Theorem nnind 8935

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 8939 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 8930	. . . . . 6 |- 1 e. NN
2		nnind.5	. . . . . 6 |- ps
3		nnind.1	. . . . . . 7 |- ( x = 1 -> ( ph <-> ps ) )
4	3	elrab 2894	. . . . . 6 |- ( 1 e. { x e. NN | ph } <-> ( 1 e. NN /\ ps ) )
5	1, 2, 4	mpbir2an 942	. . . . 5 |- 1 e. { x e. NN | ph }
6		elrabi 2891	. . . . . . 7 |- ( y e. { x e. NN | ph } -> y e. NN )
7		peano2nn 8931	. . . . . . . . . 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 2894	. . . . . . . 8 |- ( y e. { x e. NN | ph } <-> ( y e. NN /\ ch ) )
13		nnind.3	. . . . . . . . 9 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
14	13	elrab 2894	. . . . . . . 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 2530	. . . . 5 |- A. y e. { x e. NN | ph } ( y + 1 ) e. { x e. NN | ph }
18		peano5nni 8922	. . . . 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 3152	. . 3 |- ( A e. NN -> A e. { x e. NN | ph } )
21		nnind.4	. . . 4 |- ( x = A -> ( ph <-> ta ) )
22	21	elrab 2894	. . 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 1353   e. wcel 2148   A.wral 2455   {crab 2459   C_ wss 3130  (class class class)co 5875   1c1 7812   + caddc 7814   NNcn 8919

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4122  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878  df-inn 8920

This theorem is referenced by:  nnindALT  8936  nn1m1nn  8937  nnaddcl  8939  nnmulcl  8940  nnge1  8942  nn1gt1  8953  nnsub  8958  zaddcllempos  9290  zaddcllemneg  9292  nneoor  9355  peano5uzti  9361  nn0ind-raph  9370  indstr  9593  exbtwnzlemshrink  10249  exp3vallem  10521  expcllem  10531  expap0  10550  apexp1  10698  seq3coll  10822  resqrexlemover  11019  resqrexlemlo  11022  resqrexlemcalc3  11025  gcdmultiple  12021  rplpwr  12028  prmind2  12120  prmdvdsexp  12148  sqrt2irr  12162  pw2dvdslemn  12165  pcmpt  12341  prmpwdvds  12353  mulgnnass  13018  dvexp  14178  2sqlem10  14475