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Theorem addcompr 8824
Description: Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addcompr  |-  ( A  +P.  B )  =  ( B  +P.  A
)

Proof of Theorem addcompr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plpv 8813 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  =  { x  |  E. z  e.  A  E. y  e.  B  x  =  ( z  +Q  y ) } )
2 plpv 8813 . . . . 5  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  +P.  A
)  =  { x  |  E. y  e.  B  E. z  e.  A  x  =  ( y  +Q  z ) } )
3 addcomnq 8754 . . . . . . . . 9  |-  ( y  +Q  z )  =  ( z  +Q  y
)
43eqeq2i 2390 . . . . . . . 8  |-  ( x  =  ( y  +Q  z )  <->  x  =  ( z  +Q  y
) )
542rexbii 2669 . . . . . . 7  |-  ( E. y  e.  B  E. z  e.  A  x  =  ( y  +Q  z )  <->  E. y  e.  B  E. z  e.  A  x  =  ( z  +Q  y
) )
6 rexcom 2805 . . . . . . 7  |-  ( E. y  e.  B  E. z  e.  A  x  =  ( z  +Q  y )  <->  E. z  e.  A  E. y  e.  B  x  =  ( z  +Q  y
) )
75, 6bitri 241 . . . . . 6  |-  ( E. y  e.  B  E. z  e.  A  x  =  ( y  +Q  z )  <->  E. z  e.  A  E. y  e.  B  x  =  ( z  +Q  y
) )
87abbii 2492 . . . . 5  |-  { x  |  E. y  e.  B  E. z  e.  A  x  =  ( y  +Q  z ) }  =  { x  |  E. z  e.  A  E. y  e.  B  x  =  ( z  +Q  y ) }
92, 8syl6eq 2428 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  +P.  A
)  =  { x  |  E. z  e.  A  E. y  e.  B  x  =  ( z  +Q  y ) } )
109ancoms 440 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( B  +P.  A
)  =  { x  |  E. z  e.  A  E. y  e.  B  x  =  ( z  +Q  y ) } )
111, 10eqtr4d 2415 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  =  ( B  +P.  A ) )
12 dmplp 8815 . . 3  |-  dom  +P.  =  ( P.  X.  P. )
1312ndmovcom 6166 . 2  |-  ( -.  ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  =  ( B  +P.  A ) )
1411, 13pm2.61i 158 1  |-  ( A  +P.  B )  =  ( B  +P.  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2366   E.wrex 2643  (class class class)co 6013    +Q cplq 8656   P.cnp 8660    +P. cpp 8662
This theorem is referenced by:  enrer  8869  addcmpblnr  8873  mulcmpblnrlem  8874  ltsrpr  8878  addcomsr  8888  mulcomsr  8890  mulasssr  8891  distrsr  8892  ltsosr  8895  0lt1sr  8896  0idsr  8898  1idsr  8899  ltasr  8901  recexsrlem  8904  mulgt0sr  8906  ltpsrpr  8910  map2psrpr  8911
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-omul 6658  df-er 6834  df-ni 8675  df-pli 8676  df-mi 8677  df-lti 8678  df-plpq 8711  df-enq 8714  df-nq 8715  df-erq 8716  df-plq 8717  df-1nq 8719  df-np 8784  df-plp 8786
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