Theorem matplusg

Description: The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015)

	Ref	Expression
Hypotheses	matbas.a	$$⊢A=N\mathrm{Mat}R$$
	matbas.g	$$⊢G=R\mathrm{freeLMod}\left(N\times N\right)$$
Assertion	matplusg	$$⊢\left(N\in \mathrm{Fin}\wedge R\in V\right)\to {+}_G={+}_A$$

Proof

Step	Hyp	Ref	Expression
1		matbas.a	$$⊢A=N\mathrm{Mat}R$$
2		matbas.g	$$⊢G=R\mathrm{freeLMod}\left(N\times N\right)$$
3		plusgid	$$⊢{+}_{𝑔}=\mathrm{Slot}{+}_{\mathrm{ndx}}$$
4		plusgndxnmulrndx	$$⊢{+}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
5	3 4	setsnid	$$⊢{+}_G={+}_{\left(G\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},R\mathrm{maMul}⟨N,N,N⟩⟩\right)}$$
6		eqid	$$⊢R\mathrm{maMul}⟨N,N,N⟩=R\mathrm{maMul}⟨N,N,N⟩$$
7	1 2 6	matval	$$⊢\left(N\in \mathrm{Fin}\wedge R\in V\right)\to A=G\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},R\mathrm{maMul}⟨N,N,N⟩⟩$$
8	7	fveq2d	$$⊢\left(N\in \mathrm{Fin}\wedge R\in V\right)\to {+}_A={+}_{\left(G\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},R\mathrm{maMul}⟨N,N,N⟩⟩\right)}$$
9	5 8	eqtr4id	$$⊢\left(N\in \mathrm{Fin}\wedge R\in V\right)\to {+}_G={+}_A$$