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Metamath Proof Explorer

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 ) ) )