Theorem lmimgim

Description: An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015) (Revised by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	lmimgim	⊢ ( 𝐹 ∈ ( 𝑅 LMIso 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		lmimlmhm	⊢ ( 𝐹 ∈ ( 𝑅 LMIso 𝑆 ) → 𝐹 ∈ ( 𝑅 LMHom 𝑆 ) )
2		lmghm	⊢ ( 𝐹 ∈ ( 𝑅 LMHom 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )
3	1 2	syl	⊢ ( 𝐹 ∈ ( 𝑅 LMIso 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )
4		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
5		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
6	4 5	lmimf1o	⊢ ( 𝐹 ∈ ( 𝑅 LMIso 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) )
7	4 5	isgim	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) )
8	3 6 7	sylanbrc	⊢ ( 𝐹 ∈ ( 𝑅 LMIso 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) )