MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addcomnq Unicode version

Theorem addcomnq 8792
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 8791 . . . 4  |-  ( A 
+pQ  B )  =  ( B  +pQ  A
)
21fveq2i 5698 . . 3  |-  ( /Q
`  ( A  +pQ  B ) )  =  ( /Q `  ( B 
+pQ  A ) )
3 addpqnq 8779 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( /Q
`  ( A  +pQ  B ) ) )
4 addpqnq 8779 . . . 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 2470 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
7 addnqf 8789 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5563 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
98ndmovcom 6201 . 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 4843   ` cfv 5421  (class class class)co 6048    +pQ cplpq 8687   Q.cnq 8691   /Qcerq 8693    +Q cplq 8694
This theorem is referenced by:  ltaddnq  8815  addclprlem2  8858  addclpr  8859  addcompr  8862  distrlem4pr  8867  prlem934  8874  ltexprlem2  8878  ltexprlem6  8882  ltexprlem7  8883  prlem936  8888
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6872  df-ni 8713  df-pli 8714  df-mi 8715  df-lti 8716  df-plpq 8749  df-enq 8752  df-nq 8753  df-erq 8754  df-plq 8755  df-1nq 8757
  Copyright terms: Public domain W3C validator