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