Theorem zlmbas

Description: Base set of a ZZ -module. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 3-Nov-2024)

	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		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
4		scandxnbasendx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)$$
6		vscandxnbasendx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
7	6	necomi	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
8	1 3 5 7	zlmlem	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_W$$
9	2 8	eqtri	$$⊢B={\mathrm{Base}}_W$$