Metamath Proof Explorer


Theorem nmhmnghm

Description: A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015)

Ref Expression
Assertion nmhmnghm ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) )

Proof

Step Hyp Ref Expression
1 isnmhm ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) ∧ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) )
2 1 simprbi ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) → ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) )
3 2 simprd ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) )