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Metamath Proof Explorer

Theorem speccl

Description: The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	speccl	$⊢ T : ℋ ⟶ ℋ \to Lambda (T) \subseteq ℂ$

Step	Hyp	Ref	Expression
1		specval	$⊢ T : ℋ ⟶ ℋ \to Lambda (T) = \{x \in ℂ \| \neg (T -_{op} (x \cdot_{op} ({I ↾}_{ℋ}))) : ℋ \underset{1-1}{⟶} ℋ\}$
2		ssrab2	$⊢ \{x \in ℂ \| \neg (T -_{op} (x \cdot_{op} ({I ↾}_{ℋ}))) : ℋ \underset{1-1}{⟶} ℋ\} \subseteq ℂ$
3	1 2	eqsstrdi	$⊢ T : ℋ ⟶ ℋ \to Lambda (T) \subseteq ℂ$