MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulcomsr Structured version   Unicode version

Theorem mulcomsr 8964
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 8935 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 mulsrpr 8951 . . 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 8951 . . 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 8900 . . . 4  |-  ( x  .P.  z )  =  ( z  .P.  x
)
5 mulcompr 8900 . . . 4  |-  ( y  .P.  w )  =  ( w  .P.  y
)
64, 5oveq12i 6093 . . 3  |-  ( ( x  .P.  z )  +P.  ( y  .P.  w ) )  =  ( ( z  .P.  x )  +P.  (
w  .P.  y )
)
7 mulcompr 8900 . . . . 5  |-  ( x  .P.  w )  =  ( w  .P.  x
)
8 mulcompr 8900 . . . . 5  |-  ( y  .P.  z )  =  ( z  .P.  y
)
97, 8oveq12i 6093 . . . 4  |-  ( ( x  .P.  w )  +P.  ( y  .P.  z ) )  =  ( ( w  .P.  x )  +P.  (
z  .P.  y )
)
10 addcompr 8898 . . . 4  |-  ( ( w  .P.  x )  +P.  ( z  .P.  y ) )  =  ( ( z  .P.  y )  +P.  (
w  .P.  x )
)
119, 10eqtri 2456 . . 3  |-  ( ( x  .P.  w )  +P.  ( y  .P.  z ) )  =  ( ( z  .P.  y )  +P.  (
w  .P.  x )
)
121, 2, 3, 6, 11ecovcom 7015 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  =  ( B  .R  A ) )
13 dmmulsr 8961 . . 3  |-  dom  .R  =  ( R.  X.  R. )
1413ndmovcom 6234 . 2  |-  ( -.  ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  =  ( B  .R  A ) )
1512, 14pm2.61i 158 1  |-  ( A  .R  B )  =  ( B  .R  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725  (class class class)co 6081   P.cnp 8734    +P. cpp 8736    .P. cmp 8737    ~R cer 8741   R.cnr 8742    .R cmr 8747
This theorem is referenced by:  sqgt0sr  8981  mulresr  9014  axmulcom  9030  axmulass  9032  axcnre  9039
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ec 6907  df-qs 6911  df-ni 8749  df-pli 8750  df-mi 8751  df-lti 8752  df-plpq 8785  df-mpq 8786  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-plq 8791  df-mq 8792  df-1nq 8793  df-rq 8794  df-ltnq 8795  df-np 8858  df-plp 8860  df-mp 8861  df-ltp 8862  df-mpr 8933  df-enr 8934  df-nr 8935  df-mr 8937
  Copyright terms: Public domain W3C validator