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 |
|
eqid |
|- ( +g ` H ) = ( +g ` H ) |
8 |
|
eqid |
|- ( +g ` U ) = ( +g ` U ) |
9 |
|
simprl |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> X e. B ) |
10 |
|
simprr |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> Y e. B ) |
11 |
3 4 7 8 9 10
|
psradd |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` U ) Y ) = ( X oF ( +g ` H ) Y ) ) |
12 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
13 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
14 |
|
eqid |
|- ( +g ` S ) = ( +g ` S ) |
15 |
|
fvex |
|- ( Base ` R ) e. _V |
16 |
2
|
subrgbas |
|- ( T e. ( SubRing ` R ) -> T = ( Base ` H ) ) |
17 |
6 16
|
syl |
|- ( ph -> T = ( Base ` H ) ) |
18 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
19 |
18
|
subrgss |
|- ( T e. ( SubRing ` R ) -> T C_ ( Base ` R ) ) |
20 |
6 19
|
syl |
|- ( ph -> T C_ ( Base ` R ) ) |
21 |
17 20
|
eqsstrrd |
|- ( ph -> ( Base ` H ) C_ ( Base ` R ) ) |
22 |
|
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 } ) ) |
23 |
15 21 22
|
sylancr |
|- ( ph -> ( ( 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 } ) ) |
24 |
23
|
adantr |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( ( 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 } ) ) |
25 |
|
eqid |
|- ( Base ` H ) = ( Base ` H ) |
26 |
|
eqid |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
27 |
|
reldmpsr |
|- Rel dom mPwSer |
28 |
27 3 4
|
elbasov |
|- ( X e. B -> ( I e. _V /\ H e. _V ) ) |
29 |
28
|
ad2antrl |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( I e. _V /\ H e. _V ) ) |
30 |
29
|
simpld |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> I e. _V ) |
31 |
3 25 26 4 30
|
psrbas |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> B = ( ( Base ` H ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
32 |
1 18 26 12 30
|
psrbas |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( Base ` S ) = ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
33 |
24 31 32
|
3sstr4d |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> B C_ ( Base ` S ) ) |
34 |
33 9
|
sseldd |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> X e. ( Base ` S ) ) |
35 |
33 10
|
sseldd |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> Y e. ( Base ` S ) ) |
36 |
1 12 13 14 34 35
|
psradd |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` S ) Y ) = ( X oF ( +g ` R ) Y ) ) |
37 |
2 13
|
ressplusg |
|- ( T e. ( SubRing ` R ) -> ( +g ` R ) = ( +g ` H ) ) |
38 |
6 37
|
syl |
|- ( ph -> ( +g ` R ) = ( +g ` H ) ) |
39 |
38
|
adantr |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( +g ` R ) = ( +g ` H ) ) |
40 |
39
|
ofeqd |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> oF ( +g ` R ) = oF ( +g ` H ) ) |
41 |
40
|
oveqd |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X oF ( +g ` R ) Y ) = ( X oF ( +g ` H ) Y ) ) |
42 |
36 41
|
eqtrd |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` S ) Y ) = ( X oF ( +g ` H ) Y ) ) |
43 |
4
|
fvexi |
|- B e. _V |
44 |
5 14
|
ressplusg |
|- ( B e. _V -> ( +g ` S ) = ( +g ` P ) ) |
45 |
43 44
|
mp1i |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( +g ` S ) = ( +g ` P ) ) |
46 |
45
|
oveqd |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` S ) Y ) = ( X ( +g ` P ) Y ) ) |
47 |
11 42 46
|
3eqtr2d |
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` U ) Y ) = ( X ( +g ` P ) Y ) ) |