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Theorem dmaddpi 8723
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 6711 . . . . 5  |-  +o  Fn  ( On  X.  On )
3 fndm 5503 . . . . 5  |-  (  +o  Fn  ( On  X.  On )  ->  dom  +o  =  ( On  X.  On ) )
42, 3ax-mp 8 . . . 4  |-  dom  +o  =  ( On  X.  On )
54ineq2i 3499 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  +o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2424 . 2  |-  dom  (  +o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-pli 8706 . . 3  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
87dmeqi 5030 . 2  |-  dom  +N  =  dom  (  +o  |`  ( N.  X.  N. ) )
9 df-ni 8705 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3434 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3338 . . . . . 6  |-  N.  C_  om
12 omsson 4808 . . . . . 6  |-  om  C_  On
1311, 12sstri 3317 . . . . 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 4940 . . . 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 3295 . . 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 2434 1  |-  dom  +N  =  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   Oncon0 4541   omcom 4804    X. cxp 4835   dom cdm 4837    |` cres 4839    Fn wfn 5408    +o coa 6680   N.cnpi 8675    +N cpli 8676
This theorem is referenced by:  addcompi  8727  addasspi  8728  distrpi  8731  addcanpi  8732  addnidpi  8734  ltapi  8736
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-oadd 6687  df-ni 8705  df-pli 8706
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