Axiom ax-his1 8949

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that *` x is the complex conjugate cjvalt 6763 of x. In the literature, the inner product of A and B is usually written <.A, B>., but our operation notation co 3963 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 2416. Physicists use <.B | A>., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 9776.

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 8788	. . . 4 class H~
3	1, 2	wcel 958	. . 3 wff A e. H~
4		cB	. . . 4 class B
5	4, 2	wcel 958	. . 3 wff B e. H~
6	3, 5	wa 223	. 2 wff (A e. H~ /\ B e. H~)
7		csp 8793	. . . 4 class .ih
8	1, 4, 7	co 3963	. . 3 class (A .ih B)
9	4, 1, 7	co 3963	. . . 4 class (B .ih A)
10		ccj 6749	. . . 4 class *
11	9, 10	cfv 3182	. . 3 class (*` (B .ih A))
12	8, 11	wceq 956	. 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 8953  his7t 8956  his2sub2t 8959  hiret 8960  hi02t 8963  his1 8966  abshicomt 8967  hial2eq2t 8973  orthcom 8974  adjsymt 9759  cnvadj 9816  adj2t 9858

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