Step |
Hyp |
Ref |
Expression |
1 |
|
mhmcopsr.p |
|- P = ( I mPwSer R ) |
2 |
|
mhmcopsr.q |
|- Q = ( I mPwSer S ) |
3 |
|
mhmcopsr.b |
|- B = ( Base ` P ) |
4 |
|
mhmcopsr.c |
|- C = ( Base ` Q ) |
5 |
|
mhmcopsr.h |
|- ( ph -> H e. ( R MndHom S ) ) |
6 |
|
mhmcopsr.f |
|- ( ph -> F e. B ) |
7 |
|
fvexd |
|- ( ph -> ( Base ` S ) e. _V ) |
8 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
9 |
8
|
rabex |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } e. _V |
10 |
9
|
a1i |
|- ( ph -> { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } e. _V ) |
11 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
12 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
13 |
11 12
|
mhmf |
|- ( H e. ( R MndHom S ) -> H : ( Base ` R ) --> ( Base ` S ) ) |
14 |
5 13
|
syl |
|- ( ph -> H : ( Base ` R ) --> ( Base ` S ) ) |
15 |
|
eqid |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
16 |
1 11 15 3 6
|
psrelbas |
|- ( ph -> F : { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } --> ( Base ` R ) ) |
17 |
14 16
|
fcod |
|- ( ph -> ( H o. F ) : { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } --> ( Base ` S ) ) |
18 |
7 10 17
|
elmapdd |
|- ( ph -> ( H o. F ) e. ( ( Base ` S ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
19 |
|
reldmpsr |
|- Rel dom mPwSer |
20 |
19 1 3
|
elbasov |
|- ( F e. B -> ( I e. _V /\ R e. _V ) ) |
21 |
6 20
|
syl |
|- ( ph -> ( I e. _V /\ R e. _V ) ) |
22 |
21
|
simpld |
|- ( ph -> I e. _V ) |
23 |
2 12 15 4 22
|
psrbas |
|- ( ph -> C = ( ( Base ` S ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
24 |
18 23
|
eleqtrrd |
|- ( ph -> ( H o. F ) e. C ) |