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Theorem mulasspr 8901
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 8861 . 2  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  .Q  z ) } )
2 mulclnq 8824 . 2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3 dmmp 8890 . 2  |-  dom  .P.  =  ( P.  X.  P. )
4 mulclpr 8897 . 2  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  e.  P. )
5 mulassnq 8836 . 2  |-  ( ( f  .Q  g )  .Q  h )  =  ( f  .Q  (
g  .Q  h ) )
61, 2, 3, 4, 5genpass 8886 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 6081    .Q cmq 8731    .P. cmp 8737
This theorem is referenced by:  mulasssr  8965
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 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
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 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-mi 8751  df-lti 8752  df-mpq 8786  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-mq 8792  df-1nq 8793  df-rq 8794  df-ltnq 8795  df-np 8858  df-mp 8861
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