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	⊢ + = ( +g ‘ 𝑅 )
	rng1zr.t	⊢ ∗ = ( .r ‘ 𝑅 )
Assertion	rng1zr	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )

Proof

Step	Hyp	Ref	Expression
1		rng1zr.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rng1zr.p	⊢ + = ( +g ‘ 𝑅 )
3		rng1zr.t	⊢ ∗ = ( .r ‘ 𝑅 )
4		rnggrp	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Grp )
5	4	grpmgmd	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Mgm )
6		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
7	6	rngmgp	⊢ ( 𝑅 ∈ Rng → ( mulGrp ‘ 𝑅 ) ∈ Smgrp )
8		sgrpmgm	⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Smgrp → ( mulGrp ‘ 𝑅 ) ∈ Mgm )
9	7 8	syl	⊢ ( 𝑅 ∈ Rng → ( mulGrp ‘ 𝑅 ) ∈ Mgm )
10	5 9	jca	⊢ ( 𝑅 ∈ Rng → ( 𝑅 ∈ Mgm ∧ ( mulGrp ‘ 𝑅 ) ∈ Mgm ) )
11	10	3ad2ant1	⊢ ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) → ( 𝑅 ∈ Mgm ∧ ( mulGrp ‘ 𝑅 ) ∈ Mgm ) )
12	11	adantr	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝑅 ∈ Mgm ∧ ( mulGrp ‘ 𝑅 ) ∈ Mgm ) )
13		3simpc	⊢ ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) → ( + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) )
14	13	adantr	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) )
15		simpr	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → 𝑍 ∈ 𝐵 )
16	1 2 3	rng1zrlem	⊢ ( ( ( 𝑅 ∈ Mgm ∧ ( mulGrp ‘ 𝑅 ) ∈ Mgm ) ∧ ( + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )
17	12 14 15 16	syl3anc	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )