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Theorem mulcompr 9293
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 9281 . . 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 9281 . . . . 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 9223 . . . . . . . . 9  |-  ( y  .Q  z )  =  ( z  .Q  y
)
43eqeq2i 2469 . . . . . . . 8  |-  ( x  =  ( y  .Q  z )  <->  x  =  ( z  .Q  y
) )
542rexbii 2848 . . . . . . 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 2978 . . . . . . 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 249 . . . . . 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 2585 . . . . 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 2508 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  .P.  A
)  =  { x  |  E. z  e.  A  E. y  e.  B  x  =  ( z  .Q  y ) } )
109ancoms 453 . . 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 2495 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
12 dmmp 9283 . . 3  |-  dom  .P.  =  ( P.  X.  P. )
1312ndmovcom 6350 . 2  |-  ( -.  ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
1411, 13pm2.61i 164 1  |-  ( A  .P.  B )  =  ( B  .P.  A
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2436   E.wrex 2796  (class class class)co 6190    .Q cmq 9124   P.cnp 9127    .P. cmp 9130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-omul 7025  df-er 7201  df-ni 9142  df-mi 9144  df-lti 9145  df-mpq 9179  df-enq 9181  df-nq 9182  df-erq 9183  df-mq 9185  df-1nq 9186  df-np 9251  df-mp 9254
This theorem is referenced by:  mulcmpblnrlem  9341  mulcomsr  9357  mulasssr  9358  m1m1sr  9361  recexsrlem  9371  mulgt0sr  9373
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