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Theorem dmaddpi 8514
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 4976 . . 3  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  +o  )
2 fnoa 6507 . . . . 5  |-  +o  Fn  ( On  X.  On )
3 fndm 5343 . . . . 5  |-  (  +o  Fn  ( On  X.  On )  ->  dom  +o  =  ( On  X.  On ) )
42, 3ax-mp 8 . . . 4  |-  dom  +o  =  ( On  X.  On )
54ineq2i 3367 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  +o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2303 . 2  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-pli 8497 . . 3  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
87dmeqi 4880 . 2  |-  dom  +N  =  dom  (  +o  |`  ( N.  X.  N. ) )
9 df-ni 8496 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3303 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3208 . . . . . 6  |-  N.  C_  om
12 omsson 4660 . . . . . 6  |-  om  C_  On
1311, 12sstri 3188 . . . . 5  |-  N.  C_  On
14 anidm 625 . . . . 5  |-  ( ( N.  C_  On  /\  N.  C_  On )  <->  N.  C_  On )
1513, 14mpbir 200 . . . 4  |-  ( N.  C_  On  /\  N.  C_  On )
16 xpss12 4792 . . . 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 3167 . . 3  |-  ( ( N.  X.  N. )  C_  ( On  X.  On ) 
<->  ( N.  X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) ) )
1917, 18mpbi 199 . 2  |-  ( N. 
X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
206, 8, 193eqtr4i 2313 1  |-  dom  +N  =  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   Oncon0 4392   omcom 4656    X. cxp 4687   dom cdm 4689    |` cres 4691    Fn wfn 5250    +o coa 6476   N.cnpi 8466    +N cpli 8467
This theorem is referenced by:  addcompi  8518  addasspi  8519  distrpi  8522  addcanpi  8523  addnidpi  8525  ltapi  8527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-oadd 6483  df-ni 8496  df-pli 8497
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