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 = ( +v ` U )
hladdid.5
|- Z = ( 0vec ` U )
Assertion hladdid
|- ( ( U e. CHilOLD /\ A e. 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 e. CHilOLD -> U e. NrmCVec )
5 1 2 3 nv0rid
 |-  ( ( U e. NrmCVec /\ A e. X ) -> ( A G Z ) = A )
6 4 5 sylan
 |-  ( ( U e. CHilOLD /\ A e. X ) -> ( A G Z ) = A )