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Theorem addasspr 8793
Description: Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
addasspr  |-  ( ( A  +P.  B )  +P.  C )  =  ( A  +P.  ( B  +P.  C ) )

Proof of Theorem addasspr
Dummy variables  f 
g  h  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-plp 8754 . 2  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
2 addclnq 8716 . 2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
3 dmplp 8783 . 2  |-  dom  +P.  =  ( P.  X.  P. )
4 addclpr 8789 . 2  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
5 addassnq 8729 . 2  |-  ( ( f  +Q  g )  +Q  h )  =  ( f  +Q  (
g  +Q  h ) )
61, 2, 3, 4, 5genpass 8780 1  |-  ( ( A  +P.  B )  +P.  C )  =  ( A  +P.  ( B  +P.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1647  (class class class)co 5981    +Q cplq 8624    +P. cpp 8630
This theorem is referenced by:  ltaprlem  8815  enrer  8837  addcmpblnr  8841  mulcmpblnrlem  8842  ltsrpr  8846  addasssr  8857  mulasssr  8859  distrsr  8860  m1p1sr  8861  m1m1sr  8862  ltsosr  8863  0idsr  8866  1idsr  8867  ltasr  8869  recexsrlem  8872  mulgt0sr  8874  map2psrpr  8879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-omul 6626  df-er 6802  df-ni 8643  df-pli 8644  df-mi 8645  df-lti 8646  df-plpq 8679  df-mpq 8680  df-ltpq 8681  df-enq 8682  df-nq 8683  df-erq 8684  df-plq 8685  df-mq 8686  df-1nq 8687  df-rq 8688  df-ltnq 8689  df-np 8752  df-plp 8754
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