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Theorem nnaddcl 10022
Description: Closure of addition of natural numbers, 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 6089 . . . . 5  |-  ( x  =  1  ->  ( A  +  x )  =  ( A  + 
1 ) )
21eleq1d 2502 . . . 4  |-  ( x  =  1  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  1 )  e.  NN ) )
32imbi2d 308 . . 3  |-  ( x  =  1  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  1 )  e.  NN ) ) )
4 oveq2 6089 . . . . 5  |-  ( x  =  y  ->  ( A  +  x )  =  ( A  +  y ) )
54eleq1d 2502 . . . 4  |-  ( x  =  y  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  y )  e.  NN ) )
65imbi2d 308 . . 3  |-  ( x  =  y  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  y )  e.  NN ) ) )
7 oveq2 6089 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A  +  x )  =  ( A  +  ( y  +  1 ) ) )
87eleq1d 2502 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  ( y  +  1 ) )  e.  NN ) )
98imbi2d 308 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) ) )
10 oveq2 6089 . . . . 5  |-  ( x  =  B  ->  ( A  +  x )  =  ( A  +  B ) )
1110eleq1d 2502 . . . 4  |-  ( x  =  B  ->  (
( A  +  x
)  e.  NN  <->  ( A  +  B )  e.  NN ) )
1211imbi2d 308 . . 3  |-  ( x  =  B  ->  (
( A  e.  NN  ->  ( A  +  x
)  e.  NN )  <-> 
( A  e.  NN  ->  ( A  +  B
)  e.  NN ) ) )
13 peano2nn 10012 . . 3  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
14 peano2nn 10012 . . . . . 6  |-  ( ( A  +  y )  e.  NN  ->  (
( A  +  y )  +  1 )  e.  NN )
15 nncn 10008 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
16 nncn 10008 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
17 ax-1cn 9048 . . . . . . . . 9  |-  1  e.  CC
18 addass 9077 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  CC  /\  1  e.  CC )  ->  (
( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
1917, 18mp3an3 1268 . . . . . . . 8  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
2015, 16, 19syl2an 464 . . . . . . 7  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( A  +  y )  +  1 )  =  ( A  +  ( y  +  1 ) ) )
2120eleq1d 2502 . . . . . 6  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( ( A  +  y )  +  1 )  e.  NN  <->  ( A  +  ( y  +  1 ) )  e.  NN ) )
2214, 21syl5ib 211 . . . . 5  |-  ( ( A  e.  NN  /\  y  e.  NN )  ->  ( ( A  +  y )  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) )
2322expcom 425 . . . 4  |-  ( y  e.  NN  ->  ( A  e.  NN  ->  ( ( A  +  y )  e.  NN  ->  ( A  +  ( y  +  1 ) )  e.  NN ) ) )
2423a2d 24 . . 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 10018 . 2  |-  ( B  e.  NN  ->  ( A  e.  NN  ->  ( A  +  B )  e.  NN ) )
2625impcom 420 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  +  B
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993   NNcn 10000
This theorem is referenced by:  nnmulcl  10023  nnaddcld  10046  nnnn0addcl  10251  nn0addcl  10255  zaddcl  10317  xpnnenOLD  12809  pythagtriplem4  13193  vdwapun  13342  vdwap1  13345  vdwlem2  13350  mulgnndir  14912  uniioombllem3  19477  ballotlem1  24744  ballotlem2  24746  ballotlemfmpn  24752  ballotlem4  24756  ballotlemimin  24763  ballotlemsdom  24769  ballotlemsel1i  24770  ballotlemfrceq  24786  ballotlemfrcn0  24787  ballotlem1ri  24792  ballotth  24795  nndivsub  26207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-addass 9055  ax-i2m1 9058  ax-1ne0 9059  ax-rrecex 9062  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-recs 6633  df-rdg 6668  df-nn 10001
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