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 ∈ ℋ → ( 𝐴 ·ih 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝐴 ·ih 𝐴 ) = ( 𝐴 ·ih 𝐴 )
2		hire	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐴 ) ∈ ℝ ↔ ( 𝐴 ·ih 𝐴 ) = ( 𝐴 ·ih 𝐴 ) ) )
3	1 2	mpbiri	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ·ih 𝐴 ) ∈ ℝ )
4	3	anidms	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 ·ih 𝐴 ) ∈ ℝ )