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Theorem dmaddpi 8798
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 5196 . . 3  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  +o  )
2 fnoa 6781 . . . . 5  |-  +o  Fn  ( On  X.  On )
3 fndm 5573 . . . . 5  |-  (  +o  Fn  ( On  X.  On )  ->  dom  +o  =  ( On  X.  On ) )
42, 3ax-mp 5 . . . 4  |-  dom  +o  =  ( On  X.  On )
54ineq2i 3525 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  +o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2462 . 2  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-pli 8781 . . 3  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
87dmeqi 5100 . 2  |-  dom  +N  =  dom  (  +o  |`  ( N.  X.  N. ) )
9 df-ni 8780 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3460 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3364 . . . . . 6  |-  N.  C_  om
12 omsson 4878 . . . . . 6  |-  om  C_  On
1311, 12sstri 3343 . . . . 5  |-  N.  C_  On
14 anidm 627 . . . . 5  |-  ( ( N.  C_  On  /\  N.  C_  On )  <->  N.  C_  On )
1513, 14mpbir 202 . . . 4  |-  ( N.  C_  On  /\  N.  C_  On )
16 xpss12 5010 . . . 4  |-  ( ( N.  C_  On  /\  N.  C_  On )  ->  ( N.  X.  N. )  C_  ( On  X.  On ) )
1715, 16ax-mp 5 . . 3  |-  ( N. 
X.  N. )  C_  ( On  X.  On )
18 dfss 3321 . . 3  |-  ( ( N.  X.  N. )  C_  ( On  X.  On ) 
<->  ( N.  X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) ) )
1917, 18mpbi 201 . 2  |-  ( N. 
X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
206, 8, 193eqtr4i 2472 1  |-  dom  +N  =  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    \ cdif 3303    i^i cin 3305    C_ wss 3306   (/)c0 3613   {csn 3838   Oncon0 4610   omcom 4874    X. cxp 4905   dom cdm 4907    |` cres 4909    Fn wfn 5478    +o coa 6750   N.cnpi 8750    +N cpli 8751
This theorem is referenced by:  addcompi  8802  addasspi  8803  distrpi  8806  addcanpi  8807  addnidpi  8809  ltapi  8811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-oadd 6757  df-ni 8780  df-pli 8781
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