Step |
Hyp |
Ref |
Expression |
1 |
|
resrhm2b.u |
|- U = ( T |`s X ) |
2 |
|
subrgsubg |
|- ( X e. ( SubRing ` T ) -> X e. ( SubGrp ` T ) ) |
3 |
1
|
resghm2b |
|- ( ( X e. ( SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) ) |
4 |
2 3
|
sylan |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) ) |
5 |
|
eqid |
|- ( mulGrp ` T ) = ( mulGrp ` T ) |
6 |
5
|
subrgsubm |
|- ( X e. ( SubRing ` T ) -> X e. ( SubMnd ` ( mulGrp ` T ) ) ) |
7 |
|
eqid |
|- ( ( mulGrp ` T ) |`s X ) = ( ( mulGrp ` T ) |`s X ) |
8 |
7
|
resmhm2b |
|- ( ( X e. ( SubMnd ` ( mulGrp ` T ) ) /\ ran F C_ X ) -> ( F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) <-> F e. ( ( mulGrp ` S ) MndHom ( ( mulGrp ` T ) |`s X ) ) ) ) |
9 |
6 8
|
sylan |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) <-> F e. ( ( mulGrp ` S ) MndHom ( ( mulGrp ` T ) |`s X ) ) ) ) |
10 |
|
subrgrcl |
|- ( X e. ( SubRing ` T ) -> T e. Ring ) |
11 |
1 5
|
mgpress |
|- ( ( T e. Ring /\ X e. ( SubRing ` T ) ) -> ( ( mulGrp ` T ) |`s X ) = ( mulGrp ` U ) ) |
12 |
10 11
|
mpancom |
|- ( X e. ( SubRing ` T ) -> ( ( mulGrp ` T ) |`s X ) = ( mulGrp ` U ) ) |
13 |
12
|
adantr |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( ( mulGrp ` T ) |`s X ) = ( mulGrp ` U ) ) |
14 |
13
|
oveq2d |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( ( mulGrp ` S ) MndHom ( ( mulGrp ` T ) |`s X ) ) = ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) |
15 |
14
|
eleq2d |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( ( mulGrp ` S ) MndHom ( ( mulGrp ` T ) |`s X ) ) <-> F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) ) |
16 |
9 15
|
bitrd |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) <-> F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) ) |
17 |
4 16
|
anbi12d |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( ( F e. ( S GrpHom T ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) ) <-> ( F e. ( S GrpHom U ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) ) ) |
18 |
17
|
anbi2d |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( ( S e. Ring /\ ( F e. ( S GrpHom T ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) ) ) <-> ( S e. Ring /\ ( F e. ( S GrpHom U ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) ) ) ) |
19 |
10
|
adantr |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> T e. Ring ) |
20 |
19
|
biantrud |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( S e. Ring <-> ( S e. Ring /\ T e. Ring ) ) ) |
21 |
20
|
anbi1d |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( ( S e. Ring /\ ( F e. ( S GrpHom T ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) ) ) <-> ( ( S e. Ring /\ T e. Ring ) /\ ( F e. ( S GrpHom T ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) ) ) ) ) |
22 |
1
|
subrgring |
|- ( X e. ( SubRing ` T ) -> U e. Ring ) |
23 |
22
|
adantr |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> U e. Ring ) |
24 |
23
|
biantrud |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( S e. Ring <-> ( S e. Ring /\ U e. Ring ) ) ) |
25 |
24
|
anbi1d |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( ( S e. Ring /\ ( F e. ( S GrpHom U ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) ) <-> ( ( S e. Ring /\ U e. Ring ) /\ ( F e. ( S GrpHom U ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) ) ) ) |
26 |
18 21 25
|
3bitr3d |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( ( ( S e. Ring /\ T e. Ring ) /\ ( F e. ( S GrpHom T ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) ) ) <-> ( ( S e. Ring /\ U e. Ring ) /\ ( F e. ( S GrpHom U ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) ) ) ) |
27 |
|
eqid |
|- ( mulGrp ` S ) = ( mulGrp ` S ) |
28 |
27 5
|
isrhm |
|- ( F e. ( S RingHom T ) <-> ( ( S e. Ring /\ T e. Ring ) /\ ( F e. ( S GrpHom T ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` T ) ) ) ) ) |
29 |
|
eqid |
|- ( mulGrp ` U ) = ( mulGrp ` U ) |
30 |
27 29
|
isrhm |
|- ( F e. ( S RingHom U ) <-> ( ( S e. Ring /\ U e. Ring ) /\ ( F e. ( S GrpHom U ) /\ F e. ( ( mulGrp ` S ) MndHom ( mulGrp ` U ) ) ) ) ) |
31 |
26 28 30
|
3bitr4g |
|- ( ( X e. ( SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( S RingHom T ) <-> F e. ( S RingHom U ) ) ) |