Theorem zlmbas

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

	Ref	Expression
Hypotheses	zlmbas.w	$$⊢W=\mathrm{\mathbb{Z}Mod}\left(G\right)$$
	zlmbas.2	$$⊢B={\mathrm{Base}}_G$$
Assertion	zlmbas	$$⊢B={\mathrm{Base}}_W$$

Proof

Step	Hyp	Ref	Expression
1		zlmbas.w	$$⊢W=\mathrm{\mathbb{Z}Mod}\left(G\right)$$
2		zlmbas.2	$$⊢B={\mathrm{Base}}_G$$
3		df-base	$$⊢\mathrm{Base}=\mathrm{Slot}1$$
4		1nn	$$⊢1\in \mathbb{N}$$
5		1lt5	$$⊢1<5$$
6	1 3 4 5	zlmlem	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_W$$
7	2 6	eqtri	$$⊢B={\mathrm{Base}}_W$$