Theorem hvmulassi 27903

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

Step	Hyp	Ref	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 27864	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( A .h ( B .h C ) ) )
5	1, 2, 3, 4	mp3an 1424	1 |- ( ( A x. B ) .h C ) = ( A .h ( B .h C ) )

Syntax hints:   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   x. cmul 9941   ~Hchil 27776   .h csm 27778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvmulass 27864

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039

This theorem is referenced by:  hvmul2negi  27905  hvnegdii  27919  normlem0  27966  lnophmlem2  28876