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Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrrncmslem 32801* Lemma for rrncms 32802. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)    &   𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐽 = (MetOpen‘(ℝn𝐼))    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐹 ∈ (Cau‘(ℝn𝐼)))    &   (𝜑𝐹:ℕ⟶𝑋)    &   𝑃 = (𝑚𝐼 ↦ ( ⇝ ‘(𝑡 ∈ ℕ ↦ ((𝐹𝑡)‘𝑚))))       (𝜑𝐹 ∈ dom (⇝𝑡𝐽))

Theoremrrncms 32802 Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)       (𝐼 ∈ Fin → (ℝn𝐼) ∈ (CMet‘𝑋))

Theoremrepwsmet 32803 The supremum metric on ℝ↑𝐼 is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
𝑌 = ((ℂflds ℝ) ↑s 𝐼)    &   𝐷 = (dist‘𝑌)    &   𝑋 = (ℝ ↑𝑚 𝐼)       (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋))

Theoremrrnequiv 32804 The supremum metric on ℝ↑𝐼 is equivalent to the n metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
𝑌 = ((ℂflds ℝ) ↑s 𝐼)    &   𝐷 = (dist‘𝑌)    &   𝑋 = (ℝ ↑𝑚 𝐼)    &   (𝜑𝐼 ∈ Fin)       ((𝜑 ∧ (𝐹𝑋𝐺𝑋)) → ((𝐹𝐷𝐺) ≤ (𝐹(ℝn𝐼)𝐺) ∧ (𝐹(ℝn𝐼)𝐺) ≤ ((√‘(#‘𝐼)) · (𝐹𝐷𝐺))))

Theoremrrntotbnd 32805 A set in Euclidean space is totally bounded iff its is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)    &   𝑀 = ((ℝn𝐼) ↾ (𝑌 × 𝑌))       (𝐼 ∈ Fin → (𝑀 ∈ (TotBnd‘𝑌) ↔ 𝑀 ∈ (Bnd‘𝑌)))

Theoremrrnheibor 32806 Heine-Borel theorem for Euclidean space. A subset of Euclidean space is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)    &   𝑀 = ((ℝn𝐼) ↾ (𝑌 × 𝑌))    &   𝑇 = (MetOpen‘𝑀)    &   𝑈 = (MetOpen‘(ℝn𝐼))       ((𝐼 ∈ Fin ∧ 𝑌𝑋) → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌))))

21.19.12  Intervals (continued)

Theoremismrer1 32807* An isometry between and ℝ↑1. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑅 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐹 = (𝑥 ∈ ℝ ↦ ({𝐴} × {𝑥}))       (𝐴𝑉𝐹 ∈ (𝑅 Ismty (ℝn‘{𝐴})))

Theoremreheibor 32808 Heine-Borel theorem for real numbers. A subset of is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑀 = ((abs ∘ − ) ↾ (𝑌 × 𝑌))    &   𝑇 = (MetOpen‘𝑀)    &   𝑈 = (topGen‘ran (,))       (𝑌 ⊆ ℝ → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌))))

Theoremiccbnd 32809 A closed interval in is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Sep-2015.)
𝐽 = (𝐴[,]𝐵)    &   𝑀 = ((abs ∘ − ) ↾ (𝐽 × 𝐽))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑀 ∈ (Bnd‘𝐽))

TheoremicccmpALT 32810 A closed interval in is compact. Alternate proof of icccmp 22436 using the Heine-Borel theorem heibor 32790. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Aug-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐽 = (𝐴[,]𝐵)    &   𝑀 = ((abs ∘ − ) ↾ (𝐽 × 𝐽))    &   𝑇 = (MetOpen‘𝑀)       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp)

21.19.13  Operation properties

Syntaxcass 32811 Extend class notation with a device to add associativity to internal operations.
class Ass

Definitiondf-ass 32812* A device to add associativity to various sorts of internal operations. The definition is meaningful when 𝑔 is a magma at least. (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}

Syntaxcexid 32813 Extend class notation with the class of all the internal operations with an identity element.
class ExId

Definitiondf-exid 32814* A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}

Theoremisass 32815* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
𝑋 = dom dom 𝐺       (𝐺𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))

Theoremisexid 32816* The predicate 𝐺 has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = dom dom 𝐺       (𝐺𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))

21.19.14  Groups and related structures

Syntaxcmagm 32817 Extend class notation with the class of all magmas.
class Magma

Definitiondf-mgmOLD 32818* Obsolete version of df-mgm 17065 as of 3-Feb-2020. A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}

TheoremismgmOLD 32819 Obsolete version of ismgm 17066 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))

TheoremclmgmOLD 32820 Obsolete version of mgmcl 17068 as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       ((𝐺 ∈ Magma ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

TheoremopidonOLD 32821 Obsolete version of mndpfo 17137 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)

TheoremrngopidOLD 32822 Obsolete version of mndpfo 17137 as of 23-Jan-2020. Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)

Theoremopidon2OLD 32823 Obsolete version of mndpfo 17137 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)

Theoremisexid2 32824* If 𝐺 ∈ (Magma ∩ ExId ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

Theoremexidu1 32825* Unicity 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       (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

Theoremidrval 32826* 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    &   𝑈 = (GId‘𝐺)       (𝐺𝐴𝑈 = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))

Theoremiorlid 32827 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    &   𝑈 = (GId‘𝐺)       (𝐺 ∈ (Magma ∩ ExId ) → 𝑈𝑋)

Theoremcmpidelt 32828 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    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))

Syntaxcsem 32829 Extend class notation with the class of all semi-groups.
class SemiGrp

Definitiondf-sgrOLD 32830 Obsolete version of df-sgrp 17107 as of 3-Feb-2020. A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
SemiGrp = (Magma ∩ Ass)

TheoremsmgrpismgmOLD 32831 Obsolete version of sgrpmgm 17112 as of 3-Feb-2020. A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

TheoremissmgrpOLD 32832* Obsolete version of issgrp 17108 as of 3-Feb-2020. The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       (𝐺𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))

Theoremsmgrpmgm 32833 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
𝑋 = dom dom 𝐺       (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋)

TheoremsmgrpassOLD 32834* Obsolete version of sgrpass 17113 as of 3-Feb-2020. A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       (𝐺 ∈ SemiGrp → ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))

Syntaxcmndo 32835 Extend class notation with the class of all monoids.
class MndOp

Definitiondf-mndo 32836 A monoid is a semi-group with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp = (SemiGrp ∩ ExId )

TheoremmndoissmgrpOLD 32837 Obsolete version of mndsgrp 17122 as of 3-Feb-2020. A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)

Theoremmndoisexid 32838 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
(𝐺 ∈ MndOp → 𝐺 ∈ ExId )

TheoremmndoismgmOLD 32839 Obsolete version of mndmgm 17123 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 → 𝐺 ∈ Magma)

Theoremmndomgmid 32840 A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
(𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))

Theoremismndo 32841* 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       (𝐺𝐴 → (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))

Theoremismndo1 32842* 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       (𝐺𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))

Theoremismndo2 32843* 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       (𝐺𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))

Theoremgrpomndo 32844 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)

Theoremexidcl 32845 Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑋 = ran 𝐺       ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Theoremexidreslem 32846* Lemma for exidres 32847 and exidresid 32848. (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 ) ∧ 𝑌𝑋𝑈𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))

Theoremexidres 32847 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 ) ∧ 𝑌𝑋𝑈𝑌) → 𝐻 ∈ ExId )

Theoremexidresid 32848 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 ) ∧ 𝑌𝑋𝑈𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)

Theoremablo4pnp 32849 A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))

Theoremgrpoeqdivid 32850 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 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))

TheoremgrposnOLD 32851 The group operation for the singleton group. Obsolete, use grp1 17345. instead (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V       {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp

21.19.15  Group homomorphism and isomorphism

SyntaxcghomOLD 32852 Obsolete version of cghm 17480 as of 15-Mar-2020. Extend class notation to include the class of group homomorphisms. (New usage is discouraged.)
class GrpOpHom

Definitiondf-ghomOLD 32853* Obsolete version of df-ghm 17481 as of 15-Mar-2020. Define the set of group homomorphisms from 𝑔 to . (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})

Theoremelghomlem1OLD 32854* Obsolete as of 15-Mar-2020. Lemma for elghomOLD 32856. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}       ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)

Theoremelghomlem2OLD 32855* Obsolete as of 15-Mar-2020. Lemma for elghomOLD 32856. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}       ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))

TheoremelghomOLD 32856* Obsolete version of isghm 17483 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:𝑋𝑊 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))

TheoremghomlinOLD 32857 Obsolete version of ghmlin 17488 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)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵)))

TheoremghomidOLD 32858 Obsolete version of ghmid 17489 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)) → (𝐹𝑈) = 𝑇)

Theoremghomf 32859 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:𝑋𝑊)

Theoremghomco 32860 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))) → (𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾))

Theoremghomdiv 32861 Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)    &   𝐶 = ( /𝑔𝐻)       (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹𝐴)𝐶(𝐹𝐵)))

Theoremgrpokerinj 32862 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:𝑋1-1𝑌 ↔ (𝐹 “ {𝑈}) = {𝑊}))

21.19.16  Rings

Syntaxcrngo 32863 Extend class notation with the class of all unital rings.
class RingOps

Definitiondf-rngo 32864* Define the class of all unital rings. (Contributed by Jeff Hankins, 21-Nov-2006.) (New usage is discouraged.)
RingOps = {⟨𝑔, ⟩ ∣ ((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ∧ ∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦)))}

Theoremrelrngo 32865 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 32866* 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       (𝐻𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))

Theoremisrngod 32867* Conditions that determine a ring. (Changed label from isringd 18408 to isrngod 32867-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
(𝜑𝐺 ∈ AbelOp)    &   (𝜑𝑋 = ran 𝐺)    &   (𝜑𝐻:(𝑋 × 𝑋)⟶𝑋)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))    &   (𝜑𝑈𝑋)    &   ((𝜑𝑦𝑋) → (𝑈𝐻𝑦) = 𝑦)    &   ((𝜑𝑦𝑋) → (𝑦𝐻𝑈) = 𝑦)       (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)

Theoremrngoi 32868* 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))

Theoremrngosm 32869 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)

Theoremrngocl 32870 Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)

Theoremrngoid 32871* 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑢𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))

Theoremrngoideu 32872* The unit element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))

Theoremrngodi 32873 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))

Theoremrngodir 32874 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))

Theoremrngoass 32875 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))

Theoremrngo2 32876* 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑥𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))

Theoremrngoablo 32877 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 = (1st𝑅)       (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)

Theoremrngoablo2 32878 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)

Theoremrngogrpo 32879 A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)       (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Theoremrngone0 32880 The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → 𝑋 ≠ ∅)

Theoremrngogcl 32881 Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Theoremrngocom 32882 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Theoremrngoaass 32883 The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Theoremrngoa32 32884 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))

Theoremrngoa4 32885 Rearrangement of 4 terms in a sum of ring elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))

Theoremrngorcan 32886 Right cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Theoremrngolcan 32887 Left cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Theoremrngo0cl 32888 A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ RingOps → 𝑍𝑋)

Theoremrngo0rid 32889 The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Theoremrngo0lid 32890 The additive identity of a ring is a left identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)

Theoremrngolz 32891 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐻𝐴) = 𝑍)

Theoremrngorz 32892 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 = (1st𝑅)    &   𝐻 = (2nd𝑅)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑍) = 𝑍)

Theoremrngosn3 32893 Obsolete as of 25-Jan-2020. Use ring1zr 19096 or srg1zr 18352 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 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐴𝐵) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Theoremrngosn4 32894 Obsolete as of 25-Jan-2020. Use rngen1zr 19097 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 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 ≈ 1𝑜𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Theoremrngosn6 32895 Obsolete as of 25-Jan-2020. Use ringen1zr 19098 or srgen1zr 18353 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))

Theoremrngonegcl 32896 A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)

𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝑍 = (GId‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑍)