Theorem his1i

Description: Conjugate law for inner product. Postulate (S1) of Beran p. 95. (Contributed by NM, 15-May-2005) (New usage is discouraged.)

	Ref	Expression
Hypotheses	his1.1	⊢ 𝐴 ∈ ℋ
	his1.2	⊢ 𝐵 ∈ ℋ
Assertion	his1i	⊢ ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		his1.1	⊢ 𝐴 ∈ ℋ
2		his1.2	⊢ 𝐵 ∈ ℋ
3		ax-his1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) )