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Theorem mulasspr 8894
Description: Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
mulasspr  |-  ( ( A  .P.  B )  .P.  C )  =  ( A  .P.  ( B  .P.  C ) )

Proof of Theorem mulasspr
Dummy variables  f 
g  h  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mp 8854 . 2  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  .Q  z ) } )
2 mulclnq 8817 . 2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3 dmmp 8883 . 2  |-  dom  .P.  =  ( P.  X.  P. )
4 mulclpr 8890 . 2  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  e.  P. )
5 mulassnq 8829 . 2  |-  ( ( f  .Q  g )  .Q  h )  =  ( f  .Q  (
g  .Q  h ) )
61, 2, 3, 4, 5genpass 8879 1  |-  ( ( A  .P.  B )  .P.  C )  =  ( A  .P.  ( B  .P.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652  (class class class)co 6074    .Q cmq 8724    .P. cmp 8730
This theorem is referenced by:  mulasssr  8958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-omul 6722  df-er 6898  df-ni 8742  df-mi 8744  df-lti 8745  df-mpq 8779  df-ltpq 8780  df-enq 8781  df-nq 8782  df-erq 8783  df-mq 8785  df-1nq 8786  df-rq 8787  df-ltnq 8788  df-np 8851  df-mp 8854
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