P. 375 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37401-37500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grpomndo 37401	A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
⊢ (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp)
Theorem	exidcl 37402	Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Theorem	exidreslem 37403*	Lemma for exidres 37404 and exidresid 37405. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))    ⇒   ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
Theorem	exidres 37404	The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))    ⇒   ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId )
Theorem	exidresid 37405	The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))    ⇒   ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)
Theorem	ablo4pnp 37406	A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
Theorem	grpoeqdivid 37407	Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))
Theorem	grposnOLD 37408	The group operation for the singleton group. Obsolete, use grp1 19002. instead. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
21.22.15  Group homomorphism and isomorphism
Syntax	cghomOLD 37409	Obsolete version of cghm 19166 as of 15-Mar-2020. Extend class notation to include the class of group homomorphisms. (New usage is discouraged.)
class GrpOpHom
Definition	df-ghomOLD 37410*	Obsolete version of df-ghm 19167 as of 15-Mar-2020. Define the set of group homomorphisms from 𝑔 to ℎ. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
⊢ GrpOpHom = (𝑔 ∈ GrpOp, ℎ ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
Theorem	elghomlem1OLD 37411*	Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37413. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
Theorem	elghomlem2OLD 37412*	Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37413. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
Theorem	elghomOLD 37413*	Obsolete version of isghm 19169 as of 15-Mar-2020. Membership in the set of group homomorphisms from 𝐺 to 𝐻. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑊 = ran 𝐻    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋⟶𝑊 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
Theorem	ghomlinOLD 37414	Obsolete version of ghmlin 19174 as of 15-Mar-2020. Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐻(𝐹‘𝐵)) = (𝐹‘(𝐴𝐺𝐵)))
Theorem	ghomidOLD 37415	Obsolete version of ghmid 19175 as of 15-Mar-2020. A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝑇 = (GId‘𝐻)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) = 𝑇)
Theorem	ghomf 37416	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 37417	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 37418	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 37419	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}))
21.22.16  Rings
Syntax	crngo 37420	Extend class notation with the class of all unital rings.
class RingOps
Definition	df-rngo 37421*	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 37422	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 37423*	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 37424*	Conditions that determine a ring. (Changed label from isringd 20226 to isrngod 37424-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 37425*	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘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 37426	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
Theorem	rngocl 37427	Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
Theorem	rngoid 37428*	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))
Theorem	rngoideu 37429*	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
Theorem	rngodi 37430	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))
Theorem	rngodir 37431	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))
Theorem	rngoass 37432	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
Theorem	rngo2 37433*	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))
Theorem	rngoablo 37434	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 = (1st ‘𝑅)    ⇒   ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
Theorem	rngoablo2 37435	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 37436	A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    ⇒   ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
Theorem	rngone0 37437	The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)
Theorem	rngogcl 37438	Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Theorem	rngocom 37439	The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Theorem	rngoaass 37440	The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
Theorem	rngoa32 37441	The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
Theorem	rngoa4 37442	Rearrangement of 4 terms in a sum of ring elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
Theorem	rngorcan 37443	Right cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
Theorem	rngolcan 37444	Left cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
Theorem	rngo0cl 37445	A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
Theorem	rngo0rid 37446	The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)
Theorem	rngo0lid 37447	The additive identity of a ring is a left identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴)
Theorem	rngolz 37448	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = 𝑍)
Theorem	rngorz 37449	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = 𝑍)
Theorem	rngosn3 37450	Obsolete as of 25-Jan-2020. Use ring1zr 20663 or srg1zr 20154 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 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
Theorem	rngosn4 37451	Obsolete as of 25-Jan-2020. Use rngen1zr 20664 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 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
Theorem	rngosn6 37452	Obsolete as of 25-Jan-2020. Use ringen1zr 20665 or srgen1zr 20155 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
Theorem	rngonegcl 37453	A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑁 = (inv‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
Theorem	rngoaddneg1 37454	Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑁 = (inv‘𝐺)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑍)
Theorem	rngoaddneg2 37455	Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑁 = (inv‘𝐺)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑍)
Theorem	rngosub 37456	Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑁 = (inv‘𝐺)    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))
Theorem	rngmgmbs4 37457*	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 37458	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 = (2nd ‘𝑅)    &   ⊢ 𝐺 = (1st ‘𝑅)    ⇒   ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)
Theorem	rngorn1 37459	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 = (2nd ‘𝑅)    &   ⊢ 𝐺 = (1st ‘𝑅)    ⇒   ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)
Theorem	rngorn1eq 37460	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 = (2nd ‘𝑅)    &   ⊢ 𝐺 = (1st ‘𝑅)    ⇒   ⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)
Theorem	rngomndo 37461	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 = (2nd ‘𝑅)    ⇒   ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
Theorem	rngoidmlem 37462	The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran (1st ‘𝑅)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
Theorem	rngolidm 37463	The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran (1st ‘𝑅)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)
Theorem	rngoridm 37464	The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran (1st ‘𝑅)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)
Theorem	rngo1cl 37465	The unity element of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
⊢ 𝑋 = ran (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
Theorem	rngoueqz 37466	Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20468 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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
Theorem	rngonegmn1l 37467	Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑁 = (inv‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴))
Theorem	rngonegmn1r 37468	Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑁 = (inv‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)))
Theorem	rngoneglmul 37469	Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑁 = (inv‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵))
Theorem	rngonegrmul 37470	Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑁 = (inv‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁‘𝐵)))
Theorem	rngosubdi 37471	Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)))
Theorem	rngosubdir 37472	Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))
Theorem	zerdivemp1x 37473*	In a unital ring a left invertible element is not a zero divisor. See also ringinvnzdiv 20236. (Contributed by Jeff Madsen, 18-Apr-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))
21.22.17  Division Rings
Syntax	cdrng 37474	Extend class notation with the class of all division rings.
class DivRingOps
Definition	df-drngo 37475*	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 37476	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 37477	The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
Theorem	gidsn 37478	Obsolete as of 23-Jan-2020. Use mnd1id 18731 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 37479	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 37480	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 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍)
Theorem	isgrpda 37481*	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 37482	The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
Theorem	divrngcl 37483	The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ DivRingOps ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))
Theorem	isdrngo2 37484*	A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘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 37485*	A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))
21.22.18  Ring homomorphisms
Syntax	crngohom 37486	Extend class notation with the class of ring homomorphisms.
class RingOpsHom
Syntax	crngoiso 37487	Extend class notation with the class of ring isomorphisms.
class RingOpsIso
Syntax	crisc 37488	Extend class notation with the ring isomorphism relation.
class ≃𝑟
Definition	df-rngohom 37489*	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 (1st ‘𝑠) ↑m ran (1st ‘𝑟)) ∣ ((𝑓‘(GId‘(2nd ‘𝑟))) = (GId‘(2nd ‘𝑠)) ∧ ∀𝑥 ∈ ran (1st ‘𝑟)∀𝑦 ∈ ran (1st ‘𝑟)((𝑓‘(𝑥(1st ‘𝑟)𝑦)) = ((𝑓‘𝑥)(1st ‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(2nd ‘𝑟)𝑦)) = ((𝑓‘𝑥)(2nd ‘𝑠)(𝑓‘𝑦))))})
Theorem	rngohomval 37490*	The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐻)    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝐾 = (2nd ‘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 37491*	The predicate "is a ring homomorphism from 𝑅 to 𝑆". (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐻)    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝐾 = (2nd ‘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 37492	A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑌 = ran 𝐽    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)
Theorem	rngohomcl 37493	Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑌 = ran 𝐽    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌)
Theorem	rngohom1 37494	A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.)
⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑈 = (GId‘𝐻)    &   ⊢ 𝐾 = (2nd ‘𝑆)    &   ⊢ 𝑉 = (GId‘𝐾)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑈) = 𝑉)
Theorem	rngohomadd 37495	Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐽 = (1st ‘𝑆)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵)))
Theorem	rngohommul 37496	Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝐾 = (2nd ‘𝑆)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
Theorem	rngogrphom 37497	A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐽 = (1st ‘𝑆)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
Theorem	rngohom0 37498	A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑊 = (GId‘𝐽)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊)
Theorem	rngohomsub 37499	Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐻 = ( /𝑔 ‘𝐺)    &   ⊢ 𝐽 = (1st ‘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 37500	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))