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Theorem mulcompq 8785
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 8729 . . . 4  |-  ( ( 1st `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 1st `  A ) )
2 mulcompi 8729 . . . 4  |-  ( ( 2nd `  A )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  B
)  .N  ( 2nd `  A ) )
31, 2opeq12i 3949 . . 3  |-  <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  =  <. ( ( 1st `  B )  .N  ( 1st `  A ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  A ) )
>.
4 mulpipq2 8772 . . 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 8772 . . . 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 440 . . 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 2462 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  =  ( B  .pQ  A
) )
8 mulpqf 8779 . . . 4  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
98fdmi 5555 . . 3  |-  dom  .pQ  =  ( ( N. 
X.  N. )  X.  ( N.  X.  N. ) )
109ndmovcom 6193 . 2  |-  ( -.  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  =  ( B 
.pQ  A ) )
117, 10pm2.61i 158 1  |-  ( A 
.pQ  B )  =  ( B  .pQ  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3777    X. cxp 4835   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   N.cnpi 8675    .N cmi 8677    .pQ cmpq 8680
This theorem is referenced by:  mulcomnq  8786  mulerpq  8790
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-oadd 6687  df-omul 6688  df-ni 8705  df-mi 8707  df-mpq 8742
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