Theorem kbop

Description: The outer product of two vectors, expressed as | A >. <. B | in Dirac notation, is an operator. (Contributed by NM, 30-May-2006) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	kbop	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ketbra 𝐵 ) : ℋ ⟶ ℋ )

Proof

Step	Hyp	Ref	Expression
1		kbfval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ketbra 𝐵 ) = ( 𝑥 ∈ ℋ ↦ ( ( 𝑥 ·ih 𝐵 ) ·ℎ 𝐴 ) ) )
2		hicl	⊢ ( ( 𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑥 ·ih 𝐵 ) ∈ ℂ )
3		hvmulcl	⊢ ( ( ( 𝑥 ·ih 𝐵 ) ∈ ℂ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑥 ·ih 𝐵 ) ·ℎ 𝐴 ) ∈ ℋ )
4	2 3	sylan	⊢ ( ( ( 𝑥 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑥 ·ih 𝐵 ) ·ℎ 𝐴 ) ∈ ℋ )
5	4	an31s	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ 𝑥 ∈ ℋ ) → ( ( 𝑥 ·ih 𝐵 ) ·ℎ 𝐴 ) ∈ ℋ )
6	1 5	fmpt3d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ketbra 𝐵 ) : ℋ ⟶ ℋ )