Theorem nnind 9270

Description: Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 9274 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

Hypotheses

Ref	Expression
nnind.1	|- ( x = 1 -> ( ph <-> ps ) )
nnind.2	|- ( x = y -> ( ph <-> ch ) )
nnind.3	|- ( x = ( y + 1 ) -> ( ph <-> th ) )
nnind.4	|- ( x = A -> ( ph <-> ta ) )
nnind.5	|- ps
nnind.6	|- ( y e. NN -> ( ch -> th ) )

Assertion

Ref	Expression
nnind	|- ( A e. NN -> 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 nnind

Step	Hyp	Ref	Expression
1		1nn 9265	. . . . . 6 |- 1 e. NN
2		nnind.5	. . . . . 6 |- ps
3		nnind.1	. . . . . . 7 |- ( x = 1 -> ( ph <-> ps ) )
4	3	elrab 2976	. . . . . 6 |- ( 1 e. { x e. NN | ph } <-> ( 1 e. NN /\ ps ) )
5	1, 2, 4	mpbir2an 951	. . . . 5 |- 1 e. { x e. NN | ph }
6		elrabi 2973	. . . . . . 7 |- ( y e. { x e. NN | ph } -> y e. NN )
7		peano2nn 9266	. . . . . . . . . 10 |- ( y e. NN -> ( y + 1 ) e. NN )
8	7	a1d 22	. . . . . . . . 9 |- ( y e. NN -> ( y e. NN -> ( y + 1 ) e. NN ) )
9		nnind.6	. . . . . . . . 9 |- ( y e. NN -> ( ch -> th ) )
10	8, 9	anim12d 335	. . . . . . . 8 |- ( y e. NN -> ( ( y e. NN /\ ch ) -> ( ( y + 1 ) e. NN /\ th ) ) )
11		nnind.2	. . . . . . . . 9 |- ( x = y -> ( ph <-> ch ) )
12	11	elrab 2976	. . . . . . . 8 |- ( y e. { x e. NN | ph } <-> ( y e. NN /\ ch ) )
13		nnind.3	. . . . . . . . 9 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
14	13	elrab 2976	. . . . . . . 8 |- ( ( y + 1 ) e. { x e. NN | ph } <-> ( ( y + 1 ) e. NN /\ th ) )
15	10, 12, 14	3imtr4g 205	. . . . . . 7 |- ( y e. NN -> ( y e. { x e. NN | ph } -> ( y + 1 ) e. { x e. NN | ph } ) )
16	6, 15	mpcom 36	. . . . . 6 |- ( y e. { x e. NN | ph } -> ( y + 1 ) e. { x e. NN | ph } )
17	16	rgen 2597	. . . . 5 |- A. y e. { x e. NN | ph } ( y + 1 ) e. { x e. NN | ph }
18		peano5nni 9257	. . . . 5 |- ( ( 1 e. { x e. NN | ph } /\ A. y e. { x e. NN | ph } ( y + 1 ) e. { x e. NN | ph } ) -> NN C_ { x e. NN | ph } )
19	5, 17, 18	mp2an 426	. . . 4 |- NN C_ { x e. NN | ph }
20	19	sseli 3238	. . 3 |- ( A e. NN -> A e. { x e. NN | ph } )
21		nnind.4	. . . 4 |- ( x = A -> ( ph <-> ta ) )
22	21	elrab 2976	. . 3 |- ( A e. { x e. NN | ph } <-> ( A e. NN /\ ta ) )
23	20, 22	sylib 122	. 2 |- ( A e. NN -> ( A e. NN /\ ta ) )
24	23	simprd 114	1 |- ( A e. NN -> ta )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   C_ wss 3214  (class class class)co 6058   1c1 8144   + caddc 8146   NNcn 9254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-sep 4233  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-inn 9255

This theorem is referenced by:  nnindALT  9271  nn1m1nn  9272  nnaddcl  9274  nnmulcl  9275  nnge1  9277  nn1gt1  9288  nnsub  9293  zaddcllempos  9631  zaddcllemneg  9633  nneoor  9698  peano5uzti  9704  nn0ind-raph  9713  indstr  9943  exbtwnzlemshrink  10632  exp3vallem  10926  expcllem  10936  expap0  10955  apexp1  11105  seq3coll  11239  resqrexlemover  11720  resqrexlemlo  11723  resqrexlemcalc3  11726  gcdmultiple  12741  rplpwr  12748  prmind2  12842  prmdvdsexp  12870  sqrt2irr  12884  pw2dvdslemn  12887  pcmpt  13066  prmpwdvds  13078  mulgnnass  13910  dvexp  15702  plycolemc  15749  2sqlem10  16124