Theorem hvmulass 5020

Description: Scalar multiplication associative law.

Hypotheses

Ref	Expression
hvmulcom.1	⊢ A ∈ ℂ
hvmulcom.2	⊢ B ∈ ℂ
hvmulcom.3	⊢ C ∈ ℋ

Assertion

Ref	Expression
hvmulass	⊢ ((A · B) ·s C) = (A ·s (B ·s C))

Proof of Theorem hvmulass

Step	Hyp	Ref	Expression
1		hvmulcom.1	. 2 ⊢ A ∈ ℂ
2		hvmulcom.2	. 2 ⊢ B ∈ ℂ
3		hvmulcom.3	. 2 ⊢ C ∈ ℋ
4		ax-hvmulass 4992	. 2 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℋ ) → ((A · B) ·s C) = (A ·s (B ·s C)))
5	1, 2, 3, 4	mp3an 642	1 ⊢ ((A · B) ·s C) = (A ·s (B ·s C))

Syntax hints:   = wceq 1091   ∈ wcel 1092  (class class class)co 3001  ℂcc 4026   · 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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