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	$⊢ (A \in ℋ \land B \in ℋ) \to (A ketbra B) : ℋ ⟶ ℋ$

Proof

Step	Hyp	Ref	Expression
1		kbfval	$⊢ (A \in ℋ \land B \in ℋ) \to A ketbra B = (x \in ℋ ⟼ (x \cdot_{ih} B) \cdot_{ℎ} A)$
2		hicl	$⊢ (x \in ℋ \land B \in ℋ) \to x \cdot_{ih} B \in ℂ$
3		hvmulcl	$⊢ (x \cdot_{ih} B \in ℂ \land A \in ℋ) \to (x \cdot_{ih} B) \cdot_{ℎ} A \in ℋ$
4	2 3	sylan	$⊢ ((x \in ℋ \land B \in ℋ) \land A \in ℋ) \to (x \cdot_{ih} B) \cdot_{ℎ} A \in ℋ$
5	4	an31s	$⊢ ((A \in ℋ \land B \in ℋ) \land x \in ℋ) \to (x \cdot_{ih} B) \cdot_{ℎ} A \in ℋ$
6	1 5	fmpt3d	$⊢ (A \in ℋ \land B \in ℋ) \to (A ketbra B) : ℋ ⟶ ℋ$

Metamath Proof Explorer

Theorem kbop

Proof