Axiom ax-his1 31375

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 15153 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩, but our operation notation co 7411 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4601. Physicists use ⟨𝐵 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 32143. (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 31212	. . . 4 class ℋ
3	1, 2	wcel 2149	. . 3 wff 𝐴 ∈ ℋ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2149	. . 3 wff 𝐵 ∈ ℋ
6	3, 5	wa 400	. 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ)
7		csp 31215	. . . 4 class ·ih
8	1, 4, 7	co 7411	. . 3 class (𝐴 ·ih 𝐵)
9	4, 1, 7	co 7411	. . . 4 class (𝐵 ·ih 𝐴)
10		ccj 15147	. . . 4 class ∗
11	9, 10	cfv 6537	. . 3 class (∗‘(𝐵 ·ih 𝐴))
12	8, 11	wceq 1567	. 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  31379  his7  31383  his2sub2  31386  hire  31387  hi02  31390  his1i  31393  abshicom  31394  hial2eq2  31400  orthcom  31401  adjsym  32126  cnvadj  32185  adj2  32227