Theorem dsmmbase

Description: Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015)

	Ref	Expression
Hypothesis	dsmmval.b	$$⊢B=\left\{f\in {\mathrm{Base}}_{\left(S{⨉}_{𝑠}R\right)}|\left\{x\in \mathrm{dom}R|f\left(x\right)\ne 0_{R\left(x\right)}\right\}\in \mathrm{Fin}\right\}$$
Assertion	dsmmbase	$$⊢R\in V\to B={\mathrm{Base}}_{\left(S{\oplus }_{m}R\right)}$$

Proof

Step	Hyp	Ref	Expression
1		dsmmval.b	$$⊢B=\left\{f\in {\mathrm{Base}}_{\left(S{⨉}_{𝑠}R\right)}|\left\{x\in \mathrm{dom}R|f\left(x\right)\ne 0_{R\left(x\right)}\right\}\in \mathrm{Fin}\right\}$$
2		elex	$$⊢R\in V\to R\in \mathrm{V}$$
3	1	ssrab3	$$⊢B\subseteq {\mathrm{Base}}_{\left(S{⨉}_{𝑠}R\right)}$$
4		eqid	$$⊢\left(S{⨉}_{𝑠}R\right){\upharpoonright }_{𝑠}B=\left(S{⨉}_{𝑠}R\right){\upharpoonright }_{𝑠}B$$
5		eqid	$$⊢{\mathrm{Base}}_{\left(S{⨉}_{𝑠}R\right)}={\mathrm{Base}}_{\left(S{⨉}_{𝑠}R\right)}$$
6	4 5	ressbas2	$$⊢B\subseteq {\mathrm{Base}}_{\left(S{⨉}_{𝑠}R\right)}\to B={\mathrm{Base}}_{\left(\left(S{⨉}_{𝑠}R\right){\upharpoonright }_{𝑠}B\right)}$$
7	3 6	ax-mp	$$⊢B={\mathrm{Base}}_{\left(\left(S{⨉}_{𝑠}R\right){\upharpoonright }_{𝑠}B\right)}$$
8	1	dsmmval	$$⊢R\in \mathrm{V}\to S{\oplus }_{m}R=\left(S{⨉}_{𝑠}R\right){\upharpoonright }_{𝑠}B$$
9	8	fveq2d	$$⊢R\in \mathrm{V}\to {\mathrm{Base}}_{\left(S{\oplus }_{m}R\right)}={\mathrm{Base}}_{\left(\left(S{⨉}_{𝑠}R\right){\upharpoonright }_{𝑠}B\right)}$$
10	7 9	eqtr4id	$$⊢R\in \mathrm{V}\to B={\mathrm{Base}}_{\left(S{\oplus }_{m}R\right)}$$
11	2 10	syl	$$⊢R\in V\to B={\mathrm{Base}}_{\left(S{\oplus }_{m}R\right)}$$