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Theorem dmaddpi 9051
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 5126 . . 3  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  +o  )
2 fnoa 6940 . . . . 5  |-  +o  Fn  ( On  X.  On )
3 fndm 5505 . . . . 5  |-  (  +o  Fn  ( On  X.  On )  ->  dom  +o  =  ( On  X.  On ) )
42, 3ax-mp 5 . . . 4  |-  dom  +o  =  ( On  X.  On )
54ineq2i 3544 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  +o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2458 . 2  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-pli 9034 . . 3  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
87dmeqi 5036 . 2  |-  dom  +N  =  dom  (  +o  |`  ( N.  X.  N. ) )
9 df-ni 9033 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3478 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3381 . . . . . 6  |-  N.  C_  om
12 omsson 6475 . . . . . 6  |-  om  C_  On
1311, 12sstri 3360 . . . . 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 4940 . . . 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 3338 . . 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 2468 1  |-  dom  +N  =  ( N.  X.  N. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    \ cdif 3320    i^i cin 3322    C_ wss 3323   (/)c0 3632   {csn 3872   Oncon0 4714    X. cxp 4833   dom cdm 4835    |` cres 4837    Fn wfn 5408   omcom 6471    +o coa 6909   N.cnpi 9003    +N cpli 9004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-oadd 6916  df-ni 9033  df-pli 9034
This theorem is referenced by:  addcompi  9055  addasspi  9056  distrpi  9059  addcanpi  9060  addnidpi  9062  ltapi  9064
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