Theorem bra0

Description: The Dirac bra of the zero vector. (Contributed by NM, 25-May-2006) (Revised by Mario Carneiro, 23-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	bra0	⊢ ( bra ‘ 0ℎ ) = ( ℋ × { 0 } )

Proof

Step	Hyp	Ref	Expression
1		ax-hv0cl	⊢ 0ℎ ∈ ℋ
2		brafval	⊢ ( 0ℎ ∈ ℋ → ( bra ‘ 0ℎ ) = ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 0ℎ ) ) )
3		hi02	⊢ ( 𝑥 ∈ ℋ → ( 𝑥 ·ih 0ℎ ) = 0 )
4	3	mpteq2ia	⊢ ( 𝑥 ∈ ℋ ↦ ( 𝑥 ·ih 0ℎ ) ) = ( 𝑥 ∈ ℋ ↦ 0 )
5	2 4	eqtrdi	⊢ ( 0ℎ ∈ ℋ → ( bra ‘ 0ℎ ) = ( 𝑥 ∈ ℋ ↦ 0 ) )
6	1 5	ax-mp	⊢ ( bra ‘ 0ℎ ) = ( 𝑥 ∈ ℋ ↦ 0 )
7		fconstmpt	⊢ ( ℋ × { 0 } ) = ( 𝑥 ∈ ℋ ↦ 0 )
8	6 7	eqtr4i	⊢ ( bra ‘ 0ℎ ) = ( ℋ × { 0 } )