Theorem clmzss

Description: The scalar ring of a subcomplex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	clm0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	clmsub.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
Assertion	clmzss	⊢ ( 𝑊 ∈ ℂMod → ℤ ⊆ 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		clm0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		clmsub.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3	1 2	clmsubrg	⊢ ( 𝑊 ∈ ℂMod → 𝐾 ∈ ( SubRing ‘ ℂfld ) )
4		zsssubrg	⊢ ( 𝐾 ∈ ( SubRing ‘ ℂfld ) → ℤ ⊆ 𝐾 )
5	3 4	syl	⊢ ( 𝑊 ∈ ℂMod → ℤ ⊆ 𝐾 )