Step |
Hyp |
Ref |
Expression |
1 |
|
base0 |
|- (/) = ( Base ` (/) ) |
2 |
|
eqid |
|- ( +g ` (/) ) = ( +g ` (/) ) |
3 |
|
eqid |
|- ( 0g ` (/) ) = ( 0g ` (/) ) |
4 |
1 2 3
|
grpidval |
|- ( 0g ` (/) ) = ( iota e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) ) |
5 |
|
noel |
|- -. e e. (/) |
6 |
5
|
intnanr |
|- -. ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) |
7 |
6
|
nex |
|- -. E. e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) |
8 |
|
euex |
|- ( E! e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) -> E. e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) ) |
9 |
7 8
|
mto |
|- -. E! e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) |
10 |
|
iotanul |
|- ( -. E! e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) -> ( iota e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) ) = (/) ) |
11 |
9 10
|
ax-mp |
|- ( iota e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) ) = (/) |
12 |
4 11
|
eqtr2i |
|- (/) = ( 0g ` (/) ) |