Axiom ax-his1 27939

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that * ` x is the complex conjugate cjval 13842 of x. In the literature, the inner product of A and B is usually written <. A , B >., but our operation notation co 6650 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4184. Physicists use <. B | A >., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 28709. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

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 27776	. . . 4 class ~H
3	1, 2	wcel 1990	. . 3 wff A e. ~H
4		cB	. . . 4 class B
5	4, 2	wcel 1990	. . 3 wff B e. ~H
6	3, 5	wa 384	. 2 wff ( A e. ~H /\ B e. ~H )
7		csp 27779	. . . 4 class .ih
8	1, 4, 7	co 6650	. . 3 class ( A .ih B )
9	4, 1, 7	co 6650	. . . 4 class ( B .ih A )
10		ccj 13836	. . . 4 class *
11	9, 10	cfv 5888	. . 3 class ( * ` ( B .ih A ) )
12	8, 11	wceq 1483	. 2 wff ( A .ih B ) = ( * ` ( B .ih A ) )
13	6, 12	wi 4	1 wff ( ( A e. ~H /\ B e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) )

This axiom is referenced by:  his5  27943  his7  27947  his2sub2  27950  hire  27951  hi02  27954  his1i  27957  abshicom  27958  hial2eq2  27964  orthcom  27965  adjsym  28692  cnvadj  28751  adj2  28793