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Theorem dmmulpi 8760
Description: Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
dmmulpi  |-  dom  .N  =  ( N.  X.  N. )

Proof of Theorem dmmulpi
StepHypRef Expression
1 dmres 5159 . . 3  |-  dom  (  .o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  .o  )
2 fnom 6745 . . . . 5  |-  .o  Fn  ( On  X.  On )
3 fndm 5536 . . . . 5  |-  (  .o  Fn  ( On  X.  On )  ->  dom  .o  =  ( On  X.  On ) )
42, 3ax-mp 8 . . . 4  |-  dom  .o  =  ( On  X.  On )
54ineq2i 3531 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  .o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2455 . 2  |-  dom  (  .o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-mi 8743 . . 3  |-  .N  =  (  .o  |`  ( N.  X.  N. ) )
87dmeqi 5063 . 2  |-  dom  .N  =  dom  (  .o  |`  ( N.  X.  N. ) )
9 df-ni 8741 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3466 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3370 . . . . . 6  |-  N.  C_  om
12 omsson 4841 . . . . . 6  |-  om  C_  On
1311, 12sstri 3349 . . . . 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 4973 . . . 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 3327 . . 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 2465 1  |-  dom  .N  =  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   Oncon0 4573   omcom 4837    X. cxp 4868   dom cdm 4870    |` cres 4872    Fn wfn 5441    .o comu 6714   N.cnpi 8711    .N cmi 8713
This theorem is referenced by:  mulcompi  8765  mulasspi  8766  distrpi  8767  mulcanpi  8769  ltmpi  8773  ordpipq  8811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-omul 6721  df-ni 8741  df-mi 8743
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