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Theorem mulcompr 8889
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
)

Proof of Theorem mulcompr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mpv 8877 . . 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 8877 . . . . 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 8819 . . . . . . . . 9  |-  ( y  .Q  z )  =  ( z  .Q  y
)
43eqeq2i 2445 . . . . . . . 8  |-  ( x  =  ( y  .Q  z )  <->  x  =  ( z  .Q  y
) )
542rexbii 2724 . . . . . . 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 2861 . . . . . . 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 241 . . . . . 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 2547 . . . . 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 2483 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  .P.  A
)  =  { x  |  E. z  e.  A  E. y  e.  B  x  =  ( z  .Q  y ) } )
109ancoms 440 . . 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 2470 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
12 dmmp 8879 . . 3  |-  dom  .P.  =  ( P.  X.  P. )
1312ndmovcom 6225 . 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 359    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698  (class class class)co 6072    .Q cmq 8720   P.cnp 8723    .P. cmp 8726
This theorem is referenced by:  mulcmpblnrlem  8937  mulcomsr  8953  mulasssr  8954  m1m1sr  8957  recexsrlem  8967  mulgt0sr  8969
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-omul 6720  df-er 6896  df-ni 8738  df-mi 8740  df-lti 8741  df-mpq 8775  df-enq 8777  df-nq 8778  df-erq 8779  df-mq 8781  df-1nq 8782  df-np 8847  df-mp 8850
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