Axiom ax-his1 10374

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that *` x is the complex conjugate cjval 7808 of x. In the literature, the inner product of A and B is usually written <.A, B>., but our operation notation co 4695 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 2877. Physicists use <.B | A>., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 11205.

Assertion

Ref	Expression
ax-his1	|- ((A e. ~H /\ B e. ~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 10212	. . . 4 class ~H
3	1, 2	wcel 1138	. . 3 wff A e. ~H
4		cB	. . . 4 class B
5	4, 2	wcel 1138	. . 3 wff B e. ~H
6	3, 5	wa 239	. 2 wff (A e. ~H /\ B e. ~H)
7		csp 10217	. . . 4 class .ih
8	1, 4, 7	co 4695	. . 3 class (A .ih B)
9	4, 1, 7	co 4695	. . . 4 class (B .ih A)
10		ccj 7794	. . . 4 class *
11	9, 10	cfv 3809	. . 3 class (*` (B .ih A))
12	8, 11	wceq 1136	. 2 wff (A .ih B) = (*` (B .ih A))
13	6, 12	wi 3	1 wff ((A e. ~H /\ B e. ~H) -> (A .ih B) = (*` (B .ih A)))

This axiom is referenced by:  his5 10378  his7 10381  his2sub2 10384  hire 10385  hi02 10388  his1i 10391  abshicom 10392  hial2eq2 10398  orthcom 10399  adjsym 11188  cnvadj 11245  adj2 11287

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