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Metamath Proof Explorer

Theorem hiidrcl

Description: Real closure of inner product with self. (Contributed by NM, 29-May-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	hiidrcl	$⊢ A \in ℋ \to A \cdot_{ih} A \in ℝ$

Step	Hyp	Ref	Expression
1		eqid	$⊢ A \cdot_{ih} A = A \cdot_{ih} A$
2		hire	$⊢ (A \in ℋ \land A \in ℋ) \to (A \cdot_{ih} A \in ℝ \leftrightarrow A \cdot_{ih} A = A \cdot_{ih} A)$
3	1 2	mpbiri	$⊢ (A \in ℋ \land A \in ℋ) \to A \cdot_{ih} A \in ℝ$
4	3	anidms	$⊢ A \in ℋ \to A \cdot_{ih} A \in ℝ$