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

Theorem mulcompq 9319
Description: Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulcompq  |-  ( A 
.pQ  B )  =  ( B  .pQ  A
)

Proof of Theorem mulcompq
StepHypRef Expression
1 mulcompi 9263 . . . 4  |-  ( ( 1st `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 1st `  A ) )
2 mulcompi 9263 . . . 4  |-  ( ( 2nd `  A )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  B
)  .N  ( 2nd `  A ) )
31, 2opeq12i 4208 . . 3  |-  <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  =  <. ( ( 1st `  B )  .N  ( 1st `  A ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  A ) )
>.
4 mulpipq2 9306 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)
5 mulpipq2 9306 . . . 4  |-  ( ( B  e.  ( N. 
X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  ( B  .pQ  A )  = 
<. ( ( 1st `  B
)  .N  ( 1st `  A ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  A ) ) >.
)
65ancoms 451 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( B  .pQ  A )  = 
<. ( ( 1st `  B
)  .N  ( 1st `  A ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  A ) ) >.
)
73, 4, 63eqtr4a 2521 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  =  ( B  .pQ  A
) )
8 mulpqf 9313 . . . 4  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
98fdmi 5718 . . 3  |-  dom  .pQ  =  ( ( N. 
X.  N. )  X.  ( N.  X.  N. ) )
109ndmovcom 6435 . 2  |-  ( -.  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  =  ( B 
.pQ  A ) )
117, 10pm2.61i 164 1  |-  ( A 
.pQ  B )  =  ( B  .pQ  A
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022    X. cxp 4986   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   N.cnpi 9211    .N cmi 9213    .pQ cmpq 9216
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-reu 2811  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-pw 4001  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-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-oadd 7126  df-omul 7127  df-ni 9239  df-mi 9241  df-mpq 9276
This theorem is referenced by:  mulcomnq  9320  mulerpq  9324
  Copyright terms: Public domain W3C validator