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 ∈ ℋ ∧ B ∈ 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 ∈ ℋ
4		cB	. . . 4 class B
5	4, 2	wcel 1092	. . 3 wff B ∈ ℋ
6	3, 5	wa 196	. 2 wff (A ∈ ℋ ∧ B ∈ 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 ∈ ℋ ∧ B ∈ 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

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