HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvmulassi Unicode version

Theorem hvmulassi 21641
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 21603 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( A  .h  ( B  .h  C
) ) )
51, 2, 3, 4mp3an 1277 1  |-  ( ( A  x.  B )  .h  C )  =  ( A  .h  ( B  .h  C )
)
Colors of variables: wff set class
Syntax hints:    = 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:  hvmul2negi  21643  hvnegdii  21657  normlem0  21704  lnophmlem2  22613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-hvmulass 21603
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
  Copyright terms: Public domain W3C validator