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Theorem dmmulpi 9258
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 5282 . . 3  |-  dom  (  .o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  .o  )
2 fnom 7151 . . . . 5  |-  .o  Fn  ( On  X.  On )
3 fndm 5662 . . . . 5  |-  (  .o  Fn  ( On  X.  On )  ->  dom  .o  =  ( On  X.  On ) )
42, 3ax-mp 5 . . . 4  |-  dom  .o  =  ( On  X.  On )
54ineq2i 3683 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  .o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2483 . 2  |-  dom  (  .o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-mi 9241 . . 3  |-  .N  =  (  .o  |`  ( N.  X.  N. ) )
87dmeqi 5193 . 2  |-  dom  .N  =  dom  (  .o  |`  ( N.  X.  N. ) )
9 df-ni 9239 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3617 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3519 . . . . . 6  |-  N.  C_  om
12 omsson 6677 . . . . . 6  |-  om  C_  On
1311, 12sstri 3498 . . . . 5  |-  N.  C_  On
14 anidm 642 . . . . 5  |-  ( ( N.  C_  On  /\  N.  C_  On )  <->  N.  C_  On )
1513, 14mpbir 209 . . . 4  |-  ( N.  C_  On  /\  N.  C_  On )
16 xpss12 5096 . . . 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 3476 . . 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 2493 1  |-  dom  .N  =  ( N.  X.  N. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   Oncon0 4867    X. cxp 4986   dom cdm 4988    |` cres 4990    Fn wfn 5565   omcom 6673    .o comu 7120   N.cnpi 9211    .N cmi 9213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-omul 7127  df-ni 9239  df-mi 9241
This theorem is referenced by:  mulcompi  9263  mulasspi  9264  distrpi  9265  mulcanpi  9267  ltmpi  9271  ordpipq  9309
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