Theorem mnringbased

Description: The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024)

	Ref	Expression
Hypotheses	mnringbased.1	No typesetting found for |- F = ( R MndRing M ) with typecode |-
	mnringbased.2	$$⊢A={\mathrm{Base}}_M$$
	mnringbased.3	$$⊢V=R\mathrm{freeLMod}A$$
	mnringbased.4	$$⊢B={\mathrm{Base}}_V$$
	mnringbased.5	$$⊢\phi \to R\in U$$
	mnringbased.6	$$⊢\phi \to M\in W$$
Assertion	mnringbased	$$⊢\phi \to B={\mathrm{Base}}_F$$

Proof

Step	Hyp	Ref	Expression
1		mnringbased.1	Could not format F = ( R MndRing M ) : No typesetting found for |- F = ( R MndRing M ) with typecode |-
2		mnringbased.2	$$⊢A={\mathrm{Base}}_M$$
3		mnringbased.3	$$⊢V=R\mathrm{freeLMod}A$$
4		mnringbased.4	$$⊢B={\mathrm{Base}}_V$$
5		mnringbased.5	$$⊢\phi \to R\in U$$
6		mnringbased.6	$$⊢\phi \to M\in W$$
7		df-base	$$⊢\mathrm{Base}=\mathrm{Slot}1$$
8		1nn	$$⊢1\in \mathbb{N}$$
9		1re	$$⊢1\in ℝ$$
10		1lt3	$$⊢1<3$$
11	9 10	ltneii	$$⊢1\ne 3$$
12		mulrndx	$$⊢{\cdot }_{\mathrm{ndx}}=3$$
13	11 12	neeqtrri	$$⊢1\ne {\cdot }_{\mathrm{ndx}}$$
14	1 7 8 13 2 3 5 6	mnringnmulrd	$$⊢\phi \to {\mathrm{Base}}_V={\mathrm{Base}}_F$$
15	4 14	syl5eq	$$⊢\phi \to B={\mathrm{Base}}_F$$