Axiom ax-his1 9225

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that *` x is the complex conjugate cjval 6964 of x. In the literature, the inner product of A and B is usually written <.A, B>., but our operation notation co 4021 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 2474. Physicists use <.B | A>., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 10056.

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 9063	. . . 4 class H~
3	1, 2	wcel 994	. . 3 wff A e. H~
4		cB	. . . 4 class B
5	4, 2	wcel 994	. . 3 wff B e. H~
6	3, 5	wa 221	. 2 wff (A e. H~ /\ B e. H~)
7		csp 9068	. . . 4 class .ih
8	1, 4, 7	co 4021	. . 3 class (A .ih B)
9	4, 1, 7	co 4021	. . . 4 class (B .ih A)
10		ccj 6950	. . . 4 class *
11	9, 10	cfv 3263	. . 3 class (*` (B .ih A))
12	8, 11	wceq 992	. 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 9229  his7 9232  his2sub2 9235  hire 9236  hi02 9239  his1i 9242  abshicom 9243  hial2eq2 9249  orthcom 9250  adjsym 10039  cnvadj 10096  adj2 10138

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