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Theorem nn0ind-raph 8833
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 8645 . 2  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 dfsbcq2 2841 . . . 4  |-  ( z  =  1  ->  ( [ z  /  x ] ph  <->  [. 1  /  x ]. ph ) )
3 nfv 1466 . . . . 5  |-  F/ x ch
4 nn0ind-raph.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
53, 4sbhypf 2668 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ch ) )
6 nfv 1466 . . . . 5  |-  F/ x th
7 nn0ind-raph.3 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
86, 7sbhypf 2668 . . . 4  |-  ( z  =  ( y  +  1 )  ->  ( [ z  /  x ] ph  <->  th ) )
9 nfv 1466 . . . . 5  |-  F/ x ta
10 nn0ind-raph.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
119, 10sbhypf 2668 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  ta ) )
12 nfsbc1v 2856 . . . . 5  |-  F/ x [. 1  /  x ]. ph
13 1ex 7462 . . . . 5  |-  1  e.  _V
14 c0ex 7461 . . . . . . 7  |-  0  e.  _V
15 0nn0 8658 . . . . . . . . . . . 12  |-  0  e.  NN0
16 eleq1a 2159 . . . . . . . . . . . 12  |-  ( 0  e.  NN0  ->  ( y  =  0  ->  y  e.  NN0 ) )
1715, 16ax-mp 7 . . . . . . . . . . 11  |-  ( y  =  0  ->  y  e.  NN0 )
18 nn0ind-raph.5 . . . . . . . . . . . . . . 15  |-  ps
19 nn0ind-raph.1 . . . . . . . . . . . . . . 15  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2018, 19mpbiri 166 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  ph )
21 eqeq2 2097 . . . . . . . . . . . . . . . 16  |-  ( y  =  0  ->  (
x  =  y  <->  x  = 
0 ) )
2221, 4syl6bir 162 . . . . . . . . . . . . . . 15  |-  ( y  =  0  ->  (
x  =  0  -> 
( ph  <->  ch ) ) )
2322pm5.74d 180 . . . . . . . . . . . . . 14  |-  ( y  =  0  ->  (
( x  =  0  ->  ph )  <->  ( x  =  0  ->  ch ) ) )
2420, 23mpbii 146 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
x  =  0  ->  ch ) )
2524com12 30 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
y  =  0  ->  ch ) )
2614, 25vtocle 2693 . . . . . . . . . . 11  |-  ( y  =  0  ->  ch )
27 nn0ind-raph.6 . . . . . . . . . . 11  |-  ( y  e.  NN0  ->  ( ch 
->  th ) )
2817, 26, 27sylc 61 . . . . . . . . . 10  |-  ( y  =  0  ->  th )
2928adantr 270 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  th )
30 oveq1 5641 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
y  +  1 )  =  ( 0  +  1 ) )
31 0p1e1 8507 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
3230, 31syl6eq 2136 . . . . . . . . . . . 12  |-  ( y  =  0  ->  (
y  +  1 )  =  1 )
3332eqeq2d 2099 . . . . . . . . . . 11  |-  ( y  =  0  ->  (
x  =  ( y  +  1 )  <->  x  = 
1 ) )
3433, 7syl6bir 162 . . . . . . . . . 10  |-  ( y  =  0  ->  (
x  =  1  -> 
( ph  <->  th ) ) )
3534imp 122 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  ( ph  <->  th )
)
3629, 35mpbird 165 . . . . . . . 8  |-  ( ( y  =  0  /\  x  =  1 )  ->  ph )
3736ex 113 . . . . . . 7  |-  ( y  =  0  ->  (
x  =  1  ->  ph ) )
3814, 37vtocle 2693 . . . . . 6  |-  ( x  =  1  ->  ph )
39 sbceq1a 2847 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<-> 
[. 1  /  x ]. ph ) )
4038, 39mpbid 145 . . . . 5  |-  ( x  =  1  ->  [. 1  /  x ]. ph )
4112, 13, 40vtoclef 2692 . . . 4  |-  [. 1  /  x ]. ph
42 nnnn0 8650 . . . . 5  |-  ( y  e.  NN  ->  y  e.  NN0 )
4342, 27syl 14 . . . 4  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
442, 5, 8, 11, 41, 43nnind 8410 . . 3  |-  ( A  e.  NN  ->  ta )
45 nfv 1466 . . . . 5  |-  F/ x
( 0  =  A  ->  ta )
46 eqeq1 2094 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  <->  0  =  A ) )
4719bicomd 139 . . . . . . . . 9  |-  ( x  =  0  ->  ( ps 
<-> 
ph ) )
4847, 10sylan9bb 450 . . . . . . . 8  |-  ( ( x  =  0  /\  x  =  A )  ->  ( ps  <->  ta )
)
4918, 48mpbii 146 . . . . . . 7  |-  ( ( x  =  0  /\  x  =  A )  ->  ta )
5049ex 113 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  ->  ta ) )
5146, 50sylbird 168 . . . . 5  |-  ( x  =  0  ->  (
0  =  A  ->  ta ) )
5245, 14, 51vtoclef 2692 . . . 4  |-  ( 0  =  A  ->  ta )
5352eqcoms 2091 . . 3  |-  ( A  =  0  ->  ta )
5444, 53jaoi 671 . 2  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ta )
551, 54sylbi 119 1  |-  ( A  e.  NN0  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   [wsb 1692   [.wsbc 2838  (class class class)co 5634   0cc0 7329   1c1 7330    + caddc 7332   NNcn 8394   NN0cn0 8643
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-i2m1 7429  ax-0id 7432
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637  df-inn 8395  df-n0 8644
This theorem is referenced by: (None)
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