Description: Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025)
Ref | Expression | ||
---|---|---|---|
Assertion | ricsym | |- ( R ~=r S -> S ~=r R ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brric | |- ( R ~=r S <-> ( R RingIso S ) =/= (/) ) |
|
2 | n0 | |- ( ( R RingIso S ) =/= (/) <-> E. f f e. ( R RingIso S ) ) |
|
3 | rimcnv | |- ( f e. ( R RingIso S ) -> `' f e. ( S RingIso R ) ) |
|
4 | brrici | |- ( `' f e. ( S RingIso R ) -> S ~=r R ) |
|
5 | 3 4 | syl | |- ( f e. ( R RingIso S ) -> S ~=r R ) |
6 | 5 | exlimiv | |- ( E. f f e. ( R RingIso S ) -> S ~=r R ) |
7 | 2 6 | sylbi | |- ( ( R RingIso S ) =/= (/) -> S ~=r R ) |
8 | 1 7 | sylbi | |- ( R ~=r S -> S ~=r R ) |