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Theorem dmaddpi 9285
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 5304 . . 3  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  +o  )
2 fnoa 7176 . . . . 5  |-  +o  Fn  ( On  X.  On )
3 fndm 5686 . . . . 5  |-  (  +o  Fn  ( On  X.  On )  ->  dom  +o  =  ( On  X.  On ) )
42, 3ax-mp 5 . . . 4  |-  dom  +o  =  ( On  X.  On )
54ineq2i 3693 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  +o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2486 . 2  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-pli 9268 . . 3  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
87dmeqi 5214 . 2  |-  dom  +N  =  dom  (  +o  |`  ( N.  X.  N. ) )
9 df-ni 9267 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3627 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3529 . . . . . 6  |-  N.  C_  om
12 omsson 6703 . . . . . 6  |-  om  C_  On
1311, 12sstri 3508 . . . . 5  |-  N.  C_  On
14 anidm 644 . . . . 5  |-  ( ( N.  C_  On  /\  N.  C_  On )  <->  N.  C_  On )
1513, 14mpbir 209 . . . 4  |-  ( N.  C_  On  /\  N.  C_  On )
16 xpss12 5117 . . . 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 3486 . . 3  |-  ( ( N.  X.  N. )  C_  ( On  X.  On ) 
<->  ( N.  X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) ) )
1917, 18mpbi 208 . 2  |-  ( N. 
X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
206, 8, 193eqtr4i 2496 1  |-  dom  +N  =  ( N.  X.  N. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   Oncon0 4887    X. cxp 5006   dom cdm 5008    |` cres 5010    Fn wfn 5589   omcom 6699    +o coa 7145   N.cnpi 9239    +N cpli 9240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-oadd 7152  df-ni 9267  df-pli 9268
This theorem is referenced by:  addcompi  9289  addasspi  9290  distrpi  9293  addcanpi  9294  addnidpi  9296  ltapi  9298
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