Theorem his1i 31523

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

Hypotheses

Ref	Expression
his1.1	⊢ 𝐴 ∈ ℋ
his1.2	⊢ 𝐵 ∈ ℋ

Assertion

Ref	Expression
his1i	⊢ (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))

Proof of Theorem his1i

Step	Hyp	Ref	Expression
1		his1.1	. 2 ⊢ 𝐴 ∈ ℋ
2		his1.2	. 2 ⊢ 𝐵 ∈ ℋ
3		ax-his1 31505	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7419  ∗ccj 15171   ℋchba 31342   ·_ih csp 31345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-his1 31505

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  normlem2  31534  bcseqi  31543  bcsiALT  31602  pjadjii  32097  lnopunilem1  32433  lnophmlem2  32440