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Theorem nnaddcl 8414
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.
StepHypRef Expression
1 oveq2 5642 . . . . 5  |-  ( x  =  1  ->  ( A  +  x )  =  ( A  + 
1 ) )
21eleq1d 2156 . . . 4  |-  ( x  =  1  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  1 )  e.  NN ) )
32imbi2d 228 . . 3  |-  ( x  =  1  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  1 )  e.  NN ) ) )
4 oveq2 5642 . . . . 5  |-  ( x  =  y  ->  ( A  +  x )  =  ( A  +  y ) )
54eleq1d 2156 . . . 4  |-  ( x  =  y  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  y )  e.  NN ) )
65imbi2d 228 . . 3  |-  ( x  =  y  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  y )  e.  NN ) ) )
7 oveq2 5642 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A  +  x )  =  ( A  +  ( y  +  1 ) ) )
87eleq1d 2156 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  ( y  +  1 ) )  e.  NN ) )
98imbi2d 228 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) ) )
10 oveq2 5642 . . . . 5  |-  ( x  =  B  ->  ( A  +  x )  =  ( A  +  B ) )
1110eleq1d 2156 . . . 4  |-  ( x  =  B  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  B )  e.  NN ) )
1211imbi2d 228 . . 3  |-  ( x  =  B  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  B
)  e.  NN ) ) )
13 peano2nn 8406 . . 3  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
14 peano2nn 8406 . . . . . 6  |-  ( ( A  +  y )  e.  NN  ->  (
( A  +  y )  +  1 )  e.  NN )
15 nncn 8402 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
16 nncn 8402 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
17 ax-1cn 7417 . . . . . . . . 9  |-  1  e.  CC
18 addass 7451 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  CC  /\  1  e.  CC )  ->  (
( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
1917, 18mp3an3 1262 . . . . . . . 8  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
2015, 16, 19syl2an 283 . . . . . . 7  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
2120eleq1d 2156 . . . . . 6  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( ( A  +  y )  +  1 )  e.  NN  <->  ( A  +  ( y  +  1 ) )  e.  NN ) )
2214, 21syl5ib 152 . . . . 5  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( A  +  y )  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) )
2322expcom 114 . . . 4  |-  ( y  e.  NN  ->  ( A  e.  NN  ->  ( ( A  +  y )  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) ) )
2423a2d 26 . . 3  |-  ( y  e.  NN  ->  (
( A  e.  NN  ->  ( A  +  y )  e.  NN )  ->  ( A  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) ) )
253, 6, 9, 12, 13, 24nnind 8410 . 2  |-  ( B  e.  NN  ->  ( A  e.  NN  ->  ( A  +  B )  e.  NN ) )
2625impcom 123 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  +  B
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438  (class class class)co 5634   CCcc 7327   1c1 7330    + caddc 7332   NNcn 8394
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-addrcl 7421  ax-addass 7426
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637  df-inn 8395
This theorem is referenced by:  nnmulcl  8415  nn2ge  8426  nnaddcld  8441  nnnn0addcl  8673  nn0addcl  8678  9p1e10  8848
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