Metamath Proof Explorer


Theorem nvm

Description: Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008) (New usage is discouraged.)

Ref Expression
Hypotheses nvm.1 𝑋 = ( BaseSet ‘ 𝑈 )
nvm.2 𝐺 = ( +𝑣𝑈 )
nvm.3 𝑀 = ( −𝑣𝑈 )
nvm.6 𝑁 = ( /𝑔𝐺 )
Assertion nvm ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝑀 𝐵 ) = ( 𝐴 𝑁 𝐵 ) )

Proof

Step Hyp Ref Expression
1 nvm.1 𝑋 = ( BaseSet ‘ 𝑈 )
2 nvm.2 𝐺 = ( +𝑣𝑈 )
3 nvm.3 𝑀 = ( −𝑣𝑈 )
4 nvm.6 𝑁 = ( /𝑔𝐺 )
5 2 3 vsfval 𝑀 = ( /𝑔𝐺 )
6 5 4 eqtr4i 𝑀 = 𝑁
7 6 oveqi ( 𝐴 𝑀 𝐵 ) = ( 𝐴 𝑁 𝐵 )
8 7 a1i ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝑀 𝐵 ) = ( 𝐴 𝑁 𝐵 ) )