islmim2

Description: An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015)

		Ref	Expression
	Assertion	islmim2	$⊢ F \in (R LMIso S) \leftrightarrow (F \in (R LMHom S) \land F^{-1} \in (S LMHom R))$

Step	Hyp	Ref	Expression
1		eqid	$⊢ {Base}_{R} = {Base}_{R}$
2		eqid	$⊢ {Base}_{S} = {Base}_{S}$
3	1 2	islmim	$⊢ F \in (R LMIso S) \leftrightarrow (F \in (R LMHom S) \land F : {Base}_{R} \underset{1-1 onto}{⟶} {Base}_{S})$
4	1 2	lmhmf1o	$⊢ F \in (R LMHom S) \to (F : {Base}_{R} \underset{1-1 onto}{⟶} {Base}_{S} \leftrightarrow F^{-1} \in (S LMHom R))$
5	4	pm5.32i	$⊢ (F \in (R LMHom S) \land F : {Base}_{R} \underset{1-1 onto}{⟶} {Base}_{S}) \leftrightarrow (F \in (R LMHom S) \land F^{-1} \in (S LMHom R))$
6	3 5	bitri	$⊢ F \in (R LMIso S) \leftrightarrow (F \in (R LMHom S) \land F^{-1} \in (S LMHom R))$

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