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Metamath Proof Explorer

Theorem bracnln

Description: A bra is a continuous linear functional. (Contributed by NM, 30-May-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	bracnln	$⊢ A \in ℋ \to bra (A) \in (LinFn \cap ContFn)$

Step	Hyp	Ref	Expression
1		bra11	$⊢ bra : ℋ \underset{1-1 onto}{⟶} LinFn \cap ContFn$
2		f1of	$⊢ bra : ℋ \underset{1-1 onto}{⟶} LinFn \cap ContFn \to bra : ℋ ⟶ LinFn \cap ContFn$
3	1 2	ax-mp	$⊢ bra : ℋ ⟶ LinFn \cap ContFn$
4	3	ffvelcdmi	$⊢ A \in ℋ \to bra (A) \in (LinFn \cap ContFn)$