Theorem lmicrcl

Description: Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	lmicrcl	⊢ ( 𝑅 ≃𝑚 𝑆 → 𝑆 ∈ LMod )

Proof

Step	Hyp	Ref	Expression
1		brlmic	⊢ ( 𝑅 ≃𝑚 𝑆 ↔ ( 𝑅 LMIso 𝑆 ) ≠ ∅ )
2		n0	⊢ ( ( 𝑅 LMIso 𝑆 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) )
3	1 2	bitri	⊢ ( 𝑅 ≃𝑚 𝑆 ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) )
4		lmimlmhm	⊢ ( 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) → 𝑓 ∈ ( 𝑅 LMHom 𝑆 ) )
5		lmhmlmod2	⊢ ( 𝑓 ∈ ( 𝑅 LMHom 𝑆 ) → 𝑆 ∈ LMod )
6	4 5	syl	⊢ ( 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) → 𝑆 ∈ LMod )
7	6	exlimiv	⊢ ( ∃ 𝑓 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) → 𝑆 ∈ LMod )
8	3 7	sylbi	⊢ ( 𝑅 ≃𝑚 𝑆 → 𝑆 ∈ LMod )