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Theorem hvmulcom 25664
Description: Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvmulcom  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( B  .h  C ) )  =  ( B  .h  ( A  .h  C )
) )

Proof of Theorem hvmulcom
StepHypRef Expression
1 mulcom 9578 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6299 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  .h  C
)  =  ( ( B  x.  A )  .h  C ) )
323adant3 1016 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( ( B  x.  A )  .h  C ) )
4 ax-hvmulass 25628 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( A  .h  ( B  .h  C
) ) )
5 ax-hvmulass 25628 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  ~H )  ->  (
( B  x.  A
)  .h  C )  =  ( B  .h  ( A  .h  C
) ) )
653com12 1200 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( B  x.  A
)  .h  C )  =  ( B  .h  ( A  .h  C
) ) )
73, 4, 63eqtr3d 2516 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( B  .h  C ) )  =  ( B  .h  ( A  .h  C )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767  (class class class)co 6284   CCcc 9490    x. cmul 9497   ~Hchil 25540    .h csm 25542
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-mulcom 9556  ax-hvmulass 25628
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6287
This theorem is referenced by:  hvmulcomi  25668  hvsubdistr1  25670  lnopmi  26623
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