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Theorem nn0ind-raph 10128
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 hypothesis. 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 9983 . 2  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 dfsbcq2 3007 . . . 4  |-  ( z  =  1  ->  ( [ z  /  x ] ph  <->  [. 1  /  x ]. ph ) )
3 nfv 1609 . . . . 5  |-  F/ x ch
4 nn0ind-raph.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
53, 4sbhypf 2846 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ch ) )
6 nfv 1609 . . . . 5  |-  F/ x th
7 nn0ind-raph.3 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
86, 7sbhypf 2846 . . . 4  |-  ( z  =  ( y  +  1 )  ->  ( [ z  /  x ] ph  <->  th ) )
9 nfv 1609 . . . . 5  |-  F/ x ta
10 nn0ind-raph.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
119, 10sbhypf 2846 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  ta ) )
12 nfsbc1v 3023 . . . . 5  |-  F/ x [. 1  /  x ]. ph
13 1ex 8849 . . . . 5  |-  1  e.  _V
14 c0ex 8848 . . . . . . 7  |-  0  e.  _V
15 0nn0 9996 . . . . . . . . . . . 12  |-  0  e.  NN0
16 eleq1a 2365 . . . . . . . . . . . 12  |-  ( 0  e.  NN0  ->  ( y  =  0  ->  y  e.  NN0 ) )
1715, 16ax-mp 8 . . . . . . . . . . 11  |-  ( y  =  0  ->  y  e.  NN0 )
18 nn0ind-raph.5 . . . . . . . . . . . . . . 15  |-  ps
19 nn0ind-raph.1 . . . . . . . . . . . . . . 15  |-  ( x  =  0  ->  ( ph 
<->  ps ) )
2018, 19mpbiri 224 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  ph )
21 eqeq2 2305 . . . . . . . . . . . . . . . 16  |-  ( y  =  0  ->  (
x  =  y  <->  x  = 
0 ) )
2221, 4syl6bir 220 . . . . . . . . . . . . . . 15  |-  ( y  =  0  ->  (
x  =  0  -> 
( ph  <->  ch ) ) )
2322pm5.74d 238 . . . . . . . . . . . . . 14  |-  ( y  =  0  ->  (
( x  =  0  ->  ph )  <->  ( x  =  0  ->  ch ) ) )
2420, 23mpbii 202 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
x  =  0  ->  ch ) )
2524com12 27 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
y  =  0  ->  ch ) )
2614, 25vtocle 2870 . . . . . . . . . . 11  |-  ( y  =  0  ->  ch )
27 nn0ind-raph.6 . . . . . . . . . . 11  |-  ( y  e.  NN0  ->  ( ch 
->  th ) )
2817, 26, 27sylc 56 . . . . . . . . . 10  |-  ( y  =  0  ->  th )
2928adantr 451 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  th )
30 oveq1 5881 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
y  +  1 )  =  ( 0  +  1 ) )
31 0p1e1 9855 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
3230, 31syl6eq 2344 . . . . . . . . . . . 12  |-  ( y  =  0  ->  (
y  +  1 )  =  1 )
3332eqeq2d 2307 . . . . . . . . . . 11  |-  ( y  =  0  ->  (
x  =  ( y  +  1 )  <->  x  = 
1 ) )
3433, 7syl6bir 220 . . . . . . . . . 10  |-  ( y  =  0  ->  (
x  =  1  -> 
( ph  <->  th ) ) )
3534imp 418 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  ( ph  <->  th )
)
3629, 35mpbird 223 . . . . . . . 8  |-  ( ( y  =  0  /\  x  =  1 )  ->  ph )
3736ex 423 . . . . . . 7  |-  ( y  =  0  ->  (
x  =  1  ->  ph ) )
3814, 37vtocle 2870 . . . . . 6  |-  ( x  =  1  ->  ph )
39 sbceq1a 3014 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<-> 
[. 1  /  x ]. ph ) )
4038, 39mpbid 201 . . . . 5  |-  ( x  =  1  ->  [. 1  /  x ]. ph )
4112, 13, 40vtoclef 2869 . . . 4  |-  [. 1  /  x ]. ph
42 nnnn0 9988 . . . . 5  |-  ( y  e.  NN  ->  y  e.  NN0 )
4342, 27syl 15 . . . 4  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
442, 5, 8, 11, 41, 43nnind 9780 . . 3  |-  ( A  e.  NN  ->  ta )
45 nfv 1609 . . . . 5  |-  F/ x
( 0  =  A  ->  ta )
46 eqeq1 2302 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  <->  0  =  A ) )
4719bicomd 192 . . . . . . . . 9  |-  ( x  =  0  ->  ( ps 
<-> 
ph ) )
4847, 10sylan9bb 680 . . . . . . . 8  |-  ( ( x  =  0  /\  x  =  A )  ->  ( ps  <->  ta )
)
4918, 48mpbii 202 . . . . . . 7  |-  ( ( x  =  0  /\  x  =  A )  ->  ta )
5049ex 423 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  ->  ta ) )
5146, 50sylbird 226 . . . . 5  |-  ( x  =  0  ->  (
0  =  A  ->  ta ) )
5245, 14, 51vtoclef 2869 . . . 4  |-  ( 0  =  A  ->  ta )
5352eqcoms 2299 . . 3  |-  ( A  =  0  ->  ta )
5444, 53jaoi 368 . 2  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ta )
551, 54sylbi 187 1  |-  ( A  e.  NN0  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632   [wsb 1638    e. wcel 1696   [.wsbc 3004  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756   NNcn 9762   NN0cn0 9981
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-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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-nel 2462  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-nn 9763  df-n0 9982
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