Step |
Hyp |
Ref |
Expression |
1 |
|
estrcbas.c |
|- C = ( ExtStrCat ` U ) |
2 |
|
estrcbas.u |
|- ( ph -> U e. V ) |
3 |
|
estrcco.o |
|- .x. = ( comp ` C ) |
4 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
5 |
1 2 4
|
estrchomfval |
|- ( ph -> ( Hom ` C ) = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) ) |
6 |
|
eqidd |
|- ( ph -> ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) ) |
7 |
1 2 5 6
|
estrcval |
|- ( ph -> C = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } ) |
8 |
|
catstr |
|- { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } Struct <. 1 , ; 1 5 >. |
9 |
|
ccoid |
|- comp = Slot ( comp ` ndx ) |
10 |
|
snsstp3 |
|- { <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } C_ { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } |
11 |
2 2
|
xpexd |
|- ( ph -> ( U X. U ) e. _V ) |
12 |
|
mpoexga |
|- ( ( ( U X. U ) e. _V /\ U e. V ) -> ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) e. _V ) |
13 |
11 2 12
|
syl2anc |
|- ( ph -> ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) e. _V ) |
14 |
7 8 9 10 13 3
|
strfv3 |
|- ( ph -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) ) |