Metamath Proof Explorer


Theorem hladdid

Description: Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

Ref Expression
Hypotheses hladdid.1 𝑋 = ( BaseSet ‘ 𝑈 )
hladdid.2 𝐺 = ( +𝑣𝑈 )
hladdid.5 𝑍 = ( 0vec𝑈 )
Assertion hladdid ( ( 𝑈 ∈ CHilOLD𝐴𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 hladdid.1 𝑋 = ( BaseSet ‘ 𝑈 )
2 hladdid.2 𝐺 = ( +𝑣𝑈 )
3 hladdid.5 𝑍 = ( 0vec𝑈 )
4 hlnv ( 𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec )
5 1 2 3 nv0rid ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )
6 4 5 sylan ( ( 𝑈 ∈ CHilOLD𝐴𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )