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 = ( Base ‘ 𝑅 )
	ring1zr.p	⊢ + = ( +g ‘ 𝑅 )
	ring1zr.t	⊢ ∗ = ( .r ‘ 𝑅 )
Assertion	ring1zr	⊢ ( ( ( 𝑅 ∈ Ring ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )

Proof

Step	Hyp	Ref	Expression
1		ring1zr.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ring1zr.p	⊢ + = ( +g ‘ 𝑅 )
3		ring1zr.t	⊢ ∗ = ( .r ‘ 𝑅 )
4		ringsrg	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ SRing )
5	1 2 3	srg1zr	⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )
6	4 5	syl3anl1	⊢ ( ( ( 𝑅 ∈ Ring ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )