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 ‘ 0ℎ ) = 0ℎ

Proof

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