Axiom ax-his1 31011

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 15068 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩, but our operation notation co 7387 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4596. Physicists use ⟨𝐵 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 31779. (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 30848	. . . 4 class ℋ
3	1, 2	wcel 2109	. . 3 wff 𝐴 ∈ ℋ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2109	. . 3 wff 𝐵 ∈ ℋ
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ)
7		csp 30851	. . . 4 class ·ih
8	1, 4, 7	co 7387	. . 3 class (𝐴 ·ih 𝐵)
9	4, 1, 7	co 7387	. . . 4 class (𝐵 ·ih 𝐴)
10		ccj 15062	. . . 4 class ∗
11	9, 10	cfv 6511	. . 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  31015  his7  31019  his2sub2  31022  hire  31023  hi02  31026  his1i  31029  abshicom  31030  hial2eq2  31036  orthcom  31037  adjsym  31762  cnvadj  31821  adj2  31863