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Theorem dmaddpi 8731
Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
dmaddpi  |-  dom  +N  =  ( N.  X.  N. )

Proof of Theorem dmaddpi
StepHypRef Expression
1 dmres 5134 . . 3  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  +o  )
2 fnoa 6719 . . . . 5  |-  +o  Fn  ( On  X.  On )
3 fndm 5511 . . . . 5  |-  (  +o  Fn  ( On  X.  On )  ->  dom  +o  =  ( On  X.  On ) )
42, 3ax-mp 8 . . . 4  |-  dom  +o  =  ( On  X.  On )
54ineq2i 3507 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  +o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2432 . 2  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-pli 8714 . . 3  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
87dmeqi 5038 . 2  |-  dom  +N  =  dom  (  +o  |`  ( N.  X.  N. ) )
9 df-ni 8713 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3442 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3346 . . . . . 6  |-  N.  C_  om
12 omsson 4816 . . . . . 6  |-  om  C_  On
1311, 12sstri 3325 . . . . 5  |-  N.  C_  On
14 anidm 626 . . . . 5  |-  ( ( N.  C_  On  /\  N.  C_  On )  <->  N.  C_  On )
1513, 14mpbir 201 . . . 4  |-  ( N.  C_  On  /\  N.  C_  On )
16 xpss12 4948 . . . 4  |-  ( ( N.  C_  On  /\  N.  C_  On )  ->  ( N.  X.  N. )  C_  ( On  X.  On ) )
1715, 16ax-mp 8 . . 3  |-  ( N. 
X.  N. )  C_  ( On  X.  On )
18 dfss 3303 . . 3  |-  ( ( N.  X.  N. )  C_  ( On  X.  On ) 
<->  ( N.  X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) ) )
1917, 18mpbi 200 . 2  |-  ( N. 
X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
206, 8, 193eqtr4i 2442 1  |-  dom  +N  =  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    \ cdif 3285    i^i cin 3287    C_ wss 3288   (/)c0 3596   {csn 3782   Oncon0 4549   omcom 4812    X. cxp 4843   dom cdm 4845    |` cres 4847    Fn wfn 5416    +o coa 6688   N.cnpi 8683    +N cpli 8684
This theorem is referenced by:  addcompi  8735  addasspi  8736  distrpi  8739  addcanpi  8740  addnidpi  8742  ltapi  8744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-oadd 6695  df-ni 8713  df-pli 8714
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