Metamath Proof Explorer

Theorem rng1zr

Description: The only ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010) (Revised by AV, 18-Jun-2026)

		Ref	Expression
	Hypotheses	rng1zr.b	$⊢ B = {Base}_{R}$
		rng1zr.p	$⊢ \underset{˙}{+} = +_{R}$
		rng1zr.t	$⊢ \underset{˙}{*} = \cdot_{R}$
	Assertion	rng1zr	$⊢ ((R \in Rng \land \underset{˙}{+} Fn (B \times B) \land \underset{˙}{} Fn (B \times B)) \land Z \in B) \to (B = \{Z\} \leftrightarrow (\underset{˙}{+} = \{⟨⟨Z, Z⟩, Z⟩\} \land \underset{˙}{} = \{⟨⟨Z, Z⟩, Z⟩\}))$

Proof

Step	Hyp	Ref	Expression
1		rng1zr.b	$⊢ B = {Base}_{R}$
2		rng1zr.p	$⊢ \underset{˙}{+} = +_{R}$
3		rng1zr.t	$⊢ \underset{˙}{*} = \cdot_{R}$
4		rnggrp	$⊢ R \in Rng \to R \in Grp$
5	4	grpmgmd	$⊢ R \in Rng \to R \in Mgm$
6		eqid	$⊢ {mulGrp}_{R} = {mulGrp}_{R}$
7	6	rngmgp	$⊢ R \in Rng \to {mulGrp}_{R} \in Smgrp$
8		sgrpmgm	$⊢ {mulGrp}_{R} \in Smgrp \to {mulGrp}_{R} \in Mgm$
9	7 8	syl	$⊢ R \in Rng \to {mulGrp}_{R} \in Mgm$
10	5 9	jca	$⊢ R \in Rng \to (R \in Mgm \land {mulGrp}_{R} \in Mgm)$
11	10	3ad2ant1	$⊢ (R \in Rng \land \underset{˙}{+} Fn (B \times B) \land \underset{˙}{*} Fn (B \times B)) \to (R \in Mgm \land {mulGrp}_{R} \in Mgm)$
12	11	adantr	$⊢ ((R \in Rng \land \underset{˙}{+} Fn (B \times B) \land \underset{˙}{*} Fn (B \times B)) \land Z \in B) \to (R \in Mgm \land {mulGrp}_{R} \in Mgm)$
13		3simpc	$⊢ (R \in Rng \land \underset{˙}{+} Fn (B \times B) \land \underset{˙}{} Fn (B \times B)) \to (\underset{˙}{+} Fn (B \times B) \land \underset{˙}{} Fn (B \times B))$
14	13	adantr	$⊢ ((R \in Rng \land \underset{˙}{+} Fn (B \times B) \land \underset{˙}{} Fn (B \times B)) \land Z \in B) \to (\underset{˙}{+} Fn (B \times B) \land \underset{˙}{} Fn (B \times B))$
15		simpr	$⊢ ((R \in Rng \land \underset{˙}{+} Fn (B \times B) \land \underset{˙}{*} Fn (B \times B)) \land Z \in B) \to Z \in B$
16	1 2 3	rng1zrlem	$⊢ ((R \in Mgm \land {mulGrp}_{R} \in Mgm) \land (\underset{˙}{+} Fn (B \times B) \land \underset{˙}{} Fn (B \times B)) \land Z \in B) \to (B = \{Z\} \leftrightarrow (\underset{˙}{+} = \{⟨⟨Z, Z⟩, Z⟩\} \land \underset{˙}{} = \{⟨⟨Z, Z⟩, Z⟩\}))$
17	12 14 15 16	syl3anc	$⊢ ((R \in Rng \land \underset{˙}{+} Fn (B \times B) \land \underset{˙}{} Fn (B \times B)) \land Z \in B) \to (B = \{Z\} \leftrightarrow (\underset{˙}{+} = \{⟨⟨Z, Z⟩, Z⟩\} \land \underset{˙}{} = \{⟨⟨Z, Z⟩, Z⟩\}))$