Description: A ring is a field if and only if an isomorphic ring is a field. (Contributed by SN, 18-Feb-2025)
Ref | Expression | ||
---|---|---|---|
Assertion | ricfld | |- ( R ~=r S -> ( R e. Field <-> S e. Field ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ricdrng | |- ( R ~=r S -> ( R e. DivRing <-> S e. DivRing ) ) |
|
2 | riccrng | |- ( R ~=r S -> ( R e. CRing <-> S e. CRing ) ) |
|
3 | 1 2 | anbi12d | |- ( R ~=r S -> ( ( R e. DivRing /\ R e. CRing ) <-> ( S e. DivRing /\ S e. CRing ) ) ) |
4 | isfld | |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) ) |
|
5 | isfld | |- ( S e. Field <-> ( S e. DivRing /\ S e. CRing ) ) |
|
6 | 3 4 5 | 3bitr4g | |- ( R ~=r S -> ( R e. Field <-> S e. Field ) ) |