Theorem hvmul2negi

Description: Double negative in scalar multiplication. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvmulcom.1	|- A e. CC
	hvmulcom.2	|- B e. CC
	hvmulcom.3	|- C e. ~H
Assertion	hvmul2negi	|- ( -u A .h ( -u B .h C ) ) = ( A .h ( B .h C ) )

Proof

Step	Hyp	Ref	Expression
1		hvmulcom.1	|- A e. CC
2		hvmulcom.2	|- B e. CC
3		hvmulcom.3	|- C e. ~H
4	1 2	mul2negi	|- ( -u A x. -u B ) = ( A x. B )
5	4	oveq1i	|- ( ( -u A x. -u B ) .h C ) = ( ( A x. B ) .h C )
6	1	negcli	|- -u A e. CC
7	2	negcli	|- -u B e. CC
8	6 7 3	hvmulassi	|- ( ( -u A x. -u B ) .h C ) = ( -u A .h ( -u B .h C ) )
9	1 2 3	hvmulassi	|- ( ( A x. B ) .h C ) = ( A .h ( B .h C ) )
10	5 8 9	3eqtr3i	|- ( -u A .h ( -u B .h C ) ) = ( A .h ( B .h C ) )