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Theorem mulcompr 9397
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 9385 . . 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 9385 . . . . 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 9327 . . . . . . . . 9  |-  ( y  .Q  z )  =  ( z  .Q  y
)
43eqeq2i 2485 . . . . . . . 8  |-  ( x  =  ( y  .Q  z )  <->  x  =  ( z  .Q  y
) )
542rexbii 2966 . . . . . . 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 3023 . . . . . . 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 2601 . . . . 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 2524 . . . 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 2511 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
12 dmmp 9387 . . 3  |-  dom  .P.  =  ( P.  X.  P. )
1312ndmovcom 6444 . 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 1379    e. wcel 1767   {cab 2452   E.wrex 2815  (class class class)co 6282    .Q cmq 9230   P.cnp 9233    .P. cmp 9236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ni 9246  df-mi 9248  df-lti 9249  df-mpq 9283  df-enq 9285  df-nq 9286  df-erq 9287  df-mq 9289  df-1nq 9290  df-np 9355  df-mp 9358
This theorem is referenced by:  mulcmpblnrlem  9443  mulcomsr  9462  mulasssr  9463  m1m1sr  9466  recexsrlem  9476  mulgt0sr  9478
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