Theorem nnaddcl 8747

Description: Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)

Assertion

Ref	Expression
nnaddcl	|- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )

Proof of Theorem nnaddcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5782	. . . . 5 |- ( x = 1 -> ( A + x ) = ( A + 1 ) )
2	1	eleq1d 2208	. . . 4 |- ( x = 1 -> ( ( A + x ) e. NN <-> ( A + 1 ) e. NN ) )
3	2	imbi2d 229	. . 3 |- ( x = 1 -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + 1 ) e. NN ) ) )
4		oveq2 5782	. . . . 5 |- ( x = y -> ( A + x ) = ( A + y ) )
5	4	eleq1d 2208	. . . 4 |- ( x = y -> ( ( A + x ) e. NN <-> ( A + y ) e. NN ) )
6	5	imbi2d 229	. . 3 |- ( x = y -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + y ) e. NN ) ) )
7		oveq2 5782	. . . . 5 |- ( x = ( y + 1 ) -> ( A + x ) = ( A + ( y + 1 ) ) )
8	7	eleq1d 2208	. . . 4 |- ( x = ( y + 1 ) -> ( ( A + x ) e. NN <-> ( A + ( y + 1 ) ) e. NN ) )
9	8	imbi2d 229	. . 3 |- ( x = ( y + 1 ) -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + ( y + 1 ) ) e. NN ) ) )
10		oveq2 5782	. . . . 5 |- ( x = B -> ( A + x ) = ( A + B ) )
11	10	eleq1d 2208	. . . 4 |- ( x = B -> ( ( A + x ) e. NN <-> ( A + B ) e. NN ) )
12	11	imbi2d 229	. . 3 |- ( x = B -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + B ) e. NN ) ) )
13		peano2nn 8739	. . 3 |- ( A e. NN -> ( A + 1 ) e. NN )
14		peano2nn 8739	. . . . . 6 |- ( ( A + y ) e. NN -> ( ( A + y ) + 1 ) e. NN )
15		nncn 8735	. . . . . . . 8 |- ( A e. NN -> A e. CC )
16		nncn 8735	. . . . . . . 8 |- ( y e. NN -> y e. CC )
17		ax-1cn 7720	. . . . . . . . 9 |- 1 e. CC
18		addass 7757	. . . . . . . . 9 |- ( ( A e. CC /\ y e. CC /\ 1 e. CC ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
19	17, 18	mp3an3 1304	. . . . . . . 8 |- ( ( A e. CC /\ y e. CC ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
20	15, 16, 19	syl2an 287	. . . . . . 7 |- ( ( A e. NN /\ y e. NN ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
21	20	eleq1d 2208	. . . . . 6 |- ( ( A e. NN /\ y e. NN ) -> ( ( ( A + y ) + 1 ) e. NN <-> ( A + ( y + 1 ) ) e. NN ) )
22	14, 21	syl5ib 153	. . . . 5 |- ( ( A e. NN /\ y e. NN ) -> ( ( A + y ) e. NN -> ( A + ( y + 1 ) ) e. NN ) )
23	22	expcom 115	. . . 4 |- ( y e. NN -> ( A e. NN -> ( ( A + y ) e. NN -> ( A + ( y + 1 ) ) e. NN ) ) )
24	23	a2d 26	. . 3 |- ( y e. NN -> ( ( A e. NN -> ( A + y ) e. NN ) -> ( A e. NN -> ( A + ( y + 1 ) ) e. NN ) ) )
25	3, 6, 9, 12, 13, 24	nnind 8743	. 2 |- ( B e. NN -> ( A e. NN -> ( A + B ) e. NN ) )
26	25	impcom 124	1 |- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7625   1c1 7628   + caddc 7630   NNcn 8727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-addrcl 7724  ax-addass 7729

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777  df-inn 8728

This theorem is referenced by:  nnmulcl  8748  nn2ge  8760  nnaddcld  8775  nnnn0addcl  9014  nn0addcl  9019  9p1e10  9191