Theorem lmicsym

Description: Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015)

	Ref	Expression
Assertion	lmicsym	|- ( R ~=m S -> S ~=m R )

Proof

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 )