MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmmulpi Unicode version

Theorem dmmulpi 8702
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 5108 . . 3  |-  dom  (  .o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  .o  )
2 fnom 6690 . . . . 5  |-  .o  Fn  ( On  X.  On )
3 fndm 5485 . . . . 5  |-  (  .o  Fn  ( On  X.  On )  ->  dom  .o  =  ( On  X.  On ) )
42, 3ax-mp 8 . . . 4  |-  dom  .o  =  ( On  X.  On )
54ineq2i 3483 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  .o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2408 . 2  |-  dom  (  .o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-mi 8685 . . 3  |-  .N  =  (  .o  |`  ( N.  X.  N. ) )
87dmeqi 5012 . 2  |-  dom  .N  =  dom  (  .o  |`  ( N.  X.  N. ) )
9 df-ni 8683 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3418 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3322 . . . . . 6  |-  N.  C_  om
12 omsson 4790 . . . . . 6  |-  om  C_  On
1311, 12sstri 3301 . . . . 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 4922 . . . 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 3279 . . 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 2418 1  |-  dom  .N  =  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    \ cdif 3261    i^i cin 3263    C_ wss 3264   (/)c0 3572   {csn 3758   Oncon0 4523   omcom 4786    X. cxp 4817   dom cdm 4819    |` cres 4821    Fn wfn 5390    .o comu 6659   N.cnpi 8653    .N cmi 8655
This theorem is referenced by:  mulcompi  8707  mulasspi  8708  distrpi  8709  mulcanpi  8711  ltmpi  8715  ordpipq  8753
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-omul 6666  df-ni 8683  df-mi 8685
  Copyright terms: Public domain W3C validator