Metamath Proof Explorer


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 f f R LMIso S
3 lmimcnv f R LMIso S f -1 S LMIso R
4 brlmici f -1 S LMIso R S 𝑚 R
5 3 4 syl f R LMIso S S 𝑚 R
6 5 exlimiv f f 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