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Theorem mulcompq 9340
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 9284 . . . 4  |-  ( ( 1st `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 1st `  A ) )
2 mulcompi 9284 . . . 4  |-  ( ( 2nd `  A )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  B
)  .N  ( 2nd `  A ) )
31, 2opeq12i 4223 . . 3  |-  <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  =  <. ( ( 1st `  B )  .N  ( 1st `  A ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  A ) )
>.
4 mulpipq2 9327 . . 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 9327 . . . 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 453 . . 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 2534 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  =  ( B  .pQ  A
) )
8 mulpqf 9334 . . . 4  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
98fdmi 5741 . . 3  |-  dom  .pQ  =  ( ( N. 
X.  N. )  X.  ( N.  X.  N. ) )
109ndmovcom 6456 . 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 369    = wceq 1379    e. wcel 1767   <.cop 4038    X. cxp 5002   ` cfv 5593  (class class class)co 6294   1stc1st 6792   2ndc2nd 6793   N.cnpi 9232    .N cmi 9234    .pQ cmpq 9237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-oadd 7144  df-omul 7145  df-ni 9260  df-mi 9262  df-mpq 9297
This theorem is referenced by:  mulcomnq  9341  mulerpq  9345
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