Axiom ax-his1 8888

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗ ‘x is the complex conjugate cjvalt 6703 of x. In the literature, the inner product of A and B is usually written ⟨A, B⟩, but our operation notation co 3954 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 2412. Physicists use ⟨B∣A⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 9716.

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 956	. . 3 wff A ∈ ℋ
4		cB	. . . 4 class B
5	4, 2	wcel 956	. . 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 3954	. . 3 class (A ·ih B)
9	4, 1, 7	co 3954	. . . 4 class (B ·ih A)
10		ccj 6688	. . . 4 class ∗
11	9, 10	cfv 3177	. . 3 class (∗ ‘(B ·ih A))
12	8, 11	wceq 954	. 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 8892  his7t 8895  his2sub2t 8898  hiret 8899  hi02t 8902  his1 8905  abshicomt 8906  hial2eq2t 8912  orthcom 8913  adjsymt 9699  cnvadj 9756  adj2t 9797

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