Metamath Proof Explorer


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 = ( ·𝑠OLD𝑈 )
hlmul0.5 𝑍 = ( 0vec𝑈 )
Assertion hlmul0 ( ( 𝑈 ∈ CHilOLD𝐴𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 )

Proof

Step Hyp Ref Expression
1 hlmul0.1 𝑋 = ( BaseSet ‘ 𝑈 )
2 hlmul0.4 𝑆 = ( ·𝑠OLD𝑈 )
3 hlmul0.5 𝑍 = ( 0vec𝑈 )
4 hlnv ( 𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec )
5 1 2 3 nv0 ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 )
6 4 5 sylan ( ( 𝑈 ∈ CHilOLD𝐴𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 )