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Theorem mulcompr 9418
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 9406 . . 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 9406 . . . . 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 9348 . . . . . . . . 9  |-  ( y  .Q  z )  =  ( z  .Q  y
)
43eqeq2i 2475 . . . . . . . 8  |-  ( x  =  ( y  .Q  z )  <->  x  =  ( z  .Q  y
) )
542rexbii 2960 . . . . . . 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 3019 . . . . . . 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 2591 . . . . 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 2514 . . . 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 2501 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
12 dmmp 9408 . . 3  |-  dom  .P.  =  ( P.  X.  P. )
1312ndmovcom 6461 . 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808  (class class class)co 6296    .Q cmq 9251   P.cnp 9254    .P. cmp 9257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-omul 7153  df-er 7329  df-ni 9267  df-mi 9269  df-lti 9270  df-mpq 9304  df-enq 9306  df-nq 9307  df-erq 9308  df-mq 9310  df-1nq 9311  df-np 9376  df-mp 9379
This theorem is referenced by:  mulcmpblnrlem  9464  mulcomsr  9483  mulasssr  9484  m1m1sr  9487  recexsrlem  9497  mulgt0sr  9499
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