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Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexidu1 36201* Uniqueness of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    β‡’   (𝐺 ∈ (Magma ∩ ExId ) β†’ βˆƒ!𝑒 ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ (π‘₯𝐺𝑒) = π‘₯))
 
Theoremidrval 36202* The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   π‘ˆ = (GIdβ€˜πΊ)    β‡’   (𝐺 ∈ 𝐴 β†’ π‘ˆ = (℩𝑒 ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ (π‘₯𝐺𝑒) = π‘₯)))
 
Theoremiorlid 36203 A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   π‘ˆ = (GIdβ€˜πΊ)    β‡’   (𝐺 ∈ (Magma ∩ ExId ) β†’ π‘ˆ ∈ 𝑋)
 
Theoremcmpidelt 36204 A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   π‘ˆ = (GIdβ€˜πΊ)    β‡’   ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπΊπ΄) = 𝐴 ∧ (π΄πΊπ‘ˆ) = 𝐴))
 
Syntaxcsem 36205 Extend class notation with the class of all semigroups.
class SemiGrp
 
Definitiondf-sgrOLD 36206 Obsolete version of df-sgrp 18481 as of 3-Feb-2020. A semigroup is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
SemiGrp = (Magma ∩ Ass)
 
TheoremsmgrpismgmOLD 36207 Obsolete version of sgrpmgm 18486 as of 3-Feb-2020. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ SemiGrp β†’ 𝐺 ∈ Magma)
 
TheoremissmgrpOLD 36208* Obsolete version of issgrp 18482 as of 3-Feb-2020. The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺    β‡’   (𝐺 ∈ 𝐴 β†’ (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)))))
 
Theoremsmgrpmgm 36209 A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
𝑋 = dom dom 𝐺    β‡’   (𝐺 ∈ SemiGrp β†’ 𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹)
 
TheoremsmgrpassOLD 36210* Obsolete version of sgrpass 18487 as of 3-Feb-2020. A semigroup is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺    β‡’   (𝐺 ∈ SemiGrp β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)))
 
Syntaxcmndo 36211 Extend class notation with the class of all monoids.
class MndOp
 
Definitiondf-mndo 36212 A monoid is a semigroup with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp = (SemiGrp ∩ ExId )
 
TheoremmndoissmgrpOLD 36213 Obsolete version of mndsgrp 18497 as of 3-Feb-2020. A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ MndOp β†’ 𝐺 ∈ SemiGrp)
 
Theoremmndoisexid 36214 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
(𝐺 ∈ MndOp β†’ 𝐺 ∈ ExId )
 
TheoremmndoismgmOLD 36215 Obsolete version of mndmgm 18498 as of 3-Feb-2020. A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ MndOp β†’ 𝐺 ∈ Magma)
 
Theoremmndomgmid 36216 A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
(𝐺 ∈ MndOp β†’ 𝐺 ∈ (Magma ∩ ExId ))
 
Theoremismndo 36217* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = dom dom 𝐺    β‡’   (𝐺 ∈ 𝐴 β†’ (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦))))
 
Theoremismndo1 36218* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = dom dom 𝐺    β‡’   (𝐺 ∈ 𝐴 β†’ (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦))))
 
Theoremismndo2 36219* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    β‡’   (𝐺 ∈ 𝐴 β†’ (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐺𝑦) = 𝑦 ∧ (𝑦𝐺π‘₯) = 𝑦))))
 
Theoremgrpomndo 36220 A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
(𝐺 ∈ GrpOp β†’ 𝐺 ∈ MndOp)
 
Theoremexidcl 36221 Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑋 = ran 𝐺    β‡’   ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
 
Theoremexidreslem 36222* Lemma for exidres 36223 and exidresid 36224. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝑋 = ran 𝐺    &   π‘ˆ = (GIdβ€˜πΊ)    &   π» = (𝐺 β†Ύ (π‘Œ Γ— π‘Œ))    β‡’   ((𝐺 ∈ (Magma ∩ ExId ) ∧ π‘Œ βŠ† 𝑋 ∧ π‘ˆ ∈ π‘Œ) β†’ (π‘ˆ ∈ dom dom 𝐻 ∧ βˆ€π‘₯ ∈ dom dom 𝐻((π‘ˆπ»π‘₯) = π‘₯ ∧ (π‘₯π»π‘ˆ) = π‘₯)))
 
Theoremexidres 36223 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    &   π‘ˆ = (GIdβ€˜πΊ)    &   π» = (𝐺 β†Ύ (π‘Œ Γ— π‘Œ))    β‡’   ((𝐺 ∈ (Magma ∩ ExId ) ∧ π‘Œ βŠ† 𝑋 ∧ π‘ˆ ∈ π‘Œ) β†’ 𝐻 ∈ ExId )
 
Theoremexidresid 36224 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    &   π‘ˆ = (GIdβ€˜πΊ)    &   π» = (𝐺 β†Ύ (π‘Œ Γ— π‘Œ))    β‡’   (((𝐺 ∈ (Magma ∩ ExId ) ∧ π‘Œ βŠ† 𝑋 ∧ π‘ˆ ∈ π‘Œ) ∧ 𝐻 ∈ Magma) β†’ (GIdβ€˜π») = π‘ˆ)
 
Theoremablo4pnp 36225 A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)
𝑋 = ran 𝐺    &   π· = ( /𝑔 β€˜πΊ)    β‡’   ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐢 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) β†’ ((𝐴𝐺𝐡)𝐷(𝐢𝐺𝐹)) = ((𝐴𝐷𝐢)𝐺(𝐡𝐷𝐹)))
 
Theoremgrpoeqdivid 36226 Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝑋 = ran 𝐺    &   π‘ˆ = (GIdβ€˜πΊ)    &   π· = ( /𝑔 β€˜πΊ)    β‡’   ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴 = 𝐡 ↔ (𝐴𝐷𝐡) = π‘ˆ))
 
TheoremgrposnOLD 36227 The group operation for the singleton group. Obsolete, use grp1 18788. instead. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    β‡’   {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
 
21.20.15  Group homomorphism and isomorphism
 
SyntaxcghomOLD 36228 Obsolete version of cghm 18937 as of 15-Mar-2020. Extend class notation to include the class of group homomorphisms. (New usage is discouraged.)
class GrpOpHom
 
Definitiondf-ghomOLD 36229* Obsolete version of df-ghm 18938 as of 15-Mar-2020. Define the set of group homomorphisms from 𝑔 to β„Ž. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom = (𝑔 ∈ GrpOp, β„Ž ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))})
 
Theoremelghomlem1OLD 36230* Obsolete as of 15-Mar-2020. Lemma for elghomOLD 36232. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))}    β‡’   ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
 
Theoremelghomlem2OLD 36231* Obsolete as of 15-Mar-2020. Lemma for elghomOLD 36232. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))}    β‡’   ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
 
TheoremelghomOLD 36232* Obsolete version of isghm 18940 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    &   π‘Š = ran 𝐻    β‡’   ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:π‘‹βŸΆπ‘Š ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
 
TheoremghomlinOLD 36233 Obsolete version of ghmlin 18945 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    β‡’   (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((πΉβ€˜π΄)𝐻(πΉβ€˜π΅)) = (πΉβ€˜(𝐴𝐺𝐡)))
 
TheoremghomidOLD 36234 Obsolete version of ghmid 18946 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.)
π‘ˆ = (GIdβ€˜πΊ)    &   π‘‡ = (GIdβ€˜π»)    β‡’   ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (πΉβ€˜π‘ˆ) = 𝑇)
 
Theoremghomf 36235 Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.)
𝑋 = ran 𝐺    &   π‘Š = ran 𝐻    β‡’   ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ 𝐹:π‘‹βŸΆπ‘Š)
 
Theoremghomco 36236 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))) β†’ (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾))
 
Theoremghomdiv 36237 Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑋 = ran 𝐺    &   π· = ( /𝑔 β€˜πΊ)    &   πΆ = ( /𝑔 β€˜π»)    β‡’   (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜(𝐴𝐷𝐡)) = ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅)))
 
Theoremgrpokerinj 36238 A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑋 = ran 𝐺    &   π‘Š = (GIdβ€˜πΊ)    &   π‘Œ = ran 𝐻    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (𝐹:𝑋–1-1β†’π‘Œ ↔ (◑𝐹 β€œ {π‘ˆ}) = {π‘Š}))
 
21.20.16  Rings
 
Syntaxcrngo 36239 Extend class notation with the class of all unital rings.
class RingOps
 
Definitiondf-rngo 36240* Define the class of all unital rings. (Contributed by Jeff Hankins, 21-Nov-2006.) (New usage is discouraged.)
RingOps = {βŸ¨π‘”, β„ŽβŸ© ∣ ((𝑔 ∈ AbelOp ∧ β„Ž:(ran 𝑔 Γ— ran 𝑔)⟢ran 𝑔) ∧ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ∧ βˆƒπ‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦)))}
 
Theoremrelrngo 36241 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
 
Theoremisrngo 36242* 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    β‡’   (𝐻 ∈ 𝐴 β†’ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))))
 
Theoremisrngod 36243* Conditions that determine a ring. (Changed label from isringd 19932 to isrngod 36243-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
(πœ‘ β†’ 𝐺 ∈ AbelOp)    &   (πœ‘ β†’ 𝑋 = ran 𝐺)    &   (πœ‘ β†’ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))    &   (πœ‘ β†’ π‘ˆ ∈ 𝑋)    &   ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ (π‘ˆπ»π‘¦) = 𝑦)    &   ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ (π‘¦π»π‘ˆ) = 𝑦)    β‡’   (πœ‘ β†’ ⟨𝐺, 𝐻⟩ ∈ RingOps)
 
Theoremrngoi 36244* 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
 
Theoremrngosm 36245 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹)
 
Theoremrngocl 36246 Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
 
Theoremrngoid 36247* 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ βˆƒπ‘’ ∈ 𝑋 ((𝑒𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑒) = 𝐴))
 
Theoremrngoideu 36248* 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ βˆƒ!𝑒 ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐻π‘₯) = π‘₯ ∧ (π‘₯𝐻𝑒) = π‘₯))
 
Theoremrngodi 36249 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐻(𝐡𝐺𝐢)) = ((𝐴𝐻𝐡)𝐺(𝐴𝐻𝐢)))
 
Theoremrngodir 36250 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐻𝐢) = ((𝐴𝐻𝐢)𝐺(𝐡𝐻𝐢)))
 
Theoremrngoass 36251 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐡)𝐻𝐢) = (𝐴𝐻(𝐡𝐻𝐢)))
 
Theoremrngo2 36252* 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ βˆƒπ‘₯ ∈ 𝑋 (𝐴𝐺𝐴) = ((π‘₯𝐺π‘₯)𝐻𝐴))
 
Theoremrngoablo 36253 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 β€˜π‘…)    β‡’   (𝑅 ∈ RingOps β†’ 𝐺 ∈ AbelOp)
 
Theoremrngoablo2 36254 In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
(⟨𝐺, 𝐻⟩ ∈ RingOps β†’ 𝐺 ∈ AbelOp)
 
Theoremrngogrpo 36255 A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    β‡’   (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
 
Theoremrngone0 36256 The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ 𝑋 β‰  βˆ…)
 
Theoremrngogcl 36257 Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
 
Theoremrngocom 36258 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) = (𝐡𝐺𝐴))
 
Theoremrngoaass 36259 The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺𝐢) = (𝐴𝐺(𝐡𝐺𝐢)))
 
Theoremrngoa32 36260 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺𝐢) = ((𝐴𝐺𝐢)𝐺𝐡))
 
Theoremrngoa4 36261 Rearrangement of 4 terms in a sum of ring elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐢 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺(𝐢𝐺𝐷)) = ((𝐴𝐺𝐢)𝐺(𝐡𝐺𝐷)))
 
Theoremrngorcan 36262 Right cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐢) = (𝐡𝐺𝐢) ↔ 𝐴 = 𝐡))
 
Theoremrngolcan 36263 Left cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) ↔ 𝐴 = 𝐡))
 
Theoremrngo0cl 36264 A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   (𝑅 ∈ RingOps β†’ 𝑍 ∈ 𝑋)
 
Theoremrngo0rid 36265 The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺𝑍) = 𝐴)
 
Theoremrngo0lid 36266 The additive identity of a ring is a left identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝑍𝐺𝐴) = 𝐴)
 
Theoremrngolz 36267 The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
𝑍 = (GIdβ€˜πΊ)    &   π‘‹ = ran 𝐺    &   πΊ = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝑍𝐻𝐴) = 𝑍)
 
Theoremrngorz 36268 The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
𝑍 = (GIdβ€˜πΊ)    &   π‘‹ = ran 𝐺    &   πΊ = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻𝑍) = 𝑍)
 
Theoremrngosn3 36269 Obsolete as of 25-Jan-2020. Use ring1zr 20668 or srg1zr 19870 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 β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝐡) β†’ (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
 
Theoremrngosn4 36270 Obsolete as of 25-Jan-2020. Use rngen1zr 20669 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 β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝑋 β‰ˆ 1o ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
 
Theoremrngosn6 36271 Obsolete as of 25-Jan-2020. Use ringen1zr 20670 or srgen1zr 19871 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   (𝑅 ∈ RingOps β†’ (𝑋 β‰ˆ 1o ↔ 𝑅 = ⟨{βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}, {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩))
 
Theoremrngonegcl 36272 A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (invβ€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
 
Theoremrngoaddneg1 36273 Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (invβ€˜πΊ)    &   π‘ = (GIdβ€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = 𝑍)
 
Theoremrngoaddneg2 36274 Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (invβ€˜πΊ)    &   π‘ = (GIdβ€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = 𝑍)
 
Theoremrngosub 36275 Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (invβ€˜πΊ)    &   π· = ( /𝑔 β€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐴𝐺(π‘β€˜π΅)))
 
Theoremrngmgmbs4 36276* 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 Γ— 𝑋)βŸΆπ‘‹ ∧ βˆƒπ‘’ ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ (π‘₯𝐺𝑒) = π‘₯)) β†’ ran 𝐺 = 𝑋)
 
Theoremrngodm1dm2 36277 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 β€˜π‘…)    &   πΊ = (1st β€˜π‘…)    β‡’   (𝑅 ∈ RingOps β†’ dom dom 𝐺 = dom dom 𝐻)
 
Theoremrngorn1 36278 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 β€˜π‘…)    &   πΊ = (1st β€˜π‘…)    β‡’   (𝑅 ∈ RingOps β†’ ran 𝐺 = dom dom 𝐻)
 
Theoremrngorn1eq 36279 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 β€˜π‘…)    &   πΊ = (1st β€˜π‘…)    β‡’   (𝑅 ∈ RingOps β†’ ran 𝐺 = ran 𝐻)
 
Theoremrngomndo 36280 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 β€˜π‘…)    β‡’   (𝑅 ∈ RingOps β†’ 𝐻 ∈ MndOp)
 
Theoremrngoidmlem 36281 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 β€˜π‘…)    &   π‘‹ = ran (1st β€˜π‘…)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπ»π΄) = 𝐴 ∧ (π΄π»π‘ˆ) = 𝐴))
 
Theoremrngolidm 36282 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 β€˜π‘…)    &   π‘‹ = ran (1st β€˜π‘…)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘ˆπ»π΄) = 𝐴)
 
Theoremrngoridm 36283 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 β€˜π‘…)    &   π‘‹ = ran (1st β€˜π‘…)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π΄π»π‘ˆ) = 𝐴)
 
Theoremrngo1cl 36284 The unity element of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
𝑋 = ran (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
 
Theoremrngoueqz 36285 Obsolete as of 23-Jan-2020. Use 0ring01eqbi 20666 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘ = (GIdβ€˜πΊ)    &   π‘ˆ = (GIdβ€˜π»)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ (𝑋 β‰ˆ 1o ↔ π‘ˆ = 𝑍))
 
Theoremrngonegmn1l 36286 Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (invβ€˜πΊ)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = ((π‘β€˜π‘ˆ)𝐻𝐴))
 
Theoremrngonegmn1r 36287 Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (invβ€˜πΊ)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)))
 
Theoremrngoneglmul 36288 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (invβ€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((π‘β€˜π΄)𝐻𝐡))
 
Theoremrngonegrmul 36289 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (invβ€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = (𝐴𝐻(π‘β€˜π΅)))
 
Theoremrngosubdi 36290 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π· = ( /𝑔 β€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐻(𝐡𝐷𝐢)) = ((𝐴𝐻𝐡)𝐷(𝐴𝐻𝐢)))
 
Theoremrngosubdir 36291 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π· = ( /𝑔 β€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐻𝐢) = ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)))
 
Theoremzerdivemp1x 36292* In a unital ring a left invertible element is not a zero divisor. See also ringinvnzdiv 19940. (Contributed by Jeff Madsen, 18-Apr-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘ = (GIdβ€˜πΊ)    &   π‘‹ = ran 𝐺    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ βˆƒπ‘Ž ∈ 𝑋 (π‘Žπ»π΄) = π‘ˆ) β†’ (𝐡 ∈ 𝑋 β†’ ((𝐴𝐻𝐡) = 𝑍 β†’ 𝐡 = 𝑍)))
 
21.20.17  Division Rings
 
Syntaxcdrng 36293 Extend class notation with the class of all division rings.
class DivRingOps
 
Definitiondf-drngo 36294* 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 = {βŸ¨π‘”, β„ŽβŸ© ∣ (βŸ¨π‘”, β„ŽβŸ© ∈ RingOps ∧ (β„Ž β†Ύ ((ran 𝑔 βˆ– {(GIdβ€˜π‘”)}) Γ— (ran 𝑔 βˆ– {(GIdβ€˜π‘”)}))) ∈ GrpOp)}
 
Theoremisdivrngo 36295 The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
(𝐻 ∈ 𝐴 β†’ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 β†Ύ ((ran 𝐺 βˆ– {(GIdβ€˜πΊ)}) Γ— (ran 𝐺 βˆ– {(GIdβ€˜πΊ)}))) ∈ GrpOp)))
 
Theoremdrngoi 36296 The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   (𝑅 ∈ DivRingOps β†’ (𝑅 ∈ RingOps ∧ (𝐻 β†Ύ ((𝑋 βˆ– {𝑍}) Γ— (𝑋 βˆ– {𝑍}))) ∈ GrpOp))
 
Theoremgidsn 36297 Obsolete as of 23-Jan-2020. Use mnd1id 18533 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    β‡’   (GIdβ€˜{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴
 
Theoremzrdivrng 36298 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    β‡’    Β¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps
 
Theoremdvrunz 36299 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 β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   (𝑅 ∈ DivRingOps β†’ π‘ˆ β‰  𝑍)
 
Theoremisgrpda 36300* Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
(πœ‘ β†’ 𝑋 ∈ V)    &   (πœ‘ β†’ 𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)))    &   (πœ‘ β†’ π‘ˆ ∈ 𝑋)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (π‘ˆπΊπ‘₯) = π‘₯)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘› ∈ 𝑋 (𝑛𝐺π‘₯) = π‘ˆ)    β‡’   (πœ‘ β†’ 𝐺 ∈ GrpOp)
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