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Theorem mulcomsr 9483
Description: Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
mulcomsr  |-  ( A  .R  B )  =  ( B  .R  A
)

Proof of Theorem mulcomsr
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 9451 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 mulsrpr 9470 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ,  ( ( x  .P.  w )  +P.  (
y  .P.  z )
) >. ]  ~R  )
3 mulsrpr 9470 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  .R  [ <. x ,  y >. ]  ~R  )  =  [ <. (
( z  .P.  x
)  +P.  ( w  .P.  y ) ) ,  ( ( z  .P.  y )  +P.  (
w  .P.  x )
) >. ]  ~R  )
4 mulcompr 9418 . . . 4  |-  ( x  .P.  z )  =  ( z  .P.  x
)
5 mulcompr 9418 . . . 4  |-  ( y  .P.  w )  =  ( w  .P.  y
)
64, 5oveq12i 6308 . . 3  |-  ( ( x  .P.  z )  +P.  ( y  .P.  w ) )  =  ( ( z  .P.  x )  +P.  (
w  .P.  y )
)
7 mulcompr 9418 . . . . 5  |-  ( x  .P.  w )  =  ( w  .P.  x
)
8 mulcompr 9418 . . . . 5  |-  ( y  .P.  z )  =  ( z  .P.  y
)
97, 8oveq12i 6308 . . . 4  |-  ( ( x  .P.  w )  +P.  ( y  .P.  z ) )  =  ( ( w  .P.  x )  +P.  (
z  .P.  y )
)
10 addcompr 9416 . . . 4  |-  ( ( w  .P.  x )  +P.  ( z  .P.  y ) )  =  ( ( z  .P.  y )  +P.  (
w  .P.  x )
)
119, 10eqtri 2486 . . 3  |-  ( ( x  .P.  w )  +P.  ( y  .P.  z ) )  =  ( ( z  .P.  y )  +P.  (
w  .P.  x )
)
121, 2, 3, 6, 11ecovcom 7435 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  =  ( B  .R  A ) )
13 dmmulsr 9480 . . 3  |-  dom  .R  =  ( R.  X.  R. )
1413ndmovcom 6461 . 2  |-  ( -.  ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  =  ( B  .R  A ) )
1512, 14pm2.61i 164 1  |-  ( A  .R  B )  =  ( B  .R  A
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    e. wcel 1819  (class class class)co 6296   P.cnp 9254    +P. cpp 9256    .P. cmp 9257    ~R cer 9259   R.cnr 9260    .R cmr 9265
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-int 4289  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-ec 7331  df-qs 7335  df-ni 9267  df-pli 9268  df-mi 9269  df-lti 9270  df-plpq 9303  df-mpq 9304  df-ltpq 9305  df-enq 9306  df-nq 9307  df-erq 9308  df-plq 9309  df-mq 9310  df-1nq 9311  df-rq 9312  df-ltnq 9313  df-np 9376  df-plp 9378  df-mp 9379  df-ltp 9380  df-enr 9450  df-nr 9451  df-mr 9453
This theorem is referenced by:  sqgt0sr  9500  mulresr  9533  axmulcom  9549  axmulass  9551  axcnre  9558
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