Theorem lsmsnpridl

Description: The product of the ring with a single element is equal to the principal ideal generated by that element. (Contributed by Thierry Arnoux, 21-Jan-2024)

	Ref	Expression
Hypotheses	lsmsnpridl.1	$$⊢B={\mathrm{Base}}_R$$
	lsmsnpridl.2	$$⊢G={\mathrm{mulGrp}}_R$$
	lsmsnpridl.3	$$⊢\underset{\dot{}}{\times }=\mathrm{LSSum}\left(G\right)$$
	lsmsnpridl.4	$$⊢K=\mathrm{RSpan}\left(R\right)$$
	lsmsnpridl.5	$$⊢\phi \to R\in \mathrm{Ring}$$
	lsmsnpridl.6	$$⊢\phi \to X\in B$$
Assertion	lsmsnpridl	$$⊢\phi \to B\underset{\dot{}}{\times }\left\{X\right\}=K\left(\left\{X\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		lsmsnpridl.1	$$⊢B={\mathrm{Base}}_R$$
2		lsmsnpridl.2	$$⊢G={\mathrm{mulGrp}}_R$$
3		lsmsnpridl.3	$$⊢\underset{\dot{}}{\times }=\mathrm{LSSum}\left(G\right)$$
4		lsmsnpridl.4	$$⊢K=\mathrm{RSpan}\left(R\right)$$
5		lsmsnpridl.5	$$⊢\phi \to R\in \mathrm{Ring}$$
6		lsmsnpridl.6	$$⊢\phi \to X\in B$$
7	2 1	mgpbas	$$⊢B={\mathrm{Base}}_G$$
8		eqid	$$⊢{\cdot }_R={\cdot }_R$$
9	2 8	mgpplusg	$$⊢{\cdot }_R={+}_G$$
10	2	fvexi	$$⊢G\in \mathrm{V}$$
11	10	a1i	$$⊢\phi \to G\in \mathrm{V}$$
12		ssidd	$$⊢\phi \to B\subseteq B$$
13	7 9 3 11 12 6	elgrplsmsn	$$⊢\phi \to \left(x\in \left(B\underset{\dot{}}{\times }\left\{X\right\}\right)\leftrightarrow \exists y\in Bx=y{\cdot }_{R}X\right)$$
14	1 8 4	rspsnel	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\right)\to \left(x\in K\left(\left\{X\right\}\right)\leftrightarrow \exists y\in Bx=y{\cdot }_{R}X\right)$$
15	5 6 14	syl2anc	$$⊢\phi \to \left(x\in K\left(\left\{X\right\}\right)\leftrightarrow \exists y\in Bx=y{\cdot }_{R}X\right)$$
16	13 15	bitr4d	$$⊢\phi \to \left(x\in \left(B\underset{\dot{}}{\times }\left\{X\right\}\right)\leftrightarrow x\in K\left(\left\{X\right\}\right)\right)$$
17	16	eqrdv	$$⊢\phi \to B\underset{\dot{}}{\times }\left\{X\right\}=K\left(\left\{X\right\}\right)$$