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Theorem hvmulcom 21638
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 8839 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 5889 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  .h  C
)  =  ( ( B  x.  A )  .h  C ) )
323adant3 975 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( ( B  x.  A )  .h  C ) )
4 ax-hvmulass 21603 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( A  .h  ( B  .h  C
) ) )
5 ax-hvmulass 21603 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  ~H )  ->  (
( B  x.  A
)  .h  C )  =  ( B  .h  ( A  .h  C
) ) )
653com12 1155 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( B  x.  A
)  .h  C )  =  ( B  .h  ( A  .h  C
) ) )
73, 4, 63eqtr3d 2336 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( B  .h  C ) )  =  ( B  .h  ( A  .h  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751    x. cmul 8758   ~Hchil 21515    .h csm 21517
This theorem is referenced by:  hvmulcomi  21642  hvsubdistr1  21644  lnopmi  22596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-mulcom 8817  ax-hvmulass 21603
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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