Axiom ax-his1 30769

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 15056 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩, but our operation notation co 7412 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4635. Physicists use ⟨𝐵 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 31537. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-his1	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))

Detailed syntax breakdown of Axiom ax-his1

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		chba 30606	. . . 4 class ℋ
3	1, 2	wcel 2105	. . 3 wff 𝐴 ∈ ℋ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2105	. . 3 wff 𝐵 ∈ ℋ
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ)
7		csp 30609	. . . 4 class ·ih
8	1, 4, 7	co 7412	. . 3 class (𝐴 ·ih 𝐵)
9	4, 1, 7	co 7412	. . . 4 class (𝐵 ·ih 𝐴)
10		ccj 15050	. . . 4 class ∗
11	9, 10	cfv 6543	. . 3 class (∗‘(𝐵 ·ih 𝐴))
12	8, 11	wceq 1540	. 2 wff (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))
13	6, 12	wi 4	1 wff ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))

This axiom is referenced by:  his5  30773  his7  30777  his2sub2  30780  hire  30781  hi02  30784  his1i  30787  abshicom  30788  hial2eq2  30794  orthcom  30795  adjsym  31520  cnvadj  31579  adj2  31621