Theorem hvmulass 5020

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) .s C) = (A .s (B .s 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 4992	. 2 |- ((A e. CC /\ B e. CC /\ C e. H~) -> ((A x. B) .s C) = (A .s (B .s C)))
5	1, 2, 3, 4	mp3an 642	1 |- ((A x. B) .s C) = (A .s (B .s C))

Syntax hints:   = wceq 1091   e. wcel 1092  (class class class)co 3001  CCcc 4026   x. cmulc 4032  H~chil 4958   .s csm 4960

This theorem is referenced by:  hvmulcom 5021  hvmul2neg 5022  hvnegdi 5034  normlem0 5062  projlem18 5210

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-hvmulass 4992

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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