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Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpplusfo 18601 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 18602 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 18603* 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 18604 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 18605 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 18606 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 19797, 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 18607* Deduce a group from its properties. In this version of isgrpd2 18608, 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 18608* 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 2739, but we make an exception for theorems such as isgrpd2 18608, ismndd 18416, and islmodd 20138 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 18609* Deduce a group from its properties. In this version of isgrpd 18610, 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 18610* Deduce a group from its properties. Unlike isgrpd2 18608, 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 18611* 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 18612 A group is a semigroup. (Contributed by AV, 28-Aug-2021.)
(𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
 
Theoremdfgrp2 18613* 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 18589, 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 18614* 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 18615* 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 18616 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 18617 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → 𝐵 ≠ ∅)
 
Theoremgrplid 18618 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 18619 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 18620 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
(𝐺 ∈ Grp → 𝐺 ≠ ∅)
 
Theoremhashfingrpnn 18621 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 18622 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 18623* 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 18624 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 18625 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 18626* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 18610. (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑0𝐵)    &   ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)    &   (𝜑𝐺 ∈ Grp)       (𝜑0 = (0g𝐺))
 
Theoremgrpinvfval 18627* The inverse function of a group. For a shorter proof using ax-rep 5210, see grpinvfvalALT 18628. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5210. (Revised by Rohan Ridenour, 13-Aug-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
 
TheoremgrpinvfvalALT 18628* Shorter proof of grpinvfval 18627 using ax-rep 5210. (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 18629* 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 18630 Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       𝑁 Fn 𝐵
 
Theoremgrpinvfvi 18631 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 18632* Group subtraction (division) operation. For a shorter proof using ax-rep 5210, see grpsubfvalALT 18633. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5210. (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 18633* Shorter proof of grpsubfval 18632 using ax-rep 5210. (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 18634 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 18635 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 18636 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 18637 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 18638 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 18639 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 18640 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 18641* 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 18642* 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 18643* 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 18644* 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 18645 The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)
0 = (0g𝐺)    &   𝑁 = (invg𝐺)       (𝐺 ∈ Grp → (𝑁0 ) = 0 )
 
Theoremgrplcan 18646 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 18647 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 18648 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 18649 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 18650 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 18651 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 18652 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 18653 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 18654 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 18655 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 18656 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 18657 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 18658* 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 18659* 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 18660* 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 18661* 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 18662 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 18663 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)
 
Theoremgrpsubcl 18664 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
 
Theoremgrpsubrcan 18665 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 18666 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 18667 A df-neg 11217-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 18668 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 18669 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 18670 If the difference between two group elements is zero, they are equal. (subeq0 11256 analog.) (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) = 0𝑋 = 𝑌))
 
Theoremgrpsubadd0sub 18671 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 18672 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 18673 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 18674 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 18675 Cancellation law for subtraction (pncan 11236 analog). (Contributed by NM, 16-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑌) = 𝑋)
 
Theoremgrpnpcan 18676 Cancellation law for subtraction (npcan 11239 analog). (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)
 
Theoremgrpsubsub4 18677 Double group subtraction (subsub4 11263 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))
 
Theoremgrppnpcan2 18678 Cancellation law for mixed addition and subtraction. (pnpcan2 11270 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 18679 Cancellation law for group subtraction. (npncan 11251 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 18680 Cancellation law for group subtraction (npncan2 11257 analog). (Contributed by AV, 24-Nov-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )
 
Theoremgrpnnncan2 18681 Cancellation law for group subtraction. (nnncan2 11267 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 18682* Lemma for dfgrp3 18683. (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 18683* 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 18684* 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 18685* 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 18686* 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))
 
Theoremgrplactcnv 18687* The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))))
 
Theoremgrplactf1o 18688* The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴):𝑋1-1-onto𝑋)
 
Theoremgrpsubpropd 18689 Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
(𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   (𝜑 → (+g𝐺) = (+g𝐻))       (𝜑 → (-g𝐺) = (-g𝐻))
 
Theoremgrpsubpropd2 18690* Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐵 = (Base‘𝐻))    &   (𝜑𝐺 ∈ Grp)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))       (𝜑 → (-g𝐺) = (-g𝐻))
 
Theoremgrp1 18691 The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Grp)
 
Theoremgrp1inv 18692 The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Revised by AV, 26-Aug-2021.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉 → (invg𝑀) = ( I ↾ {𝐼}))
 
Theoremprdsinvlem 18693* Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Grp)    &   (𝜑𝐹𝐵)    &    0 = (0g𝑅)    &   𝑁 = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝐹𝑦)))       (𝜑 → (𝑁𝐵 ∧ (𝑁 + 𝐹) = 0 ))
 
Theoremprdsgrpd 18694 The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)       (𝜑𝑌 ∈ Grp)
 
Theoremprdsinvgd 18695* Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)    &   𝐵 = (Base‘𝑌)    &   𝑁 = (invg𝑌)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = (𝑥𝐼 ↦ ((invg‘(𝑅𝑥))‘(𝑋𝑥))))
 
Theorempwsgrp 18696 A structure power of a group is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Grp ∧ 𝐼𝑉) → 𝑌 ∈ Grp)
 
Theorempwsinvg 18697 Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (invg𝑅)    &   𝑁 = (invg𝑌)       ((𝑅 ∈ Grp ∧ 𝐼𝑉𝑋𝐵) → (𝑁𝑋) = (𝑀𝑋))
 
Theorempwssub 18698 Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (-g𝑅)    &    = (-g𝑌)       (((𝑅 ∈ Grp ∧ 𝐼𝑉) ∧ (𝐹𝐵𝐺𝐵)) → (𝐹 𝐺) = (𝐹f 𝑀𝐺))
 
Theoremimasgrp2 18699* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅𝑊)    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))    &   (𝜑0𝑉)    &   ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))    &   ((𝜑𝑥𝑉) → 𝑁𝑉)    &   ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))       (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
 
Theoremimasgrp 18700* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅 ∈ Grp)    &    0 = (0g𝑅)       (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
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