P. 208 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rngisomfv1 20701	If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is an element of the base set of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘ 1 ) ∈ 𝐵)
Theorem	rngisom1 20702*	If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is a ring unity of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ · = (.r‘𝑆)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥 ∈ 𝐵 (((𝐹‘ 1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹‘ 1 )) = 𝑥))
Theorem	rngisomring 20703	If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then both rings are unital. (Contributed by AV, 27-Feb-2025.)
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑆 ∈ Ring)
Theorem	rngisomring1 20704	If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the ring unity of the second ring is the function value of the ring unity of the first ring for the isomorphism. (Contributed by AV, 16-Mar-2025.)
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑆) = (𝐹‘(1r‘𝑅)))
10.3.10  Ring homomorphisms
Syntax	crh 20705	Extend class notation with the ring homomorphisms.
class RingHom
Syntax	crs 20706	Extend class notation with the ring isomorphisms.
class RingIso
Syntax	cric 20707	Extend class notation with the ring isomorphism relation.
class ≃𝑟
Definition	df-rhm 20708*	Define the set of ring homomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))})
Definition	df-rim 20709*	Define the set of ring isomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)})
Theorem	dfrhm2 20710*	The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
Theorem	rhmval0 20711*	The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by AV, 24-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ⨣ = (+g‘𝑆)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
Theorem	isrhm0 20712*	The predicate "is a ring homomorphism from 𝑅 to 𝑆". (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 24-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ⨣ = (+g‘𝑆)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐹‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))))))
Theorem	rhmrcl1 20713	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
Theorem	rhmrcl2 20714	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
Theorem	isrhm 20715	A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁))))
Theorem	rhmmhm 20716	A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑀 MndHom 𝑁))
Theorem	rhmisrnghm 20717	Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 RngHom 𝑆))
Theorem	rimrcl 20718	Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Theorem	isrim0 20719	A ring isomorphism is a homomorphism whose converse is also a homomorphism. Compare isgim2 19485. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
Theorem	rhmghm 20720	A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
Theorem	rhmf 20721	A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐵⟶𝐶)
Theorem	rhmcl 20722	Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.) (Revised by AV, 10-Jan-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ∈ 𝐶)
Theorem	rimcnv 20723	The converse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → ◡𝐹 ∈ (𝑆 RingIso 𝑅))
Theorem	rhmadd 20724	Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.) (Revised by AV, 10-Jan-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ⨣ = (+g‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵)))
Theorem	rhmsub 20725	Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.) (Revised by AV, 10-Jan-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑁 = (-g‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 − 𝐵)) = ((𝐹‘𝐴)𝑁(𝐹‘𝐵)))
Theorem	rhmmul 20726	A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
Theorem	isrhm2d 20727*	Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ∈ Ring)    &   ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	isrhmd 20728*	Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ∈ Ring)    &   ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ⨣ = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	rhm0 20729	A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.) (Revised by AV, 23-Jul-2026.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 0 ) = 𝑍)
Theorem	rhm1 20730	Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 1 ) = 𝑁)
Theorem	idrhm 20731	The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅))
Theorem	crngrhmfo 20732	The image of a surjective homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.) (Revised by AV, 19-Jul-2026.)
⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹–onto→𝐵) → 𝑆 ∈ CRing)
Theorem	rhmf1o 20733	A ring homomorphism is bijective iff its converse is also a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
Theorem	isrim 20734	An isomorphism of rings is a bijective homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
Theorem	rimf1o 20735	An isomorphism of rings is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 22-Oct-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶)
Theorem	rimval 20736*	The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 24-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝑅 RingIso 𝑆) = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓:𝐵–1-1-onto→𝐶}
Theorem	rimrhm 20737	A ring isomorphism is a homomorphism. Compare gimghm 19484. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	rimrcl1 20738	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ∈ Ring)
Theorem	rimrcl2 20739	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑆 ∈ Ring)
Theorem	rimgim 20740	An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))
Theorem	rimisrngim 20741	Each unital ring isomorphism is a non-unital ring isomorphism. (Contributed by AV, 30-Mar-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RngIso 𝑆))
Theorem	rhmfn 20742	The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
⊢ RingHom Fn (Ring × Ring)
Theorem	rimfn 20743	The mapping of two rings to the ring isomorphisms between them is a function. (Contributed by AV, 24-Jul-2025.)
⊢ RingIso Fn (V × V)
Theorem	rhmval 20744	The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.)
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))
Theorem	rhmco 20745	The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ ((𝐹 ∈ (𝑇 RingHom 𝑈) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 RingHom 𝑈))
Theorem	rhmkerinj 20746	A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 24-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵–1-1→𝐶 ↔ (◡𝐹 “ {𝑍}) = { 0 }))
Theorem	pwsco1rhm 20747*	Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐴)    &   ⊢ 𝑍 = (𝑅 ↑s 𝐵)    &   ⊢ 𝐶 = (Base‘𝑍)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 RingHom 𝑌))
Theorem	pwsco2rhm 20748*	Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐴)    &   ⊢ 𝑍 = (𝑆 ↑s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) ∈ (𝑌 RingHom 𝑍))
Definition	df-ric 20749	Define the ring isomorphism relation, analogous to df-gic 19480: Two (unital) rings are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic rings share all global ring properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) Adapted to df-gic 19480. (Revised by AV, 24-Dec-2019.)
⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
Theorem	ricrel 20750	The domain of the ring isomorphism relation is a relation. (Contributed by AV, 24-Jul-2026.)
⊢ Rel ≃𝑟
Theorem	brric 20751	The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Theorem	brrici 20752	Prove isomorphic by an explicit isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ≃𝑟 𝑆)
Theorem	rimco 20753	The composition of ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by SN, 17-Jan-2025.)
⊢ ((𝐹 ∈ (𝑆 RingIso 𝑇) ∧ 𝐺 ∈ (𝑅 RingIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 RingIso 𝑇))
Theorem	ricref 20754	Ring isomorphism is reflexive. (Contributed by by AV, 24-Jul-2026.)
⊢ (𝑅 ∈ Ring → 𝑅 ≃𝑟 𝑅)
Theorem	riclcl 20755	Ring isomorphism implies the left side is a ring. (Contributed by AV, 23-Jul-2026.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑅 ∈ Ring)
Theorem	ricrcl 20756	Ring isomorphism implies the right side is a ring. (Contributed by AV, 23-Jul-2026.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ∈ Ring)
Theorem	ricsym 20757	Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
Theorem	rictr 20758	Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑆 ≃𝑟 𝑇) → 𝑅 ≃𝑟 𝑇)
Theorem	isbrric2 20759*	The relation "is isomorphic to" for (unital) rings. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆)))
Theorem	brric2 20760*	The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 23-Jul-2026.)
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆)))
Theorem	ricgic 20761	If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑅 ≃𝑔 𝑆)
Theorem	ricer 20762	Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 24-Jul-2026.)
⊢ ≃𝑟 Er Ring
Theorem	dfric2 20763*	Alternate definition of the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 23-Aug-2026.)
⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingIso 𝑠))}
Theorem	rhmdvdsr 20764	A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ / = (∥r‘𝑆)    ⇒   ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) / (𝐹‘𝐵))
Theorem	rhmopp 20765	A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))
Theorem	elrhmunit 20766	Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆))
Theorem	rhmunitinv 20767	Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴)))
10.3.11  Nonzero rings and zero rings
Syntax	cnzr 20768	The class of nonzero rings.
class NzRing
Definition	df-nzr 20769	A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
Theorem	isnzr 20770	Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))
Theorem	nzrnz 20771	One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )
Theorem	nzrring 20772	A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Theorem	nzrringOLD 20773	Obsolete version of nzrring 20772 as of 23-Feb-2025. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Theorem	isnzr2 20774	Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2o ≼ 𝐵))
Theorem	drnglidl1ne0 20775	In a nonzero ring, the zero ideal is different from the unit ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝐵 ≠ { 0 })
Theorem	isnzr2hash 20776	Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 20774. (Contributed by AV, 14-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)))
Theorem	nzrpropd 20777*	If two structures have the same components (properties), one is a nonzero ring iff the other one is. (Contributed by SN, 21-Jun-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ NzRing ↔ 𝐿 ∈ NzRing))
Theorem	opprnzrb 20778	The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 20779. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing)
Theorem	opprnzr 20779	The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
Theorem	ringelnzr 20780	A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ NzRing)
Theorem	nzrunit 20781	A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝐴 ≠ 0 )
Theorem	0ringnnzr 20782	A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)
⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))
Theorem	0ring 20783	If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 })
Theorem	0ringdif 20784	A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ (Ring ∖ NzRing) ↔ (𝑅 ∈ Ring ∧ 𝐵 = { 0 }))
Theorem	0ringbas 20785	The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ (Ring ∖ NzRing) → 𝐵 = { 0 })
Theorem	0ring01eq 20786	In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 0 = 1 )
Theorem	01eq0ring 20787	If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
Theorem	01eq0ringOLD 20788	Obsolete version of 01eq0ring 20787 as of 23-Feb-2025. (Contributed by AV, 16-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
Theorem	0ring01eqbi2 20789	In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 27-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐵 = { 0 } ↔ 1 = 0 ))
Theorem	0ring01eqbi 20790	In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by AV, 23-Jan-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐵 ≈ 1o ↔ 1 = 0 ))
Theorem	0ring1eq0 20791	In a zero ring, a ring which is not a nonzero ring, the ring unity equals the zero element. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ (Ring ∖ NzRing) → 1 = 0 )
Theorem	c0rhm 20792*	The constant mapping to zero is a ring homomorphism from any ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Ring ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RingHom 𝑇))
Theorem	c0rnghm 20793*	The constant mapping to zero is a non-unital ring homomorphism from any non-unital ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHom 𝑇))
Theorem	zrrnghm 20794*	The constant mapping to zero is a non-unital ring homomorphism from the zero ring to any non-unital ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHom 𝑆))
Theorem	nrhmzr 20795	There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.)
⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (𝑍 RingHom 𝑅) = ∅)
10.3.12  Local rings
Syntax	clring 20796	Extend class notation with class of all local rings.
class LRing
Definition	df-lring 20797*	A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
Theorem	islring 20798*	The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
Theorem	lringnzr 20799	A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
Theorem	lringring 20800	A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)