Theorem nnaddcl 8059

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 5540	. . . . 5 |- ( x = 1 -> ( A + x ) = ( A + 1 ) )
2	1	eleq1d 2147	. . . 4 |- ( x = 1 -> ( ( A + x ) e. NN <-> ( A + 1 ) e. NN ) )
3	2	imbi2d 228	. . 3 |- ( x = 1 -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + 1 ) e. NN ) ) )
4		oveq2 5540	. . . . 5 |- ( x = y -> ( A + x ) = ( A + y ) )
5	4	eleq1d 2147	. . . 4 |- ( x = y -> ( ( A + x ) e. NN <-> ( A + y ) e. NN ) )
6	5	imbi2d 228	. . 3 |- ( x = y -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + y ) e. NN ) ) )
7		oveq2 5540	. . . . 5 |- ( x = ( y + 1 ) -> ( A + x ) = ( A + ( y + 1 ) ) )
8	7	eleq1d 2147	. . . 4 |- ( x = ( y + 1 ) -> ( ( A + x ) e. NN <-> ( A + ( y + 1 ) ) e. NN ) )
9	8	imbi2d 228	. . 3 |- ( x = ( y + 1 ) -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + ( y + 1 ) ) e. NN ) ) )
10		oveq2 5540	. . . . 5 |- ( x = B -> ( A + x ) = ( A + B ) )
11	10	eleq1d 2147	. . . 4 |- ( x = B -> ( ( A + x ) e. NN <-> ( A + B ) e. NN ) )
12	11	imbi2d 228	. . 3 |- ( x = B -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + B ) e. NN ) ) )
13		peano2nn 8051	. . 3 |- ( A e. NN -> ( A + 1 ) e. NN )
14		peano2nn 8051	. . . . . 6 |- ( ( A + y ) e. NN -> ( ( A + y ) + 1 ) e. NN )
15		nncn 8047	. . . . . . . 8 |- ( A e. NN -> A e. CC )
16		nncn 8047	. . . . . . . 8 |- ( y e. NN -> y e. CC )
17		ax-1cn 7069	. . . . . . . . 9 |- 1 e. CC
18		addass 7103	. . . . . . . . 9 |- ( ( A e. CC /\ y e. CC /\ 1 e. CC ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
19	17, 18	mp3an3 1257	. . . . . . . 8 |- ( ( A e. CC /\ y e. CC ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
20	15, 16, 19	syl2an 283	. . . . . . 7 |- ( ( A e. NN /\ y e. NN ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
21	20	eleq1d 2147	. . . . . 6 |- ( ( A e. NN /\ y e. NN ) -> ( ( ( A + y ) + 1 ) e. NN <-> ( A + ( y + 1 ) ) e. NN ) )
22	14, 21	syl5ib 152	. . . . 5 |- ( ( A e. NN /\ y e. NN ) -> ( ( A + y ) e. NN -> ( A + ( y + 1 ) ) e. NN ) )
23	22	expcom 114	. . . 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 8055	. 2 |- ( B e. NN -> ( A e. NN -> ( A + B ) e. NN ) )
26	25	impcom 123	1 |- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   1c1 6982   + caddc 6984   NNcn 8039

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-addrcl 7073  ax-addass 7078

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535  df-inn 8040

This theorem is referenced by:  nnmulcl  8060  nn2ge  8071  nnaddcld  8086  nnnn0addcl  8318  nn0addcl  8323  9p1e10  8479