Step |
Hyp |
Ref |
Expression |
1 |
|
dfrngc2.c |
|- C = ( RngCat ` U ) |
2 |
|
dfrngc2.u |
|- ( ph -> U e. V ) |
3 |
|
dfrngc2.b |
|- ( ph -> B = ( U i^i Rng ) ) |
4 |
|
dfrngc2.h |
|- ( ph -> H = ( RngHomo |` ( B X. B ) ) ) |
5 |
|
dfrngc2.o |
|- ( ph -> .x. = ( comp ` ( ExtStrCat ` U ) ) ) |
6 |
1 2 3 4
|
rngcval |
|- ( ph -> C = ( ( ExtStrCat ` U ) |`cat H ) ) |
7 |
|
eqid |
|- ( ( ExtStrCat ` U ) |`cat H ) = ( ( ExtStrCat ` U ) |`cat H ) |
8 |
|
fvexd |
|- ( ph -> ( ExtStrCat ` U ) e. _V ) |
9 |
|
inex1g |
|- ( U e. V -> ( U i^i Rng ) e. _V ) |
10 |
2 9
|
syl |
|- ( ph -> ( U i^i Rng ) e. _V ) |
11 |
3 10
|
eqeltrd |
|- ( ph -> B e. _V ) |
12 |
3 4
|
rnghmresfn |
|- ( ph -> H Fn ( B X. B ) ) |
13 |
7 8 11 12
|
rescval2 |
|- ( ph -> ( ( ExtStrCat ` U ) |`cat H ) = ( ( ( ExtStrCat ` U ) |`s B ) sSet <. ( Hom ` ndx ) , H >. ) ) |
14 |
|
eqid |
|- ( ExtStrCat ` U ) = ( ExtStrCat ` U ) |
15 |
|
eqidd |
|- ( ph -> ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) ) |
16 |
|
eqid |
|- ( comp ` ( ExtStrCat ` U ) ) = ( comp ` ( ExtStrCat ` U ) ) |
17 |
14 2 16
|
estrccofval |
|- ( ph -> ( comp ` ( ExtStrCat ` U ) ) = ( 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 ) ) ) ) |
18 |
5 17
|
eqtrd |
|- ( 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 ) ) ) ) |
19 |
14 2 15 18
|
estrcval |
|- ( ph -> ( ExtStrCat ` U ) = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. ( comp ` ndx ) , .x. >. } ) |
20 |
|
mpoexga |
|- ( ( U e. V /\ U e. V ) -> ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) e. _V ) |
21 |
2 2 20
|
syl2anc |
|- ( ph -> ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) e. _V ) |
22 |
|
fvexd |
|- ( ph -> ( comp ` ( ExtStrCat ` U ) ) e. _V ) |
23 |
5 22
|
eqeltrd |
|- ( ph -> .x. e. _V ) |
24 |
|
rnghmfn |
|- RngHomo Fn ( Rng X. Rng ) |
25 |
|
fnfun |
|- ( RngHomo Fn ( Rng X. Rng ) -> Fun RngHomo ) |
26 |
24 25
|
mp1i |
|- ( ph -> Fun RngHomo ) |
27 |
|
sqxpexg |
|- ( B e. _V -> ( B X. B ) e. _V ) |
28 |
11 27
|
syl |
|- ( ph -> ( B X. B ) e. _V ) |
29 |
|
resfunexg |
|- ( ( Fun RngHomo /\ ( B X. B ) e. _V ) -> ( RngHomo |` ( B X. B ) ) e. _V ) |
30 |
26 28 29
|
syl2anc |
|- ( ph -> ( RngHomo |` ( B X. B ) ) e. _V ) |
31 |
4 30
|
eqeltrd |
|- ( ph -> H e. _V ) |
32 |
|
inss1 |
|- ( U i^i Rng ) C_ U |
33 |
3 32
|
eqsstrdi |
|- ( ph -> B C_ U ) |
34 |
19 2 21 23 31 33
|
estrres |
|- ( ph -> ( ( ( ExtStrCat ` U ) |`s B ) sSet <. ( Hom ` ndx ) , H >. ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .x. >. } ) |
35 |
6 13 34
|
3eqtrd |
|- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .x. >. } ) |