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Theorem addcomnq 8833
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 8832 . . . 4  |-  ( A 
+pQ  B )  =  ( B  +pQ  A
)
21fveq2i 5734 . . 3  |-  ( /Q
`  ( A  +pQ  B ) )  =  ( /Q `  ( B 
+pQ  A ) )
3 addpqnq 8820 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( /Q
`  ( A  +pQ  B ) ) )
4 addpqnq 8820 . . . 4  |-  ( ( B  e.  Q.  /\  A  e.  Q. )  ->  ( B  +Q  A
)  =  ( /Q
`  ( B  +pQ  A ) ) )
54ancoms 441 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  A
)  =  ( /Q
`  ( B  +pQ  A ) ) )
62, 3, 53eqtr4a 2496 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
7 addnqf 8830 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5599 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
98ndmovcom 6237 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
106, 9pm2.61i 159 1  |-  ( A  +Q  B )  =  ( B  +Q  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726    X. cxp 4879   ` cfv 5457  (class class class)co 6084    +pQ cplpq 8728   Q.cnq 8732   /Qcerq 8734    +Q cplq 8735
This theorem is referenced by:  ltaddnq  8856  addclprlem2  8899  addclpr  8900  addcompr  8903  distrlem4pr  8908  prlem934  8915  ltexprlem2  8919  ltexprlem6  8923  ltexprlem7  8924  prlem936  8929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-omul 6732  df-er 6908  df-ni 8754  df-pli 8755  df-mi 8756  df-lti 8757  df-plpq 8790  df-enq 8793  df-nq 8794  df-erq 8795  df-plq 8796  df-1nq 8798
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