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Metamath Proof Explorer

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 ) )