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Axiom ax-his1 28792
Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 14456 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written 𝐴, 𝐵, but our operation notation co 7150 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4571. Physicists use 𝐵𝐴, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 29560. (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
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 chba 28629 . . . 4 class
31, 2wcel 2107 . . 3 wff 𝐴 ∈ ℋ
4 cB . . . 4 class 𝐵
54, 2wcel 2107 . . 3 wff 𝐵 ∈ ℋ
63, 5wa 396 . 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ)
7 csp 28632 . . . 4 class ·ih
81, 4, 7co 7150 . . 3 class (𝐴 ·ih 𝐵)
94, 1, 7co 7150 . . . 4 class (𝐵 ·ih 𝐴)
10 ccj 14450 . . . 4 class
119, 10cfv 6354 . . 3 class (∗‘(𝐵 ·ih 𝐴))
128, 11wceq 1530 . 2 wff (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))
136, 12wi 4 1 wff ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
Colors of variables: wff setvar class
This axiom is referenced by:  his5  28796  his7  28800  his2sub2  28803  hire  28804  hi02  28807  his1i  28810  abshicom  28811  hial2eq2  28817  orthcom  28818  adjsym  29543  cnvadj  29602  adj2  29644
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