Theorem lmicsym

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

	Ref	Expression
Assertion	lmicsym	$$⊢R{\simeq }_{𝑚}S\to S{\simeq }_{𝑚}R$$

Proof

Step	Hyp	Ref	Expression
1		brlmic	$$⊢R{\simeq }_{𝑚}S\leftrightarrow R\mathrm{LMIso}S\ne \varnothing$$
2		n0	$$⊢R\mathrm{LMIso}S\ne \varnothing \leftrightarrow \exists ff\in \left(R\mathrm{LMIso}S\right)$$
3		lmimcnv	$$⊢f\in \left(R\mathrm{LMIso}S\right)\to f^{-1}\in \left(S\mathrm{LMIso}R\right)$$
4		brlmici	$$⊢f^{-1}\in \left(S\mathrm{LMIso}R\right)\to S{\simeq }_{𝑚}R$$
5	3 4	syl	$$⊢f\in \left(R\mathrm{LMIso}S\right)\to S{\simeq }_{𝑚}R$$
6	5	exlimiv	$$⊢\exists ff\in \left(R\mathrm{LMIso}S\right)\to S{\simeq }_{𝑚}R$$
7	2 6	sylbi	$$⊢R\mathrm{LMIso}S\ne \varnothing \to S{\simeq }_{𝑚}R$$
8	1 7	sylbi	$$⊢R{\simeq }_{𝑚}S\to S{\simeq }_{𝑚}R$$