Axiom ax-his1 8870

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗ ‘x is the complex conjugate cjvalt 6695 of x. In the literature, the inner product of A and B is usually written ⟨A, B⟩, but our operation notation co 3948 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 2406. Physicists use ⟨B∣A⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 9693.

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		chil 8727	. . . 4 class ℋ
3	1, 2	wcel 955	. . 3 wff A ∈ ℋ
4		cB	. . . 4 class B
5	4, 2	wcel 955	. . 3 wff B ∈ ℋ
6	3, 5	wa 223	. 2 wff (A ∈ ℋ ⋀ B ∈ ℋ )
7		csp 8732	. . . 4 class ·ih
8	1, 4, 7	co 3948	. . 3 class (A ·ih B)
9	4, 1, 7	co 3948	. . . 4 class (B ·ih A)
10		ccj 6680	. . . 4 class ∗
11	9, 10	cfv 3172	. . 3 class (∗ ‘(B ·ih A))
12	8, 11	wceq 953	. 2 wff (A ·ih B) = (∗ ‘(B ·ih A))
13	6, 12	wi 3	1 wff ((A ∈ ℋ ⋀ B ∈ ℋ ) → (A ·ih B) = (∗ ‘(B ·ih A)))

This axiom is referenced by:  his5t 8874  his7t 8877  his2sub2t 8880  hiret 8881  hi02t 8884  his1 8887  abshicomt 8888  hial2eq2t 8894  orthcom 8895  adjsymt 9676  cnvadj 9733  adj2t 9774

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