Axiom ax-his1 5045

Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that *` x is the complex conjugate cjvalt 4799 of x. In the literature, the inner product of A and B is usually written <.A, B>. but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 1815.

Assertion

Ref	Expression
ax-his1	|- ((A e. H~ /\ B e. H~) -> (A .i B) = (*` (B .i A)))

Detailed syntax breakdown of Axiom ax-his1

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		chil 4958	. . . 4 class H~
3	1, 2	wcel 1092	. . 3 wff A e. H~
4		cB	. . . 4 class B
5	4, 2	wcel 1092	. . 3 wff B e. H~
6	3, 5	wa 196	. 2 wff (A e. H~ /\ B e. H~)
7		csp 4963	. . . 4 class .i
8	1, 4, 7	co 3001	. . 3 class (A .i B)
9	4, 1, 7	co 3001	. . . 4 class (B .i A)
10		ccj 4788	. . . 4 class *
11	9, 10	cfv 2422	. . 3 class (*` (B .i A))
12	8, 11	wceq 1091	. 2 wff (A .i B) = (*` (B .i A))
13	6, 12	wi 2	1 wff ((A e. H~ /\ B e. H~) -> (A .i B) = (*` (B .i A)))

This axiom is referenced by:  his5 5050  his7 5051  hiidrclt 5053  hizer2t 5055  orthcom 5061  normlem2 5064  bcseq 5073  bcs 5101  pjthlem5 5229  pjthlem6 5230  pjthlem13 5237  pjadj 5564

metamath.org