Step |
Hyp |
Ref |
Expression |
1 |
|
efmndtset.g |
|- G = ( EndoFMnd ` A ) |
2 |
|
efmndplusg.b |
|- B = ( Base ` G ) |
3 |
|
efmndplusg.p |
|- .+ = ( +g ` G ) |
4 |
1 2
|
efmndbas |
|- B = ( A ^m A ) |
5 |
|
eqid |
|- ( f e. B , g e. B |-> ( f o. g ) ) = ( f e. B , g e. B |-> ( f o. g ) ) |
6 |
|
eqid |
|- ( Xt_ ` ( A X. { ~P A } ) ) = ( Xt_ ` ( A X. { ~P A } ) ) |
7 |
1 4 5 6
|
efmnd |
|- ( A e. _V -> G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( f e. B , g e. B |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( A X. { ~P A } ) ) >. } ) |
8 |
7
|
fveq2d |
|- ( A e. _V -> ( +g ` G ) = ( +g ` { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( f e. B , g e. B |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( A X. { ~P A } ) ) >. } ) ) |
9 |
2
|
fvexi |
|- B e. _V |
10 |
9 9
|
mpoex |
|- ( f e. B , g e. B |-> ( f o. g ) ) e. _V |
11 |
|
eqid |
|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( f e. B , g e. B |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( A X. { ~P A } ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( f e. B , g e. B |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( A X. { ~P A } ) ) >. } |
12 |
11
|
topgrpplusg |
|- ( ( f e. B , g e. B |-> ( f o. g ) ) e. _V -> ( f e. B , g e. B |-> ( f o. g ) ) = ( +g ` { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( f e. B , g e. B |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( A X. { ~P A } ) ) >. } ) ) |
13 |
10 12
|
ax-mp |
|- ( f e. B , g e. B |-> ( f o. g ) ) = ( +g ` { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( f e. B , g e. B |-> ( f o. g ) ) >. , <. ( TopSet ` ndx ) , ( Xt_ ` ( A X. { ~P A } ) ) >. } ) |
14 |
8 3 13
|
3eqtr4g |
|- ( A e. _V -> .+ = ( f e. B , g e. B |-> ( f o. g ) ) ) |
15 |
|
fvprc |
|- ( -. A e. _V -> ( EndoFMnd ` A ) = (/) ) |
16 |
1 15
|
eqtrid |
|- ( -. A e. _V -> G = (/) ) |
17 |
16
|
fveq2d |
|- ( -. A e. _V -> ( +g ` G ) = ( +g ` (/) ) ) |
18 |
|
plusgid |
|- +g = Slot ( +g ` ndx ) |
19 |
18
|
str0 |
|- (/) = ( +g ` (/) ) |
20 |
17 3 19
|
3eqtr4g |
|- ( -. A e. _V -> .+ = (/) ) |
21 |
16
|
fveq2d |
|- ( -. A e. _V -> ( Base ` G ) = ( Base ` (/) ) ) |
22 |
|
base0 |
|- (/) = ( Base ` (/) ) |
23 |
21 2 22
|
3eqtr4g |
|- ( -. A e. _V -> B = (/) ) |
24 |
23
|
olcd |
|- ( -. A e. _V -> ( B = (/) \/ B = (/) ) ) |
25 |
|
0mpo0 |
|- ( ( B = (/) \/ B = (/) ) -> ( f e. B , g e. B |-> ( f o. g ) ) = (/) ) |
26 |
24 25
|
syl |
|- ( -. A e. _V -> ( f e. B , g e. B |-> ( f o. g ) ) = (/) ) |
27 |
20 26
|
eqtr4d |
|- ( -. A e. _V -> .+ = ( f e. B , g e. B |-> ( f o. g ) ) ) |
28 |
14 27
|
pm2.61i |
|- .+ = ( f e. B , g e. B |-> ( f o. g ) ) |