Theorem ring1zr

Description: The only unital ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rng1zr , and semirings, see srg1zr . (Contributed by FL, 13-Feb-2010) (Revised by AV, 25-Jan-2020) (Proof shortened by AV, 7-Feb-2020)

	Ref	Expression
Hypotheses	ring1zr.b	$$⊢B={\mathrm{Base}}_R$$
	ring1zr.p	$$⊢\underset{\dot{}}{+}={+}_R$$
	ring1zr.t	$$⊢\underset{\dot{}}{\ast }={\cdot }_R$$
Assertion	ring1zr	$$⊢\left(\left(R\in \mathrm{Ring}\wedge \underset{\dot{}}{+}Fn\left(B\times B\right)\wedge \underset{\dot{}}{\ast }Fn\left(B\times B\right)\right)\wedge Z\in B\right)\to \left(B=\left\{Z\right\}\leftrightarrow \left(\underset{\dot{}}{+}=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\wedge \underset{\dot{}}{\ast }=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ring1zr.b	$$⊢B={\mathrm{Base}}_R$$
2		ring1zr.p	$$⊢\underset{\dot{}}{+}={+}_R$$
3		ring1zr.t	$$⊢\underset{\dot{}}{\ast }={\cdot }_R$$
4		ringsrg	$$⊢R\in \mathrm{Ring}\to R\in \mathrm{SRing}$$
5	1 2 3	srg1zr	$$⊢\left(\left(R\in \mathrm{SRing}\wedge \underset{\dot{}}{+}Fn\left(B\times B\right)\wedge \underset{\dot{}}{\ast }Fn\left(B\times B\right)\right)\wedge Z\in B\right)\to \left(B=\left\{Z\right\}\leftrightarrow \left(\underset{\dot{}}{+}=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\wedge \underset{\dot{}}{\ast }=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\right)\right)$$
6	4 5	syl3anl1	$$⊢\left(\left(R\in \mathrm{Ring}\wedge \underset{\dot{}}{+}Fn\left(B\times B\right)\wedge \underset{\dot{}}{\ast }Fn\left(B\times B\right)\right)\wedge Z\in B\right)\to \left(B=\left\{Z\right\}\leftrightarrow \left(\underset{\dot{}}{+}=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\wedge \underset{\dot{}}{\ast }=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\right)\right)$$