P. 207 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rngimf1o 20601	An isomorphism of non-unital rings is a bijection. (Contributed by AV, 23-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶)
Theorem	rngimrnghm 20602	An isomorphism of non-unital rings is a homomorphism. (Contributed by AV, 23-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹 ∈ (𝑅 RngHom 𝑆))
Theorem	rngimcnv 20603	The converse of an isomorphism of non-unital rings is an isomorphism of non-unital rings. (Contributed by AV, 27-Feb-2025.)
⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆))
Theorem	rnghmco 20604	The composition of non-unital ring homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
⊢ ((𝐹 ∈ (𝑇 RngHom 𝑈) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 RngHom 𝑈))
Theorem	idrnghm 20605	The identity homomorphism on a non-unital ring. (Contributed by AV, 27-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → ( I ↾ 𝐵) ∈ (𝑅 RngHom 𝑅))
Theorem	c0mgm 20606*	The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))
Theorem	c0mhm 20607*	The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
Theorem	c0ghm 20608*	The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))
Theorem	c0snmgmhm 20609*	The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))
Theorem	c0snmhm 20610*	The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    &   ⊢ 𝑍 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆))
Theorem	c0snghm 20611*	The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    &   ⊢ 𝑍 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
Theorem	rngisomfv1 20612	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 20613*	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 20614	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 20615	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 20616	Extend class notation with the ring homomorphisms.
class RingHom
Syntax	crs 20617	Extend class notation with the ring isomorphisms.
class RingIso
Syntax	cric 20618	Extend class notation with the ring isomorphism relation.
class ≃𝑟
Definition	df-rhm 20619*	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 20620*	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 20621*	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 20622*	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 20623*	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 20624	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
Theorem	rhmrcl2 20625	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
Theorem	isrhm 20626	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 20627	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 20628	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 20629	Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Theorem	isrim0 20630	A ring isomorphism is a homomorphism whose converse is also a homomorphism. Compare isgim2 19398. (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 20631	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 20632	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 20633	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 20634	The converse of a ring isomorphism is a ring isomorphism. (Contributed by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → ◡𝐹 ∈ (𝑆 RingIso 𝑅))
Theorem	rhmadd 20635	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 20636	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 20637	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 20638*	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 20639*	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 20640	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 20641	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 20642	The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅))
Theorem	crngrhmfo 20643	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 20644	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 20645	An isomorphism of rings is a bijective homomorphism. (Contributed 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 20646	An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶)
Theorem	rimval 20647*	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 20648	A ring isomorphism is a homomorphism. Compare gimghm 19397. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	rimrcl1 20649	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ∈ Ring)
Theorem	rimrcl2 20650	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑆 ∈ Ring)
Theorem	rimgim 20651	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 20652	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 20653	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 20654	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 20655	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 20656	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 20657	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 20658*	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 20659*	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 20660	Define the ring isomorphism relation, analogous to df-gic 19393: 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 AV, 24-Dec-2019.)
⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
Theorem	ricrel 20661	The domain of the ring isomorphism relation is a relation. (Contributed by AV, 24-Jul-2026.)
⊢ Rel ≃𝑟
Theorem	brric 20662	The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Theorem	brrici 20663	Prove isomorphic by an explicit isomorphism. (Contributed by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ≃𝑟 𝑆)
Theorem	rimco 20664	The composition of ring isomorphisms is a ring isomorphism. (Contributed by SN, 17-Jan-2025.)
⊢ ((𝐹 ∈ (𝑆 RingIso 𝑇) ∧ 𝐺 ∈ (𝑅 RingIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 RingIso 𝑇))
Theorem	ricref 20665	Ring isomorphism is reflexive. (Contributed by by AV, 24-Jul-2026.)
⊢ (𝑅 ∈ Ring → 𝑅 ≃𝑟 𝑅)
Theorem	riclcl 20666	Ring isomorphism implies the left side is a ring. (Contributed by AV, 23-Jul-2026.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑅 ∈ Ring)
Theorem	ricrcl 20667	Ring isomorphism implies the right side is a ring. (Contributed by AV, 23-Jul-2026.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ∈ Ring)
Theorem	ricsym 20668	Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
Theorem	rictr 20669	Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑆 ≃𝑟 𝑇) → 𝑅 ≃𝑟 𝑇)
Theorem	isbrric2 20670*	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 20671*	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 20672	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 20673	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	rhmdvdsr 20674	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 20675	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 20676	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 20677	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 20678	The class of nonzero rings.
class NzRing
Definition	df-nzr 20679	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 20680	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 20681	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 20682	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 20683	Obsolete version of nzrring 20682 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 20684	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 20685	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 20686	Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 20684. (Contributed by AV, 14-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)))
Theorem	nzrpropd 20687*	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 20688	The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 20689. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing)
Theorem	opprnzr 20689	The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
Theorem	ringelnzr 20690	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 20691	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 20692	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 20693	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 20694	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 20695	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 20696	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 20697	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 20698	Obsolete version of 01eq0ring 20697 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 20699	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 20700	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 ))