Axiom ax-his1 9098

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that *` x is the complex conjugate cjvalt 6646 of x. In the literature, the inner product of A and B is usually written <.A, B>., but our operation notation co 3902 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 2387. Physicists use <.B | A>., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 9907.

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 8968	. . . 4 class H~
3	1, 2	wcel 1105	. . 3 wff A e. H~
4		cB	. . . 4 class B
5	4, 2	wcel 1105	. . 3 wff B e. H~
6	3, 5	wa 223	. 2 wff (A e. H~ /\ B e. H~)
7		csp 8973	. . . 4 class .ih
8	1, 4, 7	co 3902	. . 3 class (A .ih B)
9	4, 1, 7	co 3902	. . . 4 class (B .ih A)
10		ccj 6631	. . . 4 class *
11	9, 10	cfv 3145	. . 3 class (*` (B .ih A))
12	8, 11	wceq 1099	. 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:  his5t 9102  his7t 9105  his2sub2t 9108  hiret 9109  hi02t 9112  his1 9115  abshicomt 9116  hial2eq2t 9122  orthcom 9123  adjsymt 9890  cnvadj 9947  adj2t 9988

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