Theorem nnind 9000

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 9004 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 8995	. . . . . 6 |- 1 e. NN
2		nnind.5	. . . . . 6 |- ps
3		nnind.1	. . . . . . 7 |- ( x = 1 -> ( ph <-> ps ) )
4	3	elrab 2917	. . . . . 6 |- ( 1 e. { x e. NN | ph } <-> ( 1 e. NN /\ ps ) )
5	1, 2, 4	mpbir2an 944	. . . . 5 |- 1 e. { x e. NN | ph }
6		elrabi 2914	. . . . . . 7 |- ( y e. { x e. NN | ph } -> y e. NN )
7		peano2nn 8996	. . . . . . . . . 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 2917	. . . . . . . 8 |- ( y e. { x e. NN | ph } <-> ( y e. NN /\ ch ) )
13		nnind.3	. . . . . . . . 9 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
14	13	elrab 2917	. . . . . . . 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 2547	. . . . 5 |- A. y e. { x e. NN | ph } ( y + 1 ) e. { x e. NN | ph }
18		peano5nni 8987	. . . . 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 3176	. . 3 |- ( A e. NN -> A e. { x e. NN | ph } )
21		nnind.4	. . . 4 |- ( x = A -> ( ph <-> ta ) )
22	21	elrab 2917	. . 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 1364   e. wcel 2164   A.wral 2472   {crab 2476   C_ wss 3154  (class class class)co 5919   1c1 7875   + caddc 7877   NNcn 8984

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4148  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922  df-inn 8985

This theorem is referenced by:  nnindALT  9001  nn1m1nn  9002  nnaddcl  9004  nnmulcl  9005  nnge1  9007  nn1gt1  9018  nnsub  9023  zaddcllempos  9357  zaddcllemneg  9359  nneoor  9422  peano5uzti  9428  nn0ind-raph  9437  indstr  9661  exbtwnzlemshrink  10320  exp3vallem  10614  expcllem  10624  expap0  10643  apexp1  10792  seq3coll  10916  resqrexlemover  11157  resqrexlemlo  11160  resqrexlemcalc3  11163  gcdmultiple  12160  rplpwr  12167  prmind2  12261  prmdvdsexp  12289  sqrt2irr  12303  pw2dvdslemn  12306  pcmpt  12484  prmpwdvds  12496  mulgnnass  13230  dvexp  14890  plycolemc  14936  2sqlem10  15282