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Theorem addcomnq 8784
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
addcomnq  |-  ( A  +Q  B )  =  ( B  +Q  A
)

Proof of Theorem addcomnq
StepHypRef Expression
1 addcompq 8783 . . . 4  |-  ( A 
+pQ  B )  =  ( B  +pQ  A
)
21fveq2i 5690 . . 3  |-  ( /Q
`  ( A  +pQ  B ) )  =  ( /Q `  ( B 
+pQ  A ) )
3 addpqnq 8771 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( /Q
`  ( A  +pQ  B ) ) )
4 addpqnq 8771 . . . 4  |-  ( ( B  e.  Q.  /\  A  e.  Q. )  ->  ( B  +Q  A
)  =  ( /Q
`  ( B  +pQ  A ) ) )
54ancoms 440 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  A
)  =  ( /Q
`  ( B  +pQ  A ) ) )
62, 3, 53eqtr4a 2462 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
7 addnqf 8781 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5555 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
98ndmovcom 6193 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
106, 9pm2.61i 158 1  |-  ( A  +Q  B )  =  ( B  +Q  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721    X. cxp 4835   ` cfv 5413  (class class class)co 6040    +pQ cplpq 8679   Q.cnq 8683   /Qcerq 8685    +Q cplq 8686
This theorem is referenced by:  ltaddnq  8807  addclprlem2  8850  addclpr  8851  addcompr  8854  distrlem4pr  8859  prlem934  8866  ltexprlem2  8870  ltexprlem6  8874  ltexprlem7  8875  prlem936  8880
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-rmo 2674  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-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ni 8705  df-pli 8706  df-mi 8707  df-lti 8708  df-plpq 8741  df-enq 8744  df-nq 8745  df-erq 8746  df-plq 8747  df-1nq 8749
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