P. 385 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 38401-38500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	ghomf 38401	Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.)
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑊 = ran 𝐻 ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶𝑊)

Theorem	ghomco 38402	The composition of two group homomorphisms is a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾))

Theorem	ghomdiv 38403	Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /_𝑔 ‘𝐺) & ⊢ 𝐶 = ( /_𝑔 ‘𝐻) ⇒ ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹‘𝐴)𝐶(𝐹‘𝐵)))

Theorem	grpokerinj 38404	A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑊 = (GId‘𝐺) & ⊢ 𝑌 = ran 𝐻 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 ↔ (^◡𝐹 “ {𝑈}) = {𝑊}))

21.24.16 Rings

Syntax	crngo 38405	Extend class notation with the class of all unital rings.
class RingOps

Definition	df-rngo 38406*	Define the class of all unital rings. (Contributed by Jeff Hankins, 21-Nov-2006.) (New usage is discouraged.)
⊢ RingOps = {⟨𝑔, ℎ⟩ ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))}

Theorem	relrngo 38407	The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ Rel RingOps

Theorem	isrngo 38408*	The predicate "is a (unital) ring." Definition of "ring with unit" in [Schechter] p. 187. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))

Theorem	isrngod 38409*	Conditions that determine a ring. (Changed label from isringd 20365 to isrngod 38409-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ (𝜑 → 𝐺 ∈ AbelOp) & ⊢ (𝜑 → 𝑋 = ran 𝐺) & ⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) & ⊢ (𝜑 → 𝑈 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = 𝑦) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦) ⇒ ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)

Theorem	rngoi 38410*	The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))

Theorem	rngosm 38411	Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)

Theorem	rngocl 38412	Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)

Theorem	rngoid 38413*	The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))

Theorem	rngoideu 38414*	The unity element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))

Theorem	rngodi 38415	Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))

Theorem	rngodir 38416	Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))

Theorem	rngoass 38417	Associative law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))

Theorem	rngo2 38418*	A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))

Theorem	rngoablo 38419	A ring's addition operation is an Abelian group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) ⇒ ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)

Theorem	rngoablo2 38420	In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)

Theorem	rngogrpo 38421	A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) ⇒ ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Theorem	rngone0 38422	The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)

Theorem	rngogcl 38423	Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Theorem	rngocom 38424	The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Theorem	rngoaass 38425	The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Theorem	rngoa32 38426	The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))

Theorem	rngoa4 38427	Rearrangement of 4 terms in a sum of ring elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))

Theorem	rngorcan 38428	Right cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Theorem	rngolcan 38429	Left cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Theorem	rngo0cl 38430	A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)

Theorem	rngo0rid 38431	The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Theorem	rngo0lid 38432	The additive identity of a ring is a left identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴)

Theorem	rngolz 38433	The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = 𝑍)

Theorem	rngorz 38434	The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = 𝑍)

Theorem	rngosn3 38435	Obsolete as of 25-Jan-2020. Use ring1zr 20849 or srg1zr 20288 instead. The only unital ring with a base set consisting in one element is the zero ring. (Contributed by FL, 13-Feb-2010.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Theorem	rngosn4 38436	Obsolete as of 25-Jan-2020. Use rngen1zr 20252 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑋 ≈ 1_o ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Theorem	rngosn6 38437	Obsolete as of 25-Jan-2020. Use ringen1zr0 20850 or srgen1zr0 20289 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1_o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))

Theorem	rngonegcl 38438	A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)

Theorem	rngoaddneg1 38439	Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑍)

Theorem	rngoaddneg2 38440	Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑍)

Theorem	rngosub 38441	Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝐷 = ( /_𝑔 ‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))

Theorem	rngmgmbs4 38442*	The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋)

Theorem	rngodm1dm2 38443	In a unital ring the domain of the first variable of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝐺 = (1^st ‘𝑅) ⇒ ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)

Theorem	rngorn1 38444	In a unital ring the range of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝐺 = (1^st ‘𝑅) ⇒ ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

Theorem	rngorn1eq 38445	In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝐺 = (1^st ‘𝑅) ⇒ ⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)

Theorem	rngomndo 38446	In a unital ring the multiplication is a monoid. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (2^nd ‘𝑅) ⇒ ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)

Theorem	rngoidmlem 38447	The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran (1^st ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))

Theorem	rngolidm 38448	The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran (1^st ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)

Theorem	rngoridm 38449	The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran (1^st ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)

Theorem	rngo1cl 38450	The unity element of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
⊢ 𝑋 = ran (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)

Theorem	rngoueqz 38451	Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20608 instead. In a unital ring the zero equals the ring unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1_o ↔ 𝑈 = 𝑍))

Theorem	rngonegmn1l 38452	Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴))

Theorem	rngonegmn1r 38453	Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)))

Theorem	rngoneglmul 38454	Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵))

Theorem	rngonegrmul 38455	Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁‘𝐵)))

Theorem	rngosubdi 38456	Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /_𝑔 ‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)))

Theorem	rngosubdir 38457	Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /_𝑔 ‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))

Theorem	zerdivemp1x 38458*	In a unital ring a left invertible element is not a zero divisor. See also ringinvnzdiv 20375. (Contributed by Jeff Madsen, 18-Apr-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))

21.24.17 Division Rings

Syntax	cdrng 38459	Extend class notation with the class of all division rings.
class DivRingOps

Definition	df-drngo 38460*	Define the class of all division rings (sometimes called skew fields). A division ring is a unital ring where every element except the additive identity has a multiplicative inverse. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}

Theorem	isdivrngo 38461	The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Theorem	drngoi 38462	The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Theorem	gidsn 38463	Obsolete as of 23-Jan-2020. Use mnd1id 18828 instead. The identity element of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
⊢ 𝐴 ∈ V ⇒ ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴

Theorem	zrdivrng 38464	The zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
⊢ 𝐴 ∈ V ⇒ ⊢ ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps

Theorem	dvrunz 38465	In a division ring the ring unit is different from the zero. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍)

Theorem	isgrpda 38466*	Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) & ⊢ (𝜑 → 𝑈 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑈𝐺𝑥) = 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑛 ∈ 𝑋 (𝑛𝐺𝑥) = 𝑈) ⇒ ⊢ (𝜑 → 𝐺 ∈ GrpOp)

Theorem	isdrngo1 38467	The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Theorem	divrngcl 38468	The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ DivRingOps ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))

Theorem	isdrngo2 38469*	A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))

Theorem	isdrngo3 38470*	A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))

21.24.18 Ring homomorphisms

Syntax	crngohom 38471	Extend class notation with the class of ring homomorphisms.
class RingOpsHom

Syntax	crngoiso 38472	Extend class notation with the class of ring isomorphisms.
class RingOpsIso

Syntax	crisc 38473	Extend class notation with the ring isomorphism relation.
class ≃_𝑟

Definition	df-rngohom 38474*	Define the function which gives the set of ring homomorphisms between two given rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ RingOpsHom = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (ran (1^st ‘𝑠) ↑_m ran (1^st ‘𝑟)) ∣ ((𝑓‘(GId‘(2^nd ‘𝑟))) = (GId‘(2^nd ‘𝑠)) ∧ ∀𝑥 ∈ ran (1^st ‘𝑟)∀𝑦 ∈ ran (1^st ‘𝑟)((𝑓‘(𝑥(1^st ‘𝑟)𝑦)) = ((𝑓‘𝑥)(1^st ‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(2^nd ‘𝑟)𝑦)) = ((𝑓‘𝑥)(2^nd ‘𝑠)(𝑓‘𝑦))))})

Theorem	rngohomval 38475*	The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝐾 = (2^nd ‘𝑆) & ⊢ 𝑌 = ran 𝐽 & ⊢ 𝑉 = (GId‘𝐾) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsHom 𝑆) = {𝑓 ∈ (𝑌 ↑_m 𝑋) ∣ ((𝑓‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦))))})

Theorem	isrngohom 38476*	The predicate "is a ring homomorphism from 𝑅 to 𝑆". (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝐾 = (2^nd ‘𝑆) & ⊢ 𝑌 = ran 𝐽 & ⊢ 𝑉 = (GId‘𝐾) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))

Theorem	rngohomf 38477	A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)

Theorem	rngohomcl 38478	Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌)

Theorem	rngohom1 38479	A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.)
⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) & ⊢ 𝐾 = (2^nd ‘𝑆) & ⊢ 𝑉 = (GId‘𝐾) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑈) = 𝑉)

Theorem	rngohomadd 38480	Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1^st ‘𝑆) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵)))

Theorem	rngohommul 38481	Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐻 = (2^nd ‘𝑅) & ⊢ 𝐾 = (2^nd ‘𝑆) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

Theorem	rngogrphom 38482	A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝐽 = (1^st ‘𝑆) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))

Theorem	rngohom0 38483	A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝑊 = (GId‘𝐽) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊)

Theorem	rngohomsub 38484	Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐻 = ( /_𝑔 ‘𝐺) & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝐾 = ( /_𝑔 ‘𝐽) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

Theorem	rngohomco 38485	The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsHom 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇))

Theorem	rngokerinj 38486	A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑊 = (GId‘𝐺) & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 & ⊢ 𝑍 = (GId‘𝐽) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (^◡𝐹 “ {𝑍}) = {𝑊}))

Definition	df-rngoiso 38487*	Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})

Theorem	rngoisoval 38488*	The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})

Theorem	isrngoiso 38489	The predicate "is a ring isomorphism between 𝑅 and 𝑆". (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌)))

Theorem	rngoiso1o 38490	A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1^st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1^st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)

Theorem	rngoisohom 38491	A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))

Theorem	rngoisocnv 38492	The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → ^◡𝐹 ∈ (𝑆 RingOpsIso 𝑅))

Theorem	rngoisoco 38493	The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsIso 𝑇))

Definition	df-risc 38494*	Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ≃_𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}

Theorem	isriscg 38495*	The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃_𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))

Theorem	isrisc 38496*	The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝑅 ∈ V & ⊢ 𝑆 ∈ V ⇒ ⊢ (𝑅 ≃_𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))

Theorem	risc 38497*	The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃_𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))

Theorem	risci 38498	Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅 ≃_𝑟 𝑆)

Theorem	riscer 38499	Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ≃_𝑟 Er dom ≃_𝑟

21.24.19 Commutative rings

Syntax	ccm2 38500	Extend class notation with a class that adds commutativity to various flavors of rings.
class Com2

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