Theorem bramul

Description: Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	bramul	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( 𝐵 · ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-his3	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) = ( 𝐵 · ( 𝐶 ·ih 𝐴 ) ) )
2	1	3comr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) = ( 𝐵 · ( 𝐶 ·ih 𝐴 ) ) )
3		hvmulcl	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 ·ℎ 𝐶 ) ∈ ℋ )
4		braval	⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐵 ·ℎ 𝐶 ) ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) )
5	3 4	sylan2	⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) )
6	5	3impb	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) )
7		braval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) = ( 𝐶 ·ih 𝐴 ) )
8	7	3adant2	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) = ( 𝐶 ·ih 𝐴 ) )
9	8	oveq2d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 · ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) = ( 𝐵 · ( 𝐶 ·ih 𝐴 ) ) )
10	2 6 9	3eqtr4d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( 𝐵 · ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) )