Step |
Hyp |
Ref |
Expression |
1 |
|
cnex |
|- CC e. _V |
2 |
|
ax-addf |
|- + : ( CC X. CC ) --> CC |
3 |
|
addass |
|- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) ) |
4 |
|
0cn |
|- 0 e. CC |
5 |
|
addid2 |
|- ( x e. CC -> ( 0 + x ) = x ) |
6 |
|
negcl |
|- ( x e. CC -> -u x e. CC ) |
7 |
|
addcom |
|- ( ( x e. CC /\ -u x e. CC ) -> ( x + -u x ) = ( -u x + x ) ) |
8 |
6 7
|
mpdan |
|- ( x e. CC -> ( x + -u x ) = ( -u x + x ) ) |
9 |
|
negid |
|- ( x e. CC -> ( x + -u x ) = 0 ) |
10 |
8 9
|
eqtr3d |
|- ( x e. CC -> ( -u x + x ) = 0 ) |
11 |
1 2 3 4 5 6 10
|
isgrpoi |
|- + e. GrpOp |
12 |
2
|
fdmi |
|- dom + = ( CC X. CC ) |
13 |
|
addcom |
|- ( ( x e. CC /\ y e. CC ) -> ( x + y ) = ( y + x ) ) |
14 |
11 12 13
|
isabloi |
|- + e. AbelOp |