Theorem lmimlmhm

Description: An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015)

	Ref	Expression
Assertion	lmimlmhm	$$⊢F\in \left(R\mathrm{LMIso}S\right)\to F\in \left(R\mathrm{LMHom}S\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_R$$
2		eqid	$$⊢{\mathrm{Base}}_S={\mathrm{Base}}_S$$
3	1 2	islmim	$$⊢F\in \left(R\mathrm{LMIso}S\right)\leftrightarrow \left(F\in \left(R\mathrm{LMHom}S\right)\wedge F:{\mathrm{Base}}_R\underset{1-1\; onto}{⟶}{\mathrm{Base}}_S\right)$$
4	3	simplbi	$$⊢F\in \left(R\mathrm{LMIso}S\right)\to F\in \left(R\mathrm{LMHom}S\right)$$