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Metamath Proof Explorer

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 )