Theorem hlmul0

Description: Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlmul0.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hlmul0.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
	hlmul0.5	⊢ 𝑍 = ( 0vec ‘ 𝑈 )
Assertion	hlmul0	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		hlmul0.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlmul0.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
3		hlmul0.5	⊢ 𝑍 = ( 0vec ‘ 𝑈 )
4		hlnv	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec )
5	1 2 3	nv0	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 )
6	4 5	sylan	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 )