Theorem nnaddcl 9303

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 6083	. . . . 5 |- ( x = 1 -> ( A + x ) = ( A + 1 ) )
2	1	eleq1d 2307	. . . 4 |- ( x = 1 -> ( ( A + x ) e. NN <-> ( A + 1 ) e. NN ) )
3	2	imbi2d 230	. . 3 |- ( x = 1 -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + 1 ) e. NN ) ) )
4		oveq2 6083	. . . . 5 |- ( x = y -> ( A + x ) = ( A + y ) )
5	4	eleq1d 2307	. . . 4 |- ( x = y -> ( ( A + x ) e. NN <-> ( A + y ) e. NN ) )
6	5	imbi2d 230	. . 3 |- ( x = y -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + y ) e. NN ) ) )
7		oveq2 6083	. . . . 5 |- ( x = ( y + 1 ) -> ( A + x ) = ( A + ( y + 1 ) ) )
8	7	eleq1d 2307	. . . 4 |- ( x = ( y + 1 ) -> ( ( A + x ) e. NN <-> ( A + ( y + 1 ) ) e. NN ) )
9	8	imbi2d 230	. . 3 |- ( x = ( y + 1 ) -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + ( y + 1 ) ) e. NN ) ) )
10		oveq2 6083	. . . . 5 |- ( x = B -> ( A + x ) = ( A + B ) )
11	10	eleq1d 2307	. . . 4 |- ( x = B -> ( ( A + x ) e. NN <-> ( A + B ) e. NN ) )
12	11	imbi2d 230	. . 3 |- ( x = B -> ( ( A e. NN -> ( A + x ) e. NN ) <-> ( A e. NN -> ( A + B ) e. NN ) ) )
13		peano2nn 9295	. . 3 |- ( A e. NN -> ( A + 1 ) e. NN )
14		peano2nn 9295	. . . . . 6 |- ( ( A + y ) e. NN -> ( ( A + y ) + 1 ) e. NN )
15		nncn 9291	. . . . . . . 8 |- ( A e. NN -> A e. CC )
16		nncn 9291	. . . . . . . 8 |- ( y e. NN -> y e. CC )
17		ax-1cn 8262	. . . . . . . . 9 |- 1 e. CC
18		addass 8299	. . . . . . . . 9 |- ( ( A e. CC /\ y e. CC /\ 1 e. CC ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
19	17, 18	mp3an3 1367	. . . . . . . 8 |- ( ( A e. CC /\ y e. CC ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
20	15, 16, 19	syl2an 289	. . . . . . 7 |- ( ( A e. NN /\ y e. NN ) -> ( ( A + y ) + 1 ) = ( A + ( y + 1 ) ) )
21	20	eleq1d 2307	. . . . . 6 |- ( ( A e. NN /\ y e. NN ) -> ( ( ( A + y ) + 1 ) e. NN <-> ( A + ( y + 1 ) ) e. NN ) )
22	14, 21	imbitrid 154	. . . . 5 |- ( ( A e. NN /\ y e. NN ) -> ( ( A + y ) e. NN -> ( A + ( y + 1 ) ) e. NN ) )
23	22	expcom 116	. . . 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 9299	. 2 |- ( B e. NN -> ( A e. NN -> ( A + B ) e. NN ) )
26	25	impcom 125	1 |- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   1c1 8170   + caddc 8172   NNcn 9283

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-addrcl 8266  ax-addass 8271

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078  df-inn 9284

This theorem is referenced by:  nnmulcl  9304  nn2ge  9316  nnaddcld  9331  nnnn0addcl  9572  nn0addcl  9577  9p1e10  9758  pythagtriplem4  13025  ballotfilemofi  13197  ballotfilem1  13198  ballotfilemonn  13199  ballotfilem2  13206  ballotfilemfmpn  13212  ballotfilemefi  13215  ballotfilem4  13219  ballotfilemiex  13222  ballotfilemimin  13227  ballotfilemsval  13230  ballotfilemsdom  13233  ballotfilemsel1i  13234  ballotfilemrval  13239  ballotfilemfrceq  13250  ballotfilemfrcn0  13251  ballotfilem1ri  13256  ballotfilemth  13259  mulgnndir  13931