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Theorem nnaddcl 8697
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 5748 . . . . 5  |-  ( x  =  1  ->  ( A  +  x )  =  ( A  + 
1 ) )
21eleq1d 2184 . . . 4  |-  ( x  =  1  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  1 )  e.  NN ) )
32imbi2d 229 . . 3  |-  ( x  =  1  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  1 )  e.  NN ) ) )
4 oveq2 5748 . . . . 5  |-  ( x  =  y  ->  ( A  +  x )  =  ( A  +  y ) )
54eleq1d 2184 . . . 4  |-  ( x  =  y  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  y )  e.  NN ) )
65imbi2d 229 . . 3  |-  ( x  =  y  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  y )  e.  NN ) ) )
7 oveq2 5748 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A  +  x )  =  ( A  +  ( y  +  1 ) ) )
87eleq1d 2184 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  ( y  +  1 ) )  e.  NN ) )
98imbi2d 229 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) ) )
10 oveq2 5748 . . . . 5  |-  ( x  =  B  ->  ( A  +  x )  =  ( A  +  B ) )
1110eleq1d 2184 . . . 4  |-  ( x  =  B  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  B )  e.  NN ) )
1211imbi2d 229 . . 3  |-  ( x  =  B  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  B
)  e.  NN ) ) )
13 peano2nn 8689 . . 3  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
14 peano2nn 8689 . . . . . 6  |-  ( ( A  +  y )  e.  NN  ->  (
( A  +  y )  +  1 )  e.  NN )
15 nncn 8685 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
16 nncn 8685 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
17 ax-1cn 7677 . . . . . . . . 9  |-  1  e.  CC
18 addass 7714 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  CC  /\  1  e.  CC )  ->  (
( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
1917, 18mp3an3 1287 . . . . . . . 8  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
2015, 16, 19syl2an 285 . . . . . . 7  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
2120eleq1d 2184 . . . . . 6  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( ( A  +  y )  +  1 )  e.  NN  <->  ( A  +  ( y  +  1 ) )  e.  NN ) )
2214, 21syl5ib 153 . . . . 5  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( A  +  y )  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) )
2322expcom 115 . . . 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 8693 . 2  |-  ( B  e.  NN  ->  ( A  e.  NN  ->  ( A  +  B )  e.  NN ) )
2625impcom 124 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  +  B
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463  (class class class)co 5740   CCcc 7582   1c1 7585    + caddc 7587   NNcn 8677
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-addrcl 7681  ax-addass 7686
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743  df-inn 8678
This theorem is referenced by:  nnmulcl  8698  nn2ge  8710  nnaddcld  8725  nnnn0addcl  8958  nn0addcl  8963  9p1e10  9135
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