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Theorem dmmulpi 8531
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 4992 . . 3  |-  dom  (  .o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  dom  .o  )
2 fnom 6524 . . . . 5  |-  .o  Fn  ( On  X.  On )
3 fndm 5359 . . . . 5  |-  (  .o  Fn  ( On  X.  On )  ->  dom  .o  =  ( On  X.  On ) )
42, 3ax-mp 8 . . . 4  |-  dom  .o  =  ( On  X.  On )
54ineq2i 3380 . . 3  |-  ( ( N.  X.  N. )  i^i  dom  .o  )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
61, 5eqtri 2316 . 2  |-  dom  (  .o  |`  ( N.  X.  N. ) )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
7 df-mi 8514 . . 3  |-  .N  =  (  .o  |`  ( N.  X.  N. ) )
87dmeqi 4896 . 2  |-  dom  .N  =  dom  (  .o  |`  ( N.  X.  N. ) )
9 df-ni 8512 . . . . . . 7  |-  N.  =  ( om  \  { (/) } )
10 difss 3316 . . . . . . 7  |-  ( om 
\  { (/) } ) 
C_  om
119, 10eqsstri 3221 . . . . . 6  |-  N.  C_  om
12 omsson 4676 . . . . . 6  |-  om  C_  On
1311, 12sstri 3201 . . . . 5  |-  N.  C_  On
14 anidm 625 . . . . 5  |-  ( ( N.  C_  On  /\  N.  C_  On )  <->  N.  C_  On )
1513, 14mpbir 200 . . . 4  |-  ( N.  C_  On  /\  N.  C_  On )
16 xpss12 4808 . . . 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 3180 . . 3  |-  ( ( N.  X.  N. )  C_  ( On  X.  On ) 
<->  ( N.  X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) ) )
1917, 18mpbi 199 . 2  |-  ( N. 
X.  N. )  =  ( ( N.  X.  N. )  i^i  ( On  X.  On ) )
206, 8, 193eqtr4i 2326 1  |-  dom  .N  =  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   Oncon0 4408   omcom 4672    X. cxp 4703   dom cdm 4705    |` cres 4707    Fn wfn 5266    .o comu 6493   N.cnpi 8482    .N cmi 8484
This theorem is referenced by:  mulcompi  8536  mulasspi  8537  distrpi  8538  mulcanpi  8540  ltmpi  8544  ordpipq  8582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-omul 6500  df-ni 8512  df-mi 8514
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