Theorem nnind 9054

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 9058 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 9049	. . . . . 6 |- 1 e. NN
2		nnind.5	. . . . . 6 |- ps
3		nnind.1	. . . . . . 7 |- ( x = 1 -> ( ph <-> ps ) )
4	3	elrab 2929	. . . . . 6 |- ( 1 e. { x e. NN | ph } <-> ( 1 e. NN /\ ps ) )
5	1, 2, 4	mpbir2an 945	. . . . 5 |- 1 e. { x e. NN | ph }
6		elrabi 2926	. . . . . . 7 |- ( y e. { x e. NN | ph } -> y e. NN )
7		peano2nn 9050	. . . . . . . . . 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 2929	. . . . . . . 8 |- ( y e. { x e. NN | ph } <-> ( y e. NN /\ ch ) )
13		nnind.3	. . . . . . . . 9 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
14	13	elrab 2929	. . . . . . . 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 2559	. . . . 5 |- A. y e. { x e. NN | ph } ( y + 1 ) e. { x e. NN | ph }
18		peano5nni 9041	. . . . 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 3189	. . 3 |- ( A e. NN -> A e. { x e. NN | ph } )
21		nnind.4	. . . 4 |- ( x = A -> ( ph <-> ta ) )
22	21	elrab 2929	. . 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 1373   e. wcel 2176   A.wral 2484   {crab 2488   C_ wss 3166  (class class class)co 5946   1c1 7928   + caddc 7930   NNcn 9038

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-sep 4163  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949  df-inn 9039

This theorem is referenced by:  nnindALT  9055  nn1m1nn  9056  nnaddcl  9058  nnmulcl  9059  nnge1  9061  nn1gt1  9072  nnsub  9077  zaddcllempos  9411  zaddcllemneg  9413  nneoor  9477  peano5uzti  9483  nn0ind-raph  9492  indstr  9716  exbtwnzlemshrink  10393  exp3vallem  10687  expcllem  10697  expap0  10716  apexp1  10865  seq3coll  10989  resqrexlemover  11354  resqrexlemlo  11357  resqrexlemcalc3  11360  gcdmultiple  12374  rplpwr  12381  prmind2  12475  prmdvdsexp  12503  sqrt2irr  12517  pw2dvdslemn  12520  pcmpt  12699  prmpwdvds  12711  mulgnnass  13526  dvexp  15216  plycolemc  15263  2sqlem10  15635