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Theorem nn0ind-raph 10742
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 10581 . 2  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 dfsbcq2 3189 . . . 4  |-  ( z  =  1  ->  ( [ z  /  x ] ph  <->  [. 1  /  x ]. ph ) )
3 nfv 1673 . . . . 5  |-  F/ x ch
4 nn0ind-raph.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
53, 4sbhypf 3019 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ch ) )
6 nfv 1673 . . . . 5  |-  F/ x th
7 nn0ind-raph.3 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
86, 7sbhypf 3019 . . . 4  |-  ( z  =  ( y  +  1 )  ->  ( [ z  /  x ] ph  <->  th ) )
9 nfv 1673 . . . . 5  |-  F/ x ta
10 nn0ind-raph.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
119, 10sbhypf 3019 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  ta ) )
12 nfsbc1v 3206 . . . . 5  |-  F/ x [. 1  /  x ]. ph
13 1ex 9381 . . . . 5  |-  1  e.  _V
14 c0ex 9380 . . . . . . 7  |-  0  e.  _V
15 0nn0 10594 . . . . . . . . . . . 12  |-  0  e.  NN0
16 eleq1a 2512 . . . . . . . . . . . 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 2452 . . . . . . . . . . . . . . . 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 3046 . . . . . . . . . . 11  |-  ( y  =  0  ->  ch )
27 nn0ind-raph.6 . . . . . . . . . . 11  |-  ( y  e.  NN0  ->  ( ch 
->  th ) )
2817, 26, 27sylc 60 . . . . . . . . . 10  |-  ( y  =  0  ->  th )
2928adantr 465 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  th )
30 oveq1 6098 . . . . . . . . . . . . 13  |-  ( y  =  0  ->  (
y  +  1 )  =  ( 0  +  1 ) )
31 0p1e1 10433 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
3230, 31syl6eq 2491 . . . . . . . . . . . 12  |-  ( y  =  0  ->  (
y  +  1 )  =  1 )
3332eqeq2d 2454 . . . . . . . . . . 11  |-  ( y  =  0  ->  (
x  =  ( y  +  1 )  <->  x  = 
1 ) )
3433, 7syl6bir 229 . . . . . . . . . 10  |-  ( y  =  0  ->  (
x  =  1  -> 
( ph  <->  th ) ) )
3534imp 429 . . . . . . . . 9  |-  ( ( y  =  0  /\  x  =  1 )  ->  ( ph  <->  th )
)
3629, 35mpbird 232 . . . . . . . 8  |-  ( ( y  =  0  /\  x  =  1 )  ->  ph )
3736ex 434 . . . . . . 7  |-  ( y  =  0  ->  (
x  =  1  ->  ph ) )
3814, 37vtocle 3046 . . . . . 6  |-  ( x  =  1  ->  ph )
39 sbceq1a 3197 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<-> 
[. 1  /  x ]. ph ) )
4038, 39mpbid 210 . . . . 5  |-  ( x  =  1  ->  [. 1  /  x ]. ph )
4112, 13, 40vtoclef 3045 . . . 4  |-  [. 1  /  x ]. ph
42 nnnn0 10586 . . . . 5  |-  ( y  e.  NN  ->  y  e.  NN0 )
4342, 27syl 16 . . . 4  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
442, 5, 8, 11, 41, 43nnind 10340 . . 3  |-  ( A  e.  NN  ->  ta )
45 nfv 1673 . . . . 5  |-  F/ x
( 0  =  A  ->  ta )
46 eqeq1 2449 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  <->  0  =  A ) )
4719bicomd 201 . . . . . . . . 9  |-  ( x  =  0  ->  ( ps 
<-> 
ph ) )
4847, 10sylan9bb 699 . . . . . . . 8  |-  ( ( x  =  0  /\  x  =  A )  ->  ( ps  <->  ta )
)
4918, 48mpbii 211 . . . . . . 7  |-  ( ( x  =  0  /\  x  =  A )  ->  ta )
5049ex 434 . . . . . 6  |-  ( x  =  0  ->  (
x  =  A  ->  ta ) )
5146, 50sylbird 235 . . . . 5  |-  ( x  =  0  ->  (
0  =  A  ->  ta ) )
5245, 14, 51vtoclef 3045 . . . 4  |-  ( 0  =  A  ->  ta )
5352eqcoms 2446 . . 3  |-  ( A  =  0  ->  ta )
5444, 53jaoi 379 . 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 368    /\ wa 369    = wceq 1369   [wsb 1700    e. wcel 1756   [.wsbc 3186  (class class class)co 6091   0cc0 9282   1c1 9283    + caddc 9285   NNcn 10322   NN0cn0 10579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-nn 10323  df-n0 10580
This theorem is referenced by: (None)
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