . . . . 7  |-  ( ( x  =  0  /\  x  =  A )  ->  ta )
5049ex 435 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  ->  ta ) )
5146, 50sylbird 238 . . . . 5  |-  ( x  =  0  ->  (
0  =  A  ->  ta ) )
5245, 14, 51vtoclef 3092 . . . 4  |-  ( 0  =  A  ->  ta )
5352eqcoms 2431 . . 3  |-  ( A  =  0  ->  ta )
5444, 53jaoi 380 . 2  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ta )
551, 54sylbi 198 1  |-  ( A  e.  NN0  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437   [wsb 1790    e. wcel 1872   [.wsbc 3237  (class class class)co 6244   0cc0 9485   1c1 9486    + caddc 9488   NNcn 10555   NN0cn0 10815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-ov 6247  df-om 6646  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-pnf 9623  df-mnf 9624  df-ltxr 9626  df-nn 10556  df-n0 10816
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator