Step |
Hyp |
Ref |
Expression |
1 |
|
axhil.1 |
|- U = <. <. +h , .h >. , normh >. |
2 |
|
axhil.2 |
|- U e. CHilOLD |
3 |
|
df-hba |
|- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. ) |
4 |
1
|
fveq2i |
|- ( BaseSet ` U ) = ( BaseSet ` <. <. +h , .h >. , normh >. ) |
5 |
3 4
|
eqtr4i |
|- ~H = ( BaseSet ` U ) |
6 |
2
|
hlnvi |
|- U e. NrmCVec |
7 |
1 6
|
h2hva |
|- +h = ( +v ` U ) |
8 |
|
df-h0v |
|- 0h = ( 0vec ` <. <. +h , .h >. , normh >. ) |
9 |
1
|
fveq2i |
|- ( 0vec ` U ) = ( 0vec ` <. <. +h , .h >. , normh >. ) |
10 |
8 9
|
eqtr4i |
|- 0h = ( 0vec ` U ) |
11 |
5 7 10
|
hladdid |
|- ( ( U e. CHilOLD /\ A e. ~H ) -> ( A +h 0h ) = A ) |
12 |
2 11
|
mpan |
|- ( A e. ~H -> ( A +h 0h ) = A ) |