Theorem hlmulid

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

	Ref	Expression
Hypotheses	hlmulf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hlmulf.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
Assertion	hlmulid	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ) → ( 1 𝑆 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		hlmulf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlmulf.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
3		hlnv	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec )
4	1 2	nvsid	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 1 𝑆 𝐴 ) = 𝐴 )
5	3 4	sylan	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ) → ( 1 𝑆 𝐴 ) = 𝐴 )