Theorem matvsca

Description: The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015) (Proof shortened by AV, 12-Nov-2024)

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

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		vscaid	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		vscandxnmulrndx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
5	3 4	setsnid	$$⊢{\cdot }_G={\cdot }_{\left(G\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},R\mathrm{maMul}⟨N,N,N⟩⟩\right)}$$
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	$$⊢\left(N\in \mathrm{Fin}\wedge R\in V\right)\to {\cdot }_A={\cdot }_{\left(G\mathrm{sSet}⟨{\cdot }_{\mathrm{ndx}},R\mathrm{maMul}⟨N,N,N⟩⟩\right)}$$
9	5 8	eqtr4id	$$⊢\left(N\in \mathrm{Fin}\wedge R\in V\right)\to {\cdot }_G={\cdot }_A$$