Metamath Proof Explorer


Theorem nvmid

Description: A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)

Ref Expression
Hypotheses nvmeq0.1 𝑋 = ( BaseSet ‘ 𝑈 )
nvmeq0.3 𝑀 = ( −𝑣𝑈 )
nvmeq0.5 𝑍 = ( 0vec𝑈 )
Assertion nvmid ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ) → ( 𝐴 𝑀 𝐴 ) = 𝑍 )

Proof

Step Hyp Ref Expression
1 nvmeq0.1 𝑋 = ( BaseSet ‘ 𝑈 )
2 nvmeq0.3 𝑀 = ( −𝑣𝑈 )
3 nvmeq0.5 𝑍 = ( 0vec𝑈 )
4 eqid 𝐴 = 𝐴
5 1 2 3 nvmeq0 ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐴𝑋 ) → ( ( 𝐴 𝑀 𝐴 ) = 𝑍𝐴 = 𝐴 ) )
6 5 3anidm23 ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ) → ( ( 𝐴 𝑀 𝐴 ) = 𝑍𝐴 = 𝐴 ) )
7 4 6 mpbiri ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ) → ( 𝐴 𝑀 𝐴 ) = 𝑍 )