Theorem brlmic

Description: The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015)

	Ref	Expression
Assertion	brlmic	⊢ ( 𝑅 ≃𝑚 𝑆 ↔ ( 𝑅 LMIso 𝑆 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-lmic	⊢ ≃𝑚 = ( ◡ LMIso “ ( V ∖ 1o ) )
2		lmimfn	⊢ LMIso Fn ( LMod × LMod )
3	1 2	brwitnlem	⊢ ( 𝑅 ≃𝑚 𝑆 ↔ ( 𝑅 LMIso 𝑆 ) ≠ ∅ )