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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 ‾