Theorem zlmplusg

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

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

Proof

Step	Hyp	Ref	Expression
1		zlmbas.w	$$⊢W=\mathrm{\mathbb{Z}Mod}\left(G\right)$$
2		zlmplusg.2	$$⊢\underset{\dot{}}{+}={+}_G$$
3		plusgid	$$⊢{+}_{𝑔}=\mathrm{Slot}{+}_{\mathrm{ndx}}$$
4		scandxnplusgndx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{+}_{\mathrm{ndx}}\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)$$
6		vscandxnplusgndx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {+}_{\mathrm{ndx}}$$
7	6	necomi	$$⊢{+}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
8	1 3 5 7	zlmlem	$$⊢{+}_G={+}_W$$
9	2 8	eqtri	$$⊢\underset{\dot{}}{+}={+}_W$$