Theorem zlmmulr

Description: Ring operation of a ZZ -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 3-Nov-2024)

	Ref	Expression
Hypotheses	zlmbas.w	$$⊢W=\mathrm{\mathbb{Z}Mod}\left(G\right)$$
	zlmmulr.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
Assertion	zlmmulr	$$⊢\underset{\dot{}}{·}={\cdot }_W$$

Proof

Step	Hyp	Ref	Expression
1		zlmbas.w	$$⊢W=\mathrm{\mathbb{Z}Mod}\left(G\right)$$
2		zlmmulr.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
3		mulridx	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		scandxnmulrndx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{\cdot }_{\mathrm{ndx}}\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)$$
6		vscandxnmulrndx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
7	6	necomi	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
8	1 3 5 7	zlmlem	$$⊢{\cdot }_G={\cdot }_W$$
9	2 8	eqtri	$$⊢\underset{\dot{}}{·}={\cdot }_W$$