Theorem lmhmsca

Description: A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015)

	Ref	Expression
Hypotheses	lmhmlem.k	⊢ 𝐾 = ( Scalar ‘ 𝑆 )
	lmhmlem.l	⊢ 𝐿 = ( Scalar ‘ 𝑇 )
Assertion	lmhmsca	⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐿 = 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		lmhmlem.k	⊢ 𝐾 = ( Scalar ‘ 𝑆 )
2		lmhmlem.l	⊢ 𝐿 = ( Scalar ‘ 𝑇 )
3	1 2	lmhmlem	⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐿 = 𝐾 ) ) )
4	3	simprrd	⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐿 = 𝐾 )