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Theorem hvmulassi 22398
Description: Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvmulcom.1  |-  A  e.  CC
hvmulcom.2  |-  B  e.  CC
hvmulcom.3  |-  C  e. 
~H
Assertion
Ref Expression
hvmulassi  |-  ( ( A  x.  B )  .h  C )  =  ( A  .h  ( B  .h  C )
)

Proof of Theorem hvmulassi
StepHypRef Expression
1 hvmulcom.1 . 2  |-  A  e.  CC
2 hvmulcom.2 . 2  |-  B  e.  CC
3 hvmulcom.3 . 2  |-  C  e. 
~H
4 ax-hvmulass 22360 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( A  .h  ( B  .h  C
) ) )
51, 2, 3, 4mp3an 1279 1  |-  ( ( A  x.  B )  .h  C )  =  ( A  .h  ( B  .h  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717  (class class class)co 6022   CCcc 8923    x. cmul 8930   ~Hchil 22272    .h csm 22274
This theorem is referenced by:  hvmul2negi  22400  hvnegdii  22414  normlem0  22461  lnophmlem2  23370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-hvmulass 22360
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938
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