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Theorem addcomnq 8665
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 8664 . . . 4  |-  ( A 
+pQ  B )  =  ( B  +pQ  A
)
21fveq2i 5611 . . 3  |-  ( /Q
`  ( A  +pQ  B ) )  =  ( /Q `  ( B 
+pQ  A ) )
3 addpqnq 8652 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( /Q
`  ( A  +pQ  B ) ) )
4 addpqnq 8652 . . . 4  |-  ( ( B  e.  Q.  /\  A  e.  Q. )  ->  ( B  +Q  A
)  =  ( /Q
`  ( B  +pQ  A ) ) )
54ancoms 439 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  A
)  =  ( /Q
`  ( B  +pQ  A ) ) )
62, 3, 53eqtr4a 2416 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
7 addnqf 8662 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5477 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
98ndmovcom 6094 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
106, 9pm2.61i 156 1  |-  ( A  +Q  B )  =  ( B  +Q  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1642    e. wcel 1710    X. cxp 4769   ` cfv 5337  (class class class)co 5945    +pQ cplpq 8560   Q.cnq 8564   /Qcerq 8566    +Q cplq 8567
This theorem is referenced by:  ltaddnq  8688  addclprlem2  8731  addclpr  8732  addcompr  8735  distrlem4pr  8740  prlem934  8747  ltexprlem2  8751  ltexprlem6  8755  ltexprlem7  8756  prlem936  8761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-omul 6571  df-er 6747  df-ni 8586  df-pli 8587  df-mi 8588  df-lti 8589  df-plpq 8622  df-enq 8625  df-nq 8626  df-erq 8627  df-plq 8628  df-1nq 8630
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