Step |
Hyp |
Ref |
Expression |
1 |
|
rnghmf1o.b |
|- B = ( Base ` R ) |
2 |
|
rnghmf1o.c |
|- C = ( Base ` S ) |
3 |
|
rnghmrcl |
|- ( F e. ( R RngHomo S ) -> ( R e. Rng /\ S e. Rng ) ) |
4 |
3
|
ancomd |
|- ( F e. ( R RngHomo S ) -> ( S e. Rng /\ R e. Rng ) ) |
5 |
4
|
adantr |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> ( S e. Rng /\ R e. Rng ) ) |
6 |
|
simpr |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> F : B -1-1-onto-> C ) |
7 |
|
rnghmghm |
|- ( F e. ( R RngHomo S ) -> F e. ( R GrpHom S ) ) |
8 |
7
|
adantr |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> F e. ( R GrpHom S ) ) |
9 |
1 2
|
ghmf1o |
|- ( F e. ( R GrpHom S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S GrpHom R ) ) ) |
10 |
9
|
bicomd |
|- ( F e. ( R GrpHom S ) -> ( `' F e. ( S GrpHom R ) <-> F : B -1-1-onto-> C ) ) |
11 |
8 10
|
syl |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> ( `' F e. ( S GrpHom R ) <-> F : B -1-1-onto-> C ) ) |
12 |
6 11
|
mpbird |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> `' F e. ( S GrpHom R ) ) |
13 |
|
eqidd |
|- ( F e. ( R RngHomo S ) -> F = F ) |
14 |
|
eqid |
|- ( mulGrp ` R ) = ( mulGrp ` R ) |
15 |
14 1
|
mgpbas |
|- B = ( Base ` ( mulGrp ` R ) ) |
16 |
15
|
a1i |
|- ( F e. ( R RngHomo S ) -> B = ( Base ` ( mulGrp ` R ) ) ) |
17 |
|
eqid |
|- ( mulGrp ` S ) = ( mulGrp ` S ) |
18 |
17 2
|
mgpbas |
|- C = ( Base ` ( mulGrp ` S ) ) |
19 |
18
|
a1i |
|- ( F e. ( R RngHomo S ) -> C = ( Base ` ( mulGrp ` S ) ) ) |
20 |
13 16 19
|
f1oeq123d |
|- ( F e. ( R RngHomo S ) -> ( F : B -1-1-onto-> C <-> F : ( Base ` ( mulGrp ` R ) ) -1-1-onto-> ( Base ` ( mulGrp ` S ) ) ) ) |
21 |
20
|
biimpa |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> F : ( Base ` ( mulGrp ` R ) ) -1-1-onto-> ( Base ` ( mulGrp ` S ) ) ) |
22 |
14 17
|
rnghmmgmhm |
|- ( F e. ( R RngHomo S ) -> F e. ( ( mulGrp ` R ) MgmHom ( mulGrp ` S ) ) ) |
23 |
22
|
adantr |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> F e. ( ( mulGrp ` R ) MgmHom ( mulGrp ` S ) ) ) |
24 |
|
eqid |
|- ( Base ` ( mulGrp ` R ) ) = ( Base ` ( mulGrp ` R ) ) |
25 |
|
eqid |
|- ( Base ` ( mulGrp ` S ) ) = ( Base ` ( mulGrp ` S ) ) |
26 |
24 25
|
mgmhmf1o |
|- ( F e. ( ( mulGrp ` R ) MgmHom ( mulGrp ` S ) ) -> ( F : ( Base ` ( mulGrp ` R ) ) -1-1-onto-> ( Base ` ( mulGrp ` S ) ) <-> `' F e. ( ( mulGrp ` S ) MgmHom ( mulGrp ` R ) ) ) ) |
27 |
26
|
bicomd |
|- ( F e. ( ( mulGrp ` R ) MgmHom ( mulGrp ` S ) ) -> ( `' F e. ( ( mulGrp ` S ) MgmHom ( mulGrp ` R ) ) <-> F : ( Base ` ( mulGrp ` R ) ) -1-1-onto-> ( Base ` ( mulGrp ` S ) ) ) ) |
28 |
23 27
|
syl |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> ( `' F e. ( ( mulGrp ` S ) MgmHom ( mulGrp ` R ) ) <-> F : ( Base ` ( mulGrp ` R ) ) -1-1-onto-> ( Base ` ( mulGrp ` S ) ) ) ) |
29 |
21 28
|
mpbird |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> `' F e. ( ( mulGrp ` S ) MgmHom ( mulGrp ` R ) ) ) |
30 |
12 29
|
jca |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> ( `' F e. ( S GrpHom R ) /\ `' F e. ( ( mulGrp ` S ) MgmHom ( mulGrp ` R ) ) ) ) |
31 |
17 14
|
isrnghmmul |
|- ( `' F e. ( S RngHomo R ) <-> ( ( S e. Rng /\ R e. Rng ) /\ ( `' F e. ( S GrpHom R ) /\ `' F e. ( ( mulGrp ` S ) MgmHom ( mulGrp ` R ) ) ) ) ) |
32 |
5 30 31
|
sylanbrc |
|- ( ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) -> `' F e. ( S RngHomo R ) ) |
33 |
1 2
|
rnghmf |
|- ( F e. ( R RngHomo S ) -> F : B --> C ) |
34 |
33
|
adantr |
|- ( ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) -> F : B --> C ) |
35 |
34
|
ffnd |
|- ( ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) -> F Fn B ) |
36 |
2 1
|
rnghmf |
|- ( `' F e. ( S RngHomo R ) -> `' F : C --> B ) |
37 |
36
|
adantl |
|- ( ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) -> `' F : C --> B ) |
38 |
37
|
ffnd |
|- ( ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) -> `' F Fn C ) |
39 |
|
dff1o4 |
|- ( F : B -1-1-onto-> C <-> ( F Fn B /\ `' F Fn C ) ) |
40 |
35 38 39
|
sylanbrc |
|- ( ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) -> F : B -1-1-onto-> C ) |
41 |
32 40
|
impbida |
|- ( F e. ( R RngHomo S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S RngHomo R ) ) ) |