ax-his1

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

		Ref	Expression
	Assertion	ax-his1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·_ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·_ih 𝐴 ) ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	⊢ 𝐴
1		chba	⊢ ℋ
2	0 1	wcel	⊢ 𝐴 ∈ ℋ
3		cB	⊢ 𝐵
4	3 1	wcel	⊢ 𝐵 ∈ ℋ
5	2 4	wa	⊢ ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ )
6		csp	⊢ ·_ih
7	0 3 6	co	⊢ ( 𝐴 ·_ih 𝐵 )
8		ccj	⊢ ∗
9	3 0 6	co	⊢ ( 𝐵 ·_ih 𝐴 )
10	9 8	cfv	⊢ ( ∗ ‘ ( 𝐵 ·_ih 𝐴 ) )
11	7 10	wceq	⊢ ( 𝐴 ·_ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·_ih 𝐴 ) )
12	5 11	wi	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·_ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·_ih 𝐴 ) ) )

Metamath Proof Explorer

Axiom ax-his1

Detailed syntax breakdown