Description: Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | lmicsym | |- ( R ~=m S -> S ~=m R ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brlmic | |- ( R ~=m S <-> ( R LMIso S ) =/= (/) ) |
|
2 | n0 | |- ( ( R LMIso S ) =/= (/) <-> E. f f e. ( R LMIso S ) ) |
|
3 | lmimcnv | |- ( f e. ( R LMIso S ) -> `' f e. ( S LMIso R ) ) |
|
4 | brlmici | |- ( `' f e. ( S LMIso R ) -> S ~=m R ) |
|
5 | 3 4 | syl | |- ( f e. ( R LMIso S ) -> S ~=m R ) |
6 | 5 | exlimiv | |- ( E. f f e. ( R LMIso S ) -> S ~=m R ) |
7 | 2 6 | sylbi | |- ( ( R LMIso S ) =/= (/) -> S ~=m R ) |
8 | 1 7 | sylbi | |- ( R ~=m S -> S ~=m R ) |