Theorem brafval

Description: The bra of a vector, expressed as <. A | in Dirac notation. See df-bra . (Contributed by NM, 15-May-2006) (Revised by Mario Carneiro, 23-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	brafval	⊢ ( 𝐴 ∈ ℋ → ( bra ‘ 𝐴 ) = ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝑦 = 𝐴 → ( 𝑥 ·ih 𝑦 ) = ( 𝑥 ·ih 𝐴 ) )
2	1	mpteq2dv	⊢ ( 𝑦 = 𝐴 → ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝑦 ) ) = ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐴 ) ) )
3		df-bra	⊢ bra = ( 𝑦 ∈ ℋ ↦ ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝑦 ) ) )
4		ax-hilex	⊢ ℋ ∈ V
5	4	mptex	⊢ ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐴 ) ) ∈ V
6	2 3 5	fvmpt	⊢ ( 𝐴 ∈ ℋ → ( bra ‘ 𝐴 ) = ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 𝐴 ) ) )