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Theorem nnaddcl 7932
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 5520 . . . . 5  |-  ( x  =  1  ->  ( A  +  x )  =  ( A  + 
1 ) )
21eleq1d 2106 . . . 4  |-  ( x  =  1  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  1 )  e.  NN ) )
32imbi2d 219 . . 3  |-  ( x  =  1  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  1 )  e.  NN ) ) )
4 oveq2 5520 . . . . 5  |-  ( x  =  y  ->  ( A  +  x )  =  ( A  +  y ) )
54eleq1d 2106 . . . 4  |-  ( x  =  y  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  y )  e.  NN ) )
65imbi2d 219 . . 3  |-  ( x  =  y  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  y )  e.  NN ) ) )
7 oveq2 5520 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A  +  x )  =  ( A  +  ( y  +  1 ) ) )
87eleq1d 2106 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  ( y  +  1 ) )  e.  NN ) )
98imbi2d 219 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) ) )
10 oveq2 5520 . . . . 5  |-  ( x  =  B  ->  ( A  +  x )  =  ( A  +  B ) )
1110eleq1d 2106 . . . 4  |-  ( x  =  B  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  B )  e.  NN ) )
1211imbi2d 219 . . 3  |-  ( x  =  B  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  B
)  e.  NN ) ) )
13 peano2nn 7924 . . 3  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
14 peano2nn 7924 . . . . . 6  |-  ( ( A  +  y )  e.  NN  ->  (
( A  +  y )  +  1 )  e.  NN )
15 nncn 7920 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
16 nncn 7920 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
17 ax-1cn 6975 . . . . . . . . 9  |-  1  e.  CC
18 addass 7009 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  CC  /\  1  e.  CC )  ->  (
( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
1917, 18mp3an3 1221 . . . . . . . 8  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
2015, 16, 19syl2an 273 . . . . . . 7  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
2120eleq1d 2106 . . . . . 6  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( ( A  +  y )  +  1 )  e.  NN  <->  ( A  +  ( y  +  1 ) )  e.  NN ) )
2214, 21syl5ib 143 . . . . 5  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( A  +  y )  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) )
2322expcom 109 . . . 4  |-  ( y  e.  NN  ->  ( A  e.  NN  ->  ( ( A  +  y )  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) ) )
2423a2d 23 . . 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 7928 . 2  |-  ( B  e.  NN  ->  ( A  e.  NN  ->  ( A  +  B )  e.  NN ) )
2625impcom 116 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  +  B
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393  (class class class)co 5512   CCcc 6885   1c1 6888    + caddc 6890   NNcn 7912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6973  ax-resscn 6974  ax-1cn 6975  ax-1re 6976  ax-addrcl 6979  ax-addass 6984
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-inn 7913
This theorem is referenced by:  nnmulcl  7933  nn2ge  7944  nnaddcld  7959  nnnn0addcl  8210  nn0addcl  8215
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