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Theorem mulasspr 8650
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 8610 . 2  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  .Q  z ) } )
2 mulclnq 8573 . 2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3 dmmp 8639 . 2  |-  dom  .P.  =  ( P.  X.  P. )
4 mulclpr 8646 . 2  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  e.  P. )
5 mulassnq 8585 . 2  |-  ( ( f  .Q  g )  .Q  h )  =  ( f  .Q  (
g  .Q  h ) )
61, 2, 3, 4, 5genpass 8635 1  |-  ( ( A  .P.  B )  .P.  C )  =  ( A  .P.  ( B  .P.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1625  (class class class)co 5860    .Q cmq 8480    .P. cmp 8486
This theorem is referenced by:  mulasssr  8714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-ni 8498  df-mi 8500  df-lti 8501  df-mpq 8535  df-ltpq 8536  df-enq 8537  df-nq 8538  df-erq 8539  df-mq 8541  df-1nq 8542  df-rq 8543  df-ltnq 8544  df-np 8607  df-mp 8610
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