Theorem hire

Description: A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	hire	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) ∈ ℝ ↔ ( 𝐴 ·ih 𝐵 ) = ( 𝐵 ·ih 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hicl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ih 𝐵 ) ∈ ℂ )
2		cjreb	⊢ ( ( 𝐴 ·ih 𝐵 ) ∈ ℂ → ( ( 𝐴 ·ih 𝐵 ) ∈ ℝ ↔ ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) = ( 𝐴 ·ih 𝐵 ) ) )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) ∈ ℝ ↔ ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) = ( 𝐴 ·ih 𝐵 ) ) )
4		eqcom	⊢ ( ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) = ( 𝐴 ·ih 𝐵 ) ↔ ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) )
5	3 4	bitrdi	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) ∈ ℝ ↔ ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) ) )
6		ax-his1	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐵 ·ih 𝐴 ) = ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) )
7	6	ancoms	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐵 ·ih 𝐴 ) = ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) )
8	7	eqeq2d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) = ( 𝐵 ·ih 𝐴 ) ↔ ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) ) )
9	5 8	bitr4d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) ∈ ℝ ↔ ( 𝐴 ·ih 𝐵 ) = ( 𝐵 ·ih 𝐴 ) ) )