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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempwmnd 18101* The power set of a class 𝐴 is a monoid under union. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))       𝑀 ∈ Mnd

10.2  Groups

10.2.1  Definition and basic properties

Syntaxcgrp 18102 Extend class notation with class of all groups.
class Grp

Syntaxcminusg 18103 Extend class notation with inverse of group element.
class invg

Syntaxcsg 18104 Extend class notation with group subtraction (or division) operation.
class -g

Definitiondf-grp 18105* Define class of all groups. A group is a monoid (df-mnd 17911) whose internal operation is such that every element admits a left inverse (which can be proven to be a two-sided inverse). Thus, a group 𝐺 is an algebraic structure formed from a base set of elements (notated (Base‘𝐺) per df-base 16488) and an internal group operation (notated (+g𝐺) per df-plusg 16577). The operation combines any two elements of the group base set and must satisfy the 4 group axioms: closure (the result of the group operation must always be a member of the base set, see grpcl 18110), associativity (so ((𝑎+g𝑏)+g𝑐) = (𝑎+g(𝑏+g𝑐)) for any a, b, c, see grpass 18111), identity (there must be an element 𝑒 = (0g𝐺) such that 𝑒+g𝑎 = 𝑎+g𝑒 = 𝑎 for any a), and inverse (for each element a in the base set, there must be an element 𝑏 = invg𝑎 in the base set such that 𝑎+g𝑏 = 𝑏+g𝑎 = 𝑒). It can be proven that the identity element is unique (grpideu 18113). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 18908). Subgroups can often be formed from groups, see df-subg 18275. An example of an (Abelian) group is the set of complex numbers over the group operation + (addition), as proven in cnaddablx 18987; an Abelian group is a group as proven in ablgrp 18910. Other structures include groups, including unital rings (df-ring 19298) and fields (df-field 19504). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}

Definitiondf-minusg 18106* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)
invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑤 ∈ (Base‘𝑔)(𝑤(+g𝑔)𝑥) = (0g𝑔))))

Definitiondf-sbg 18107* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
-g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))

Theoremisgrp 18108* The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))

Theoremgrpmnd 18109 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝐺 ∈ Grp → 𝐺 ∈ Mnd)

Theoremgrpcl 18110 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Theoremgrpass 18111 A group operation is associative. (Contributed by NM, 14-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theoremgrpinvex 18112* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )

Theoremgrpideu 18113* The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

Theoremgrpplusf 18114 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &   𝐹 = (+𝑓𝐺)       (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)⟶𝐵)

Theoremgrpplusfo 18115 The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &   𝐹 = (+𝑓𝐺)       (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)–onto𝐵)

Theoremresgrpplusfrn 18116 The underlying set of a group operation which is a restriction of a structure. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &   𝐻 = (𝐺s 𝑆)    &   𝐹 = (+𝑓𝐻)       ((𝐻 ∈ Grp ∧ 𝑆𝐵) → 𝑆 = ran 𝐹)

Theoremgrppropd 18117* If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))

Theoremgrpprop 18118 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)       (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Theoremgrppropstr 18119 Generalize a specific 2-element group 𝐿 to show that any set 𝐾 with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
(Base‘𝐾) = 𝐵    &   (+g𝐾) = +    &   𝐿 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}       (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Theoremgrpss 18120 Show that a structure extending a constructed group (e.g., a ring) is also a group. This allows us to prove that a constructed potential ring 𝑅 is a group before we know that it is also a ring. (Theorem ringgrp 19301, on the other hand, requires that we know in advance that 𝑅 is a ring.) (Contributed by NM, 11-Oct-2013.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    &   𝑅 ∈ V    &   𝐺𝑅    &   Fun 𝑅       (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)

Theoremisgrpd2e 18121* Deduce a group from its properties. In this version of isgrpd2 18122, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 10-Aug-2013.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑0 = (0g𝐺))    &   (𝜑𝐺 ∈ Mnd)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )       (𝜑𝐺 ∈ Grp)

Theoremisgrpd2 18122* Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2821, but we make an exception for theorems such as isgrpd2 18122, ismndd 17932, and islmodd 19639 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑0 = (0g𝐺))    &   (𝜑𝐺 ∈ Mnd)    &   ((𝜑𝑥𝐵) → 𝑁𝐵)    &   ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )       (𝜑𝐺 ∈ Grp)

Theoremisgrpde 18123* Deduce a group from its properties. In this version of isgrpd 18124, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑0𝐵)    &   ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )       (𝜑𝐺 ∈ Grp)

Theoremisgrpd 18124* Deduce a group from its properties. Unlike isgrpd2 18122, this one goes straight from the base properties rather than going through Mnd. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑0𝐵)    &   ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → 𝑁𝐵)    &   ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )       (𝜑𝐺 ∈ Grp)

Theoremisgrpi 18125* Properties that determine a group. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 3-Sep-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &    0𝐵    &   (𝑥𝐵 → ( 0 + 𝑥) = 𝑥)    &   (𝑥𝐵𝑁𝐵)    &   (𝑥𝐵 → (𝑁 + 𝑥) = 0 )       𝐺 ∈ Grp

Theoremgrpsgrp 18126 A group is a semigroup. (Contributed by AV, 28-Aug-2021.)
(𝐺 ∈ Grp → 𝐺 ∈ Smgrp)

Theoremdfgrp2 18127* Alternate definition of a group as semigroup with a left identity and a left inverse for each element. This "definition" is weaker than df-grp 18105, based on the definition of a monoid which provides a left and a right identity. (Contributed by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))

Theoremdfgrp2e 18128* Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp ↔ (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))

Theoremisgrpix 18129* Properties that determine a group. Read 𝑁 as 𝑁(𝑥). Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
𝐵 ∈ V    &    + ∈ V    &   𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    &   ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &    0𝐵    &   (𝑥𝐵 → ( 0 + 𝑥) = 𝑥)    &   (𝑥𝐵𝑁𝐵)    &   (𝑥𝐵 → (𝑁 + 𝑥) = 0 )       𝐺 ∈ Grp

Theoremgrpidcl 18130 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp → 0𝐵)

Theoremgrpbn0 18131 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → 𝐵 ≠ ∅)

Theoremgrplid 18132 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)

Theoremgrprid 18133 The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + 0 ) = 𝑋)

Theoremgrpn0 18134 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
(𝐺 ∈ Grp → 𝐺 ≠ ∅)

Theoremhashfingrpnn 18135 A finite group has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → (♯‘𝐵) ∈ ℕ)

Theoremgrprcan 18136 Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))

Theoremgrpinveu 18137* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃!𝑦𝐵 (𝑦 + 𝑋) = 0 )

Theoremgrpid 18138 Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))

Theoremisgrpid2 18139 Properties showing that an element 𝑍 is the identity element of a group. (Contributed by NM, 7-Aug-2013.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp → ((𝑍𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))

Theoremgrpidd2 18140* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 18124. (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑0𝐵)    &   ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)    &   (𝜑𝐺 ∈ Grp)       (𝜑0 = (0g𝐺))

Theoremgrpinvfval 18141* The inverse function of a group. For a shorter proof using ax-rep 5189, see grpinvfvalALT 18142. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5189. (Revised by Rohan Ridenour, 13-Aug-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))

TheoremgrpinvfvalALT 18142* Shorter proof of grpinvfval 18141 using ax-rep 5189. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))

Theoremgrpinvval 18143* The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       (𝑋𝐵 → (𝑁𝑋) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))

Theoremgrpinvfn 18144 Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       𝑁 Fn 𝐵

Theoremgrpinvfvi 18145 The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝑁 = (invg𝐺)       𝑁 = (invg‘( I ‘𝐺))

Theoremgrpsubfval 18146* Group subtraction (division) operation. For a shorter proof using ax-rep 5189, see grpsubfvalALT 18147. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5189. (Revised by Rohan Ridenour, 17-Aug-2023.) (Proof shortened by AV, 19-Feb-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)    &    = (-g𝐺)        = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))

TheoremgrpsubfvalALT 18147* Shorter proof of grpsubfval 18146 using ax-rep 5189. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)    &    = (-g𝐺)        = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))

Theoremgrpsubval 18148 Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)    &    = (-g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + (𝐼𝑌)))

Theoremgrpinvf 18149 The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (𝐺 ∈ Grp → 𝑁:𝐵𝐵)

Theoremgrpinvcl 18150 A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)

Theoremgrplinv 18151 The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )

Theoremgrprinv 18152 The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )

Theoremgrpinvid1 18153 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))

Theoremgrpinvid2 18154 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 ))

Theoremisgrpinv 18155* Properties showing that a function 𝑀 is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       (𝐺 ∈ Grp → ((𝑀:𝐵𝐵 ∧ ∀𝑥𝐵 ((𝑀𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))

Theoremgrplrinv 18156* In a group, every member has a left and right inverse. (Contributed by AV, 1-Sep-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))

Theoremgrpidinv2 18157* A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by AV, 1-Sep-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))

Theoremgrpidinv 18158* A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (Revised by AV, 1-Sep-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp → ∃𝑢𝐵𝑥𝐵 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢)))

Theoremgrpinvid 18159 The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)
0 = (0g𝐺)    &   𝑁 = (invg𝐺)       (𝐺 ∈ Grp → (𝑁0 ) = 0 )

Theoremgrplcan 18160 Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

Theoremgrpasscan1 18161 An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑋) + 𝑌)) = 𝑌)

Theoremgrpasscan2 18162 An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)

Theoremgrpidrcan 18163 If right adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = 𝑋𝑍 = 0 ))

Theoremgrpidlcan 18164 If left adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = 𝑋𝑍 = 0 ))

Theoremgrpinvinv 18165 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)

Theoremgrpinvcnv 18166 The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (𝐺 ∈ Grp → 𝑁 = 𝑁)

Theoremgrpinv11 18167 The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) = (𝑁𝑌) ↔ 𝑋 = 𝑌))

Theoremgrpinvf1o 18168 The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)       (𝜑𝑁:𝐵1-1-onto𝐵)

Theoremgrpinvnz 18169 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑋0 ) → (𝑁𝑋) ≠ 0 )

Theoremgrpinvnzcl 18170 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → (𝑁𝑋) ∈ (𝐵 ∖ { 0 }))

Theoremgrpsubinv 18171 Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 (𝑁𝑌)) = (𝑋 + 𝑌))

Theoremgrplmulf1o 18172* Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐹 = (𝑥𝐵 ↦ (𝑋 + 𝑥))       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → 𝐹:𝐵1-1-onto𝐵)

Theoremgrpinvpropd 18173* If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (invg𝐾) = (invg𝐿))

Theoremgrpidssd 18174* If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then both groups have the same identity element. (Contributed by AV, 15-Mar-2019.)
(𝜑𝑀 ∈ Grp)    &   (𝜑𝑆 ∈ Grp)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐵 ⊆ (Base‘𝑀))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (0g𝑀) = (0g𝑆))

Theoremgrpinvssd 18175* If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the elements of the first group have the same inverses in both groups. (Contributed by AV, 15-Mar-2019.)
(𝜑𝑀 ∈ Grp)    &   (𝜑𝑆 ∈ Grp)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐵 ⊆ (Base‘𝑀))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (𝑋𝐵 → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))

Theoremgrpinvadd 18176 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)))

Theoremgrpsubf 18177 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)

Theoremgrpsubcl 18178 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)

Theoremgrpsubrcan 18179 Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))

Theoremgrpinvsub 18180 Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))

Theoremgrpinvval2 18181 A df-neg 10872-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) = ( 0 𝑋))

Theoremgrpsubid 18182 Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = 0 )

Theoremgrpsubid1 18183 Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 0 ) = 𝑋)

Theoremgrpsubeq0 18184 If the difference between two group elements is zero, they are equal. (subeq0 10911 analog.) (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) = 0𝑋 = 𝑌))

Theoremgrpsubadd0sub 18185 Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ( 0 𝑌)))

Theoremgrpsubadd 18186 Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))

Theoremgrpsubsub 18187 Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑋 + (𝑍 𝑌)))

Theoremgrpaddsubass 18188 Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = (𝑋 + (𝑌 𝑍)))

Theoremgrppncan 18189 Cancellation law for subtraction (pncan 10891 analog). (Contributed by NM, 16-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑌) = 𝑋)

Theoremgrpnpcan 18190 Cancellation law for subtraction (npcan 10894 analog). (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)

Theoremgrpsubsub4 18191 Double group subtraction (subsub4 10918 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))

Theoremgrppnpcan2 18192 Cancellation law for mixed addition and subtraction. (pnpcan2 10925 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) (𝑌 + 𝑍)) = (𝑋 𝑌))

Theoremgrpnpncan 18193 Cancellation law for group subtraction. (npncan 10906 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + (𝑌 𝑍)) = (𝑋 𝑍))

Theoremgrpnpncan0 18194 Cancellation law for group subtraction (npncan2 10912 analog). (Contributed by AV, 24-Nov-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )

Theoremgrpnnncan2 18195 Cancellation law for group subtraction. (nnncan2 10922 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 𝑌))

Theoremdfgrp3lem 18196* Lemma for dfgrp3 18197. (Contributed by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))

Theoremdfgrp3 18197* Alternate definition of a group as semigroup (with at least one element) which is also a quasigroup, i.e. a magma in which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Theorem 3.2 of [Bruck] p. 28. (Contributed by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))

Theoremdfgrp3e 18198* Alternate definition of a group as a set with a closed, associative operation, for which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp ↔ (𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))))

Theoremgrplactfval 18199* The left group action of element 𝐴 of group 𝐺. (Contributed by Paul Chapman, 18-Mar-2008.)
𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑋 = (Base‘𝐺)       (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))

Theoremgrplactval 18200* The value of the left group action of element 𝐴 of group 𝐺 at 𝐵. (Contributed by Paul Chapman, 18-Mar-2008.)
𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑋 = (Base‘𝐺)       ((𝐴𝑋𝐵𝑋) → ((𝐹𝐴)‘𝐵) = (𝐴 + 𝐵))

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