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Theorem nn0ind-raph 8131
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  0
nn0ind-raph.2
nn0ind-raph.3  + 
1
nn0ind-raph.4
nn0ind-raph.5
nn0ind-raph.6  NN0
Assertion
Ref Expression
nn0ind-raph  NN0
Distinct variable groups:   ,   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()   ()   ()

Proof of Theorem nn0ind-raph
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elnn0 7959 . 2  NN0  NN  0
2 dfsbcq2 2761 . . . 4  1  [. 1  ].
3 nfv 1418 . . . . 5  F/
4 nn0ind-raph.2 . . . . 5
53, 4sbhypf 2597 . . . 4
6 nfv 1418 . . . . 5  F/
7 nn0ind-raph.3 . . . . 5  + 
1
86, 7sbhypf 2597 . . . 4  + 
1
9 nfv 1418 . . . . 5  F/
10 nn0ind-raph.4 . . . . 5
119, 10sbhypf 2597 . . . 4
12 nfsbc1v 2776 . . . . 5  F/ [. 1  ].
13 1ex 6820 . . . . 5  1  _V
14 c0ex 6819 . . . . . . 7  0  _V
15 0nn0 7972 . . . . . . . . . . . 12  0  NN0
16 eleq1a 2106 . . . . . . . . . . . 12  0  NN0  0  NN0
1715, 16ax-mp 7 . . . . . . . . . . 11  0  NN0
18 nn0ind-raph.5 . . . . . . . . . . . . . . 15
19 nn0ind-raph.1 . . . . . . . . . . . . . . 15  0
2018, 19mpbiri 157 . . . . . . . . . . . . . 14  0
21 eqeq2 2046 . . . . . . . . . . . . . . . 16  0  0
2221, 4syl6bir 153 . . . . . . . . . . . . . . 15  0  0
2322pm5.74d 171 . . . . . . . . . . . . . 14  0  0  0
2420, 23mpbii 136 . . . . . . . . . . . . 13  0  0
2524com12 27 . . . . . . . . . . . 12  0  0
2614, 25vtocle 2621 . . . . . . . . . . 11  0
27 nn0ind-raph.6 . . . . . . . . . . 11  NN0
2817, 26, 27sylc 56 . . . . . . . . . 10  0
2928adantr 261 . . . . . . . . 9  0  1
30 oveq1 5462 . . . . . . . . . . . . 13  0  +  1  0  +  1
31 0p1e1 7809 . . . . . . . . . . . . 13  0  +  1  1
3230, 31syl6eq 2085 . . . . . . . . . . . 12  0  +  1  1
3332eqeq2d 2048 . . . . . . . . . . 11  0  +  1  1
3433, 7syl6bir 153 . . . . . . . . . 10  0  1
3534imp 115 . . . . . . . . 9  0  1
3629, 35mpbird 156 . . . . . . . 8  0  1
3736ex 108 . . . . . . 7  0  1
3814, 37vtocle 2621 . . . . . 6  1
39 sbceq1a 2767 . . . . . 6  1 
[. 1  ].
4038, 39mpbid 135 . . . . 5  1  [. 1  ].
4112, 13, 40vtoclef 2620 . . . 4  [. 1  ].
42 nnnn0 7964 . . . . 5  NN  NN0
4342, 27syl 14 . . . 4  NN
442, 5, 8, 11, 41, 43nnind 7711 . . 3  NN
45 nfv 1418 . . . . 5  F/ 0
46 eqeq1 2043 . . . . . 6  0  0
4719bicomd 129 . . . . . . . . 9  0
4847, 10sylan9bb 435 . . . . . . . 8  0
4918, 48mpbii 136 . . . . . . 7  0
5049ex 108 . . . . . 6  0
5146, 50sylbird 159 . . . . 5  0 
0
5245, 14, 51vtoclef 2620 . . . 4  0
5352eqcoms 2040 . . 3  0
5444, 53jaoi 635 . 2  NN  0
551, 54sylbi 114 1  NN0
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 628   wceq 1242   wcel 1390  wsb 1642   [.wsbc 2758  (class class class)co 5455   0cc0 6711   1c1 6712    + caddc 6714   NNcn 7695   NN0cn0 7957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-i2m1 6788  ax-0id 6791
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-inn 7696  df-n0 7958
This theorem is referenced by: (None)
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