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Theorem mulcompr 8643
Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulcompr  |-  ( A  .P.  B )  =  ( B  .P.  A
)
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem mulcompr
StepHypRef Expression
1 mpv 8631 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  { x  |  E. z  e.  A  E. y  e.  B  x  =  ( z  .Q  y ) } )
2 mpv 8631 . . . . 5  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  .P.  A
)  =  { x  |  E. y  e.  B  E. z  e.  A  x  =  ( y  .Q  z ) } )
3 mulcomnq 8573 . . . . . . . . 9  |-  ( y  .Q  z )  =  ( z  .Q  y
)
43eqeq2i 2295 . . . . . . . 8  |-  ( x  =  ( y  .Q  z )  <->  x  =  ( z  .Q  y
) )
542rexbii 2572 . . . . . . 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 2703 . . . . . . 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 242 . . . . . 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 2397 . . . . 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 2333 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  .P.  A
)  =  { x  |  E. z  e.  A  E. y  e.  B  x  =  ( z  .Q  y ) } )
109ancoms 441 . . 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 2320 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
12 dmmp 8633 . . 3  |-  dom  .P.  =  ( P.  X.  P. )
1312ndmovcom 5969 . 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 360    = wceq 1624    e. wcel 1685   {cab 2271   E.wrex 2546  (class class class)co 5820    .Q cmq 8474   P.cnp 8477    .P. cmp 8480
This theorem is referenced by:  mulcmpblnrlem  8691  mulcomsr  8707  mulasssr  8708  m1m1sr  8711  recexsrlem  8721  mulgt0sr  8723
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  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 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-omul 6480  df-er 6656  df-ni 8492  df-mi 8494  df-lti 8495  df-mpq 8529  df-enq 8531  df-nq 8532  df-erq 8533  df-mq 8535  df-1nq 8536  df-np 8601  df-mp 8604
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