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Theorem addcompq 8827
Description: Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addcompq  |-  ( A 
+pQ  B )  =  ( B  +pQ  A
)

Proof of Theorem addcompq
StepHypRef Expression
1 addcompi 8771 . . . 4  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 1st `  B )  .N  ( 2nd `  A
) )  +N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) )
2 mulcompi 8773 . . . 4  |-  ( ( 2nd `  A )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  B
)  .N  ( 2nd `  A ) )
31, 2opeq12i 3989 . . 3  |-  <. (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  =  <. ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  A ) )
>.
4 addpipq2 8813 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  +pQ  B )  = 
<. ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
5 addpipq2 8813 . . . 4  |-  ( ( B  e.  ( N. 
X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  ( B  +pQ  A )  = 
<. ( ( ( 1st `  B )  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  A
) ) >. )
65ancoms 440 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( B  +pQ  A )  = 
<. ( ( ( 1st `  B )  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  A
) ) >. )
73, 4, 63eqtr4a 2494 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  +pQ  B )  =  ( B  +pQ  A
) )
8 addpqf 8821 . . . 4  |-  +pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
98fdmi 5596 . . 3  |-  dom  +pQ  =  ( ( N. 
X.  N. )  X.  ( N.  X.  N. ) )
109ndmovcom 6234 . 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 1652    e. wcel 1725   <.cop 3817    X. cxp 4876   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   N.cnpi 8719    +N cpli 8720    .N cmi 8721    +pQ cplpq 8723
This theorem is referenced by:  addcomnq  8828  adderpq  8833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-oadd 6728  df-omul 6729  df-ni 8749  df-pli 8750  df-mi 8751  df-plpq 8785
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