Theorem hvmul2negi

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

	Ref	Expression
Hypotheses	hvmulcom.1	⊢ 𝐴 ∈ ℂ
	hvmulcom.2	⊢ 𝐵 ∈ ℂ
	hvmulcom.3	⊢ 𝐶 ∈ ℋ
Assertion	hvmul2negi	⊢ ( - 𝐴 ·ℎ ( - 𝐵 ·ℎ 𝐶 ) ) = ( 𝐴 ·ℎ ( 𝐵 ·ℎ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		hvmulcom.1	⊢ 𝐴 ∈ ℂ
2		hvmulcom.2	⊢ 𝐵 ∈ ℂ
3		hvmulcom.3	⊢ 𝐶 ∈ ℋ
4	1 2	mul2negi	⊢ ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 )
5	4	oveq1i	⊢ ( ( - 𝐴 · - 𝐵 ) ·ℎ 𝐶 ) = ( ( 𝐴 · 𝐵 ) ·ℎ 𝐶 )
6	1	negcli	⊢ - 𝐴 ∈ ℂ
7	2	negcli	⊢ - 𝐵 ∈ ℂ
8	6 7 3	hvmulassi	⊢ ( ( - 𝐴 · - 𝐵 ) ·ℎ 𝐶 ) = ( - 𝐴 ·ℎ ( - 𝐵 ·ℎ 𝐶 ) )
9	1 2 3	hvmulassi	⊢ ( ( 𝐴 · 𝐵 ) ·ℎ 𝐶 ) = ( 𝐴 ·ℎ ( 𝐵 ·ℎ 𝐶 ) )
10	5 8 9	3eqtr3i	⊢ ( - 𝐴 ·ℎ ( - 𝐵 ·ℎ 𝐶 ) ) = ( 𝐴 ·ℎ ( 𝐵 ·ℎ 𝐶 ) )