Theorem zlmbas

Description: Base set of a ZZ -module. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	zlmbas.w	⊢ 𝑊 = ( ℤMod ‘ 𝐺 )
	zlmbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	zlmbas	⊢ 𝐵 = ( Base ‘ 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		zlmbas.w	⊢ 𝑊 = ( ℤMod ‘ 𝐺 )
2		zlmbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
3		df-base	⊢ Base = Slot 1
4		1nn	⊢ 1 ∈ ℕ
5		1lt5	⊢ 1 < 5
6	1 3 4 5	zlmlem	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝑊 )
7	2 6	eqtri	⊢ 𝐵 = ( Base ‘ 𝑊 )