Description: Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | lmicrcl | |- ( R ~=m S -> S e. LMod ) |
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 | 1 2 | bitri | |- ( R ~=m S <-> E. f f e. ( R LMIso S ) ) |
4 | lmimlmhm | |- ( f e. ( R LMIso S ) -> f e. ( R LMHom S ) ) |
|
5 | lmhmlmod2 | |- ( f e. ( R LMHom S ) -> S e. LMod ) |
|
6 | 4 5 | syl | |- ( f e. ( R LMIso S ) -> S e. LMod ) |
7 | 6 | exlimiv | |- ( E. f f e. ( R LMIso S ) -> S e. LMod ) |
8 | 3 7 | sylbi | |- ( R ~=m S -> S e. LMod ) |