Axiom ax-his1 29345

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 14741 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩, but our operation notation co 7255 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4565. Physicists use ⟨𝐵 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 30113. (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 29182	. . . 4 class ℋ
3	1, 2	wcel 2108	. . 3 wff 𝐴 ∈ ℋ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2108	. . 3 wff 𝐵 ∈ ℋ
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ)
7		csp 29185	. . . 4 class ·ih
8	1, 4, 7	co 7255	. . 3 class (𝐴 ·ih 𝐵)
9	4, 1, 7	co 7255	. . . 4 class (𝐵 ·ih 𝐴)
10		ccj 14735	. . . 4 class ∗
11	9, 10	cfv 6418	. . 3 class (∗‘(𝐵 ·ih 𝐴))
12	8, 11	wceq 1539	. 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  29349  his7  29353  his2sub2  29356  hire  29357  hi02  29360  his1i  29363  abshicom  29364  hial2eq2  29370  orthcom  29371  adjsym  30096  cnvadj  30155  adj2  30197