Theorem hvmulass 8897

Description: Scalar multiplication associative law.

Hypotheses

Ref	Expression
hvmulcom.1	|- A e. CC
hvmulcom.2	|- B e. CC
hvmulcom.3	|- C e. H~

Assertion

Ref	Expression
hvmulass	|- ((A x. B) .h C) = (A .h (B .h C))

Proof of Theorem hvmulass

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 8861	. 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 915	1 |- ((A x. B) .h C) = (A .h (B .h C))

Syntax hints:   = wceq 955   e. wcel 957  (class class class)co 3960  CCcc 5219   x. cmul 5226  H~chil 8772   .h csm 8774

This theorem is referenced by:  hvmul2neg 8899  hvnegdi 8913  normlem0 8959  projlem18 9191  lnophmlem2 9933

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-hvmulass 8861

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776

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