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Theorem nn0ind-raph 10960
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 10793 . 2  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 dfsbcq2 3327 . . . 4  |-  ( z  =  1  ->  ( [ z  /  x ] ph  <->  [. 1  /  x ]. ph ) )
3 nfv 1712 . . . . 5  |-  F/ x ch
4 nn0ind-raph.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
53, 4sbhypf 3153 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ch ) )
6 nfv 1712 . . . . 5  |-  F/ x th
7 nn0ind-raph.3 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
86, 7sbhypf 3153 . . . 4  |-  ( z  =  ( y  +  1 )  ->  ( [ z  /  x ] ph  <->  th ) )
9 nfv 1712 . . . . 5  |-  F/ x ta
10 nn0ind-raph.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
119, 10sbhypf 3153 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  ta ) )
12 nfsbc1v 3344 . . . . 5  |-  F/ x [. 1  /  x ]. ph
13 1ex 9580 . . . . 5  |-  1  e.  _V
14 c0ex 9579 . . . . . . 7  |-  0  e.  _V
15 0nn0 10806 . . . . . . . . . . . 12  |-  0  e.  NN0
16 eleq1a 2537 . . . . . . . . . . . 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 233 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  ph )
21 eqeq2 2469 . . . . . . . . . . . . . . . 16  |-  ( y  =  0  ->  (
x  =  y  <->  x  = 
0 ) )
2221, 4syl6bir 229 . . . . . . . . . . . . . . 15  |-  ( y  =  0  ->  (
x  =  0  -> 
( ph  <->  ch ) ) )
2322pm5.74d 247 . . . . . . . . . . . . . 14  |-  ( y  =  0  ->  (
( x  =  0  ->  ph )  <->  ( x  =  0  ->  ch ) ) )
2420, 23mpbii 211 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
x  =  0  ->  ch ) )
2524com12 31 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
y  =  0  ->  ch ) )
2614, 25vtocle 3180 . . . . . . . . . . 11  |-  ( y  =  0  ->  ch )
27 nn0ind-raph.6 . . . . . . . . . . 11  |-  ( y  e.  NN0  ->  ( ch 
->  th ) )
2817, 26, 27sylc 60 . . . . . . . . . 10  |-  ( y  =  0  ->  th )
2928adantr 463 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  th )
30 oveq1 6277 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
y  +  1 )  =  ( 0  +  1 ) )
31 0p1e1 10643 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
3230, 31syl6eq 2511 . . . . . . . . . . . 12  |-  ( y  =  0  ->  (
y  +  1 )  =  1 )
3332eqeq2d 2468 . . . . . . . . . . 11  |-  ( y  =  0  ->  (
x  =  ( y  +  1 )  <->  x  = 
1 ) )
3433, 7syl6bir 229 . . . . . . . . . 10  |-  ( y  =  0  ->  (
x  =  1  -> 
( ph  <->  th ) ) )
3534imp 427 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  ( ph  <->  th )
)
3629, 35mpbird 232 . . . . . . . 8  |-  ( ( y  =  0  /\  x  =  1 )  ->  ph )
3736ex 432 . . . . . . 7  |-  ( y  =  0  ->  (
x  =  1  ->  ph ) )
3814, 37vtocle 3180 . . . . . 6  |-  ( x  =  1  ->  ph )
39 sbceq1a 3335 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<-> 
[. 1  /  x ]. ph ) )
4038, 39mpbid 210 . . . . 5  |-  ( x  =  1  ->  [. 1  /  x ]. ph )
4112, 13, 40vtoclef 3179 . . . 4  |-  [. 1  /  x ]. ph
42 nnnn0 10798 . . . . 5  |-  ( y  e.  NN  ->  y  e.  NN0 )
4342, 27syl 16 . . . 4  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
442, 5, 8, 11, 41, 43nnind 10549 . . 3  |-  ( A  e.  NN  ->  ta )
45 nfv 1712 . . . . 5  |-  F/ x
( 0  =  A  ->  ta )
46 eqeq1 2458 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  <->  0  =  A ) )
4719bicomd 201 . . . . . . . . 9  |-  ( x  =  0  ->  ( ps 
<-> 
ph ) )
4847, 10sylan9bb 697 . . . . . . . 8  |-  ( ( x  =  0  /\  x  =  A )  ->  ( ps  <->  ta )
)
4918, 48mpbii 211 . . . . . . 7  |-  ( ( x  =  0  /\  x  =  A )  ->  ta )
5049ex 432 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  ->  ta ) )
5146, 50sylbird 235 . . . . 5  |-  ( x  =  0  ->  (
0  =  A  ->  ta ) )
5245, 14, 51vtoclef 3179 . . . 4  |-  ( 0  =  A  ->  ta )
5352eqcoms 2466 . . 3  |-  ( A  =  0  ->  ta )
5444, 53jaoi 377 . 2  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ta )
551, 54sylbi 195 1  |-  ( A  e.  NN0  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398   [wsb 1744    e. wcel 1823   [.wsbc 3324  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484   NNcn 10531   NN0cn0 10791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-nn 10532  df-n0 10792
This theorem is referenced by: (None)
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