Step |
Hyp |
Ref |
Expression |
1 |
|
resspsr.s |
|- S = ( I mPwSer R ) |
2 |
|
resspsr.h |
|- H = ( R |`s T ) |
3 |
|
resspsr.u |
|- U = ( I mPwSer H ) |
4 |
|
resspsr.b |
|- B = ( Base ` U ) |
5 |
|
resspsr.p |
|- P = ( S |`s B ) |
6 |
|
resspsr.2 |
|- ( ph -> T e. ( SubRing ` R ) ) |
7 |
|
fvex |
|- ( Base ` R ) e. _V |
8 |
2
|
subrgbas |
|- ( T e. ( SubRing ` R ) -> T = ( Base ` H ) ) |
9 |
6 8
|
syl |
|- ( ph -> T = ( Base ` H ) ) |
10 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
11 |
10
|
subrgss |
|- ( T e. ( SubRing ` R ) -> T C_ ( Base ` R ) ) |
12 |
6 11
|
syl |
|- ( ph -> T C_ ( Base ` R ) ) |
13 |
9 12
|
eqsstrrd |
|- ( ph -> ( Base ` H ) C_ ( Base ` R ) ) |
14 |
13
|
adantr |
|- ( ( ph /\ I e. _V ) -> ( Base ` H ) C_ ( Base ` R ) ) |
15 |
|
mapss |
|- ( ( ( Base ` R ) e. _V /\ ( Base ` H ) C_ ( Base ` R ) ) -> ( ( Base ` H ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) C_ ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
16 |
7 14 15
|
sylancr |
|- ( ( ph /\ I e. _V ) -> ( ( Base ` H ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) C_ ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
17 |
|
eqid |
|- ( Base ` H ) = ( Base ` H ) |
18 |
|
eqid |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
19 |
|
simpr |
|- ( ( ph /\ I e. _V ) -> I e. _V ) |
20 |
3 17 18 4 19
|
psrbas |
|- ( ( ph /\ I e. _V ) -> B = ( ( Base ` H ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
21 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
22 |
1 10 18 21 19
|
psrbas |
|- ( ( ph /\ I e. _V ) -> ( Base ` S ) = ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
23 |
16 20 22
|
3sstr4d |
|- ( ( ph /\ I e. _V ) -> B C_ ( Base ` S ) ) |
24 |
|
reldmpsr |
|- Rel dom mPwSer |
25 |
24
|
ovprc1 |
|- ( -. I e. _V -> ( I mPwSer H ) = (/) ) |
26 |
3 25
|
syl5eq |
|- ( -. I e. _V -> U = (/) ) |
27 |
26
|
adantl |
|- ( ( ph /\ -. I e. _V ) -> U = (/) ) |
28 |
27
|
fveq2d |
|- ( ( ph /\ -. I e. _V ) -> ( Base ` U ) = ( Base ` (/) ) ) |
29 |
|
base0 |
|- (/) = ( Base ` (/) ) |
30 |
28 4 29
|
3eqtr4g |
|- ( ( ph /\ -. I e. _V ) -> B = (/) ) |
31 |
|
0ss |
|- (/) C_ ( Base ` S ) |
32 |
30 31
|
eqsstrdi |
|- ( ( ph /\ -. I e. _V ) -> B C_ ( Base ` S ) ) |
33 |
23 32
|
pm2.61dan |
|- ( ph -> B C_ ( Base ` S ) ) |
34 |
5 21
|
ressbas2 |
|- ( B C_ ( Base ` S ) -> B = ( Base ` P ) ) |
35 |
33 34
|
syl |
|- ( ph -> B = ( Base ` P ) ) |