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Metamath Proof Explorer

Theorem lnop0i

Description: The value of a linear Hilbert space operator at zero is zero. Remark in Beran p. 99. (Contributed by NM, 11-May-2005) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	lnopl.1	⊢ 𝑇 ∈ LinOp
	Assertion	lnop0i	⊢ ( 𝑇 ‘ 0_ℎ ) = 0_ℎ

Proof

Step	Hyp	Ref	Expression
1		lnopl.1	⊢ 𝑇 ∈ LinOp
2		lnop0	⊢ ( 𝑇 ∈ LinOp → ( 𝑇 ‘ 0_ℎ ) = 0_ℎ )
3	1 2	ax-mp	⊢ ( 𝑇 ‘ 0_ℎ ) = 0_ℎ