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 = ( -v ` U ) |
||
nvmeq0.5 | |- Z = ( 0vec ` U ) |
||
Assertion | nvmid | |- ( ( U e. NrmCVec /\ A e. X ) -> ( A M A ) = Z ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvmeq0.1 | |- X = ( BaseSet ` U ) |
|
2 | nvmeq0.3 | |- M = ( -v ` U ) |
|
3 | nvmeq0.5 | |- Z = ( 0vec ` U ) |
|
4 | eqid | |- A = A |
|
5 | 1 2 3 | nvmeq0 | |- ( ( U e. NrmCVec /\ A e. X /\ A e. X ) -> ( ( A M A ) = Z <-> A = A ) ) |
6 | 5 | 3anidm23 | |- ( ( U e. NrmCVec /\ A e. X ) -> ( ( A M A ) = Z <-> A = A ) ) |
7 | 4 6 | mpbiri | |- ( ( U e. NrmCVec /\ A e. X ) -> ( A M A ) = Z ) |