Theorem lmhmlmod1

Description: A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015)

	Ref	Expression
Assertion	lmhmlmod1	⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝑆 ∈ LMod )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Scalar ‘ 𝑆 ) = ( Scalar ‘ 𝑆 )
2		eqid	⊢ ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑇 )
3	1 2	lmhmlem	⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑆 ) ) ) )
4	3	simplld	⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝑆 ∈ LMod )