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 = ( +𝑣 ‘ 𝑈 )
	hladdid.5	⊢ 𝑍 = ( 0vec ‘ 𝑈 )
Assertion	hladdid	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		hladdid.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hladdid.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		hladdid.5	⊢ 𝑍 = ( 0vec ‘ 𝑈 )
4		hlnv	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec )
5	1 2 3	nv0rid	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )
6	4 5	sylan	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )