P. 189 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	frmdup1 18801*	Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺))
Theorem	frmdup2 18802*	The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)    &   ⊢ 𝑈 = (varFMnd‘𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌))
Theorem	frmdup3lem 18803*	Lemma for frmdup3 18804. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑈 = (varFMnd‘𝐼)    ⇒   ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ (𝐹 ∈ (𝑀 MndHom 𝐺) ∧ (𝐹 ∘ 𝑈) = 𝐴)) → 𝐹 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))))
Theorem	frmdup3 18804*	Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑈 = (varFMnd‘𝐼)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → ∃!𝑚 ∈ (𝑀 MndHom 𝐺)(𝑚 ∘ 𝑈) = 𝐴)
10.1.9.1  Monoid of endofunctions According to Wikipedia ("Endomorphism", 25-Jan-2024, https://en.wikipedia.org/wiki/Endomorphism) "An endofunction is a function whose domain is equal to its codomain.". An endofunction is sometimes also called "self-mapping" (see https://www.wikidata.org/wiki/Q1691962) or "self-map" (see https://mathworld.wolfram.com/Self-Map.html), in German "Selbstabbildung" (see https://de.wikipedia.org/wiki/Selbstabbildung).
Syntax	cefmnd 18805	Extend class notation to include the class of monoids of endofunctions.
class EndoFMnd
Definition	df-efmnd 18806*	Define the monoid of endofunctions on set 𝑥. We represent the monoid as the set of functions from 𝑥 to itself ((𝑥 ↑m 𝑥)) under function composition, and topologize it as a function space assuming the set is discrete. Analogous to the former definition of SymGrp, see df-symg 19311 and symgvalstruct 19338. (Contributed by AV, 25-Jan-2024.)
⊢ EndoFMnd = (𝑥 ∈ V ↦ ⦋(𝑥 ↑m 𝑥) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))⟩})
Theorem	efmnd 18807*	The monoid of endofunctions on set 𝐴. (Contributed by AV, 25-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (𝐴 ↑m 𝐴)    &   ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))    &   ⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Theorem	efmndbas 18808	The base set of the monoid of endofunctions on class 𝐴. (Contributed by AV, 25-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ 𝐵 = (𝐴 ↑m 𝐴)
Theorem	efmndbasabf 18809*	The base set of the monoid of endofunctions on class 𝐴 is the set of functions from 𝐴 into itself. (Contributed by AV, 29-Mar-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ 𝐵 = {𝑓 ∣ 𝑓:𝐴⟶𝐴}
Theorem	elefmndbas 18810	Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐴))
Theorem	elefmndbas2 18811	Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 29-Mar-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐴))
Theorem	efmndbasf 18812	Elements in the monoid of endofunctions on 𝐴 are functions from 𝐴 into itself. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴⟶𝐴)
Theorem	efmndhash 18813	The monoid of endofunctions on 𝑛 objects has cardinality 𝑛↑𝑛. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ Fin → (♯‘𝐵) = ((♯‘𝐴)↑(♯‘𝐴)))
Theorem	efmndbasfi 18814	The monoid of endofunctions on a finite set 𝐴 is finite. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ Fin → 𝐵 ∈ Fin)
Theorem	efmndfv 18815	The function value of an endofunction. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐴)
Theorem	efmndtset 18816	The topology of the monoid of endofunctions on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just endofunctions - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by AV, 25-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))
Theorem	efmndplusg 18817*	The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
Theorem	efmndov 18818	The value of the group operation of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌))
Theorem	efmndcl 18819	The group operation of the monoid of endofunctions on 𝐴 is closed. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	efmndtopn 18820	The topology of the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺))
Theorem	symggrplem 18821*	Lemma for symggrp 19341 and efmndsgrp 18823. Conditions for an operation to be associative. Formerly part of proof for symggrp 19341. (Contributed by AV, 28-Jan-2024.)
⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ∘ 𝑦))    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	efmndmgm 18822	The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 ∈ Mgm
Theorem	efmndsgrp 18823	The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 ∈ Smgrp
Theorem	ielefmnd 18824	The identity function restricted to a set 𝐴 is an element of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))
Theorem	efmndid 18825	The identity function restricted to a set 𝐴 is the identity element of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝐺))
Theorem	efmndmnd 18826	The monoid of endofunctions on a set 𝐴 is actually a monoid. (Contributed by AV, 31-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd)
Theorem	efmnd0nmnd 18827	Even the monoid of endofunctions on the empty set is actually a monoid. (Contributed by AV, 31-Jan-2024.)
⊢ (EndoFMnd‘∅) ∈ Mnd
Theorem	efmndbas0 18828	The base set of the monoid of endofunctions on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 31-Mar-2024.)
⊢ (Base‘(EndoFMnd‘∅)) = {∅}
Theorem	efmnd1hash 18829	The monoid of endofunctions on a singleton has cardinality 1. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (♯‘𝐵) = 1)
Theorem	efmnd1bas 18830	The monoid of endofunctions on a singleton consists of the identity only. (Contributed by AV, 31-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{⟨𝐼, 𝐼⟩}})
Theorem	efmnd2hash 18831	The monoid of endofunctions on a (proper) pair has cardinality 4. (Contributed by AV, 18-Feb-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼, 𝐽}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐵) = 4)
Theorem	submefmnd 18832*	If the base set of a monoid is contained in the base set of the monoid of endofunctions on a set 𝐴, contains the identity function and has the function composition as group operation, then its base set is a submonoid of the monoid of endofunctions on set 𝐴. Analogous to pgrpsubgsymg 19350. (Contributed by AV, 17-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐹 = (Base‘𝑆)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (((𝑆 ∈ Mnd ∧ 𝐹 ⊆ 𝐵 ∧ 0 ∈ 𝐹) ∧ (+g‘𝑆) = (𝑓 ∈ 𝐹, 𝑔 ∈ 𝐹 ↦ (𝑓 ∘ 𝑔))) → 𝐹 ∈ (SubMnd‘𝑀)))
Theorem	sursubmefmnd 18833*	The set of surjective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–onto→𝐴} ∈ (SubMnd‘𝑀))
Theorem	injsubmefmnd 18834*	The set of injective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀))
Theorem	idressubmefmnd 18835	The singleton containing only the identity function restricted to a set is a submonoid of the monoid of endofunctions on this set. (Contributed by AV, 17-Feb-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {( I ↾ 𝐴)} ∈ (SubMnd‘𝐺))
Theorem	idresefmnd 18836	The structure with the singleton containing only the identity function restricted to a set 𝐴 as base set and the function composition as group operation, constructed by (structure) restricting the monoid of endofunctions on 𝐴 to that singleton, is a monoid whose base set is a subset of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 17-Feb-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐸 = (𝐺 ↾s {( I ↾ 𝐴)})    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐸 ∈ Mnd ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))
Theorem	smndex1ibas 18837	The modulo function 𝐼 is an endofunction on ℕ0. (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    ⇒   ⊢ 𝐼 ∈ (Base‘𝑀)
Theorem	smndex1iidm 18838*	The modulo function 𝐼 is idempotent. (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    ⇒   ⊢ (𝐼 ∘ 𝐼) = 𝐼
Theorem	smndex1gbas 18839*	The constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   ⊢ (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾) ∈ (Base‘𝑀))
Theorem	smndex1gid 18840*	The composition of a constant function (𝐺‘𝐾) with another endofunction on ℕ0 results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺‘𝐾) ∘ 𝐹) = (𝐺‘𝐾))
Theorem	smndex1igid 18841*	The composition of the modulo function 𝐼 and a constant function (𝐺‘𝐾) results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   ⊢ (𝐾 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝐾)) = (𝐺‘𝐾))
Theorem	smndex1basss 18842*	The modulo function 𝐼 and the constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    ⇒   ⊢ 𝐵 ⊆ (Base‘𝑀)
Theorem	smndex1bas 18843*	The base set of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ (Base‘𝑆) = 𝐵
Theorem	smndex1mgm 18844*	The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a magma. (Contributed by AV, 14-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝑆 ∈ Mgm
Theorem	smndex1sgrp 18845*	The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a semigroup. (Contributed by AV, 14-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝑆 ∈ Smgrp
Theorem	smndex1mndlem 18846*	Lemma for smndex1mnd 18847 and smndex1id 18848. (Contributed by AV, 16-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ (𝑋 ∈ 𝐵 → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋))
Theorem	smndex1mnd 18847*	The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a monoid. (Contributed by AV, 16-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝑆 ∈ Mnd
Theorem	smndex1id 18848*	The modulo function 𝐼 is the identity of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 16-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝐼 = (0g‘𝑆)
Theorem	smndex1n0mnd 18849*	The identity of the monoid 𝑀 of endofunctions on set ℕ0 is not contained in the base set of the constructed monoid 𝑆. (Contributed by AV, 17-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ (0g‘𝑀) ∉ 𝐵
Theorem	nsmndex1 18850*	The base set 𝐵 of the constructed monoid 𝑆 is not a submonoid of the monoid 𝑀 of endofunctions on set ℕ0, although 𝑀 ∈ Mnd and 𝑆 ∈ Mnd and 𝐵 ⊆ (Base‘𝑀) hold. (Contributed by AV, 17-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝐵 ∉ (SubMnd‘𝑀)
Theorem	smndex2dbas 18851	The doubling function 𝐷 is an endofunction on ℕ0. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    ⇒   ⊢ 𝐷 ∈ 𝐵
Theorem	smndex2dnrinv 18852	The doubling function 𝐷 has no right inverse in the monoid of endofunctions on ℕ0. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    ⇒   ⊢ ∀𝑓 ∈ 𝐵 (𝐷 ∘ 𝑓) ≠ 0
Theorem	smndex2hbas 18853	The halving functions 𝐻 are endofunctions on ℕ0. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))    ⇒   ⊢ 𝐻 ∈ 𝐵
Theorem	smndex2dlinvh 18854*	The halving functions 𝐻 are left inverses of the doubling function 𝐷. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))    ⇒   ⊢ (𝐻 ∘ 𝐷) = 0
10.1.10  Examples and counterexamples for magmas, semigroups and monoids
Theorem	mgm2nsgrplem1 18855*	Lemma 1 for mgm2nsgrp 18859: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18592). (Contributed by AV, 27-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm)
Theorem	mgm2nsgrplem2 18856*	Lemma 2 for mgm2nsgrp 18859. (Contributed by AV, 27-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ⚬ 𝐴) ⚬ 𝐵) = 𝐴)
Theorem	mgm2nsgrplem3 18857*	Lemma 3 for mgm2nsgrp 18859. (Contributed by AV, 28-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⚬ (𝐴 ⚬ 𝐵)) = 𝐵)
Theorem	mgm2nsgrplem4 18858*	Lemma 4 for mgm2nsgrp 18859: M is not a semigroup. (Contributed by AV, 28-Jan-2020.) (Proof shortened by AV, 31-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    ⇒   ⊢ ((♯‘𝑆) = 2 → 𝑀 ∉ Smgrp)
Theorem	mgm2nsgrp 18859*	A small magma (with two elements) which is not a semigroup. (Contributed by AV, 28-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    ⇒   ⊢ ((♯‘𝑆) = 2 → (𝑀 ∈ Mgm ∧ 𝑀 ∉ Smgrp))
Theorem	sgrp2nmndlem1 18860*	Lemma 1 for sgrp2nmnd 18867: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18592). (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm)
Theorem	sgrp2nmndlem2 18861*	Lemma 2 for sgrp2nmnd 18867. (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ⚬ 𝐶) = 𝐴)
Theorem	sgrp2nmndlem3 18862*	Lemma 3 for sgrp2nmnd 18867. (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵)
Theorem	sgrp2rid2 18863*	A small semigroup (with two elements) with two right identities which are different if 𝐴 ≠ 𝐵. (Contributed by AV, 10-Feb-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦)
Theorem	sgrp2rid2ex 18864*	A small semigroup (with two elements) with two right identities which are different. (Contributed by AV, 10-Feb-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((♯‘𝑆) = 2 → ∃𝑥 ∈ 𝑆 ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ≠ 𝑧 ∧ (𝑦 ⚬ 𝑥) = 𝑦 ∧ (𝑦 ⚬ 𝑧) = 𝑦))
Theorem	sgrp2nmndlem4 18865*	Lemma 4 for sgrp2nmnd 18867: M is a semigroup. (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    ⇒   ⊢ ((♯‘𝑆) = 2 → 𝑀 ∈ Smgrp)
Theorem	sgrp2nmndlem5 18866*	Lemma 5 for sgrp2nmnd 18867: M is not a monoid. (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    ⇒   ⊢ ((♯‘𝑆) = 2 → 𝑀 ∉ Mnd)
Theorem	sgrp2nmnd 18867*	A small semigroup (with two elements) which is not a monoid. (Contributed by AV, 26-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    ⇒   ⊢ ((♯‘𝑆) = 2 → (𝑀 ∈ Smgrp ∧ 𝑀 ∉ Mnd))
Theorem	mgmnsgrpex 18868	There is a magma which is not a semigroup. (Contributed by AV, 29-Jan-2020.)
⊢ ∃𝑚 ∈ Mgm 𝑚 ∉ Smgrp
Theorem	sgrpnmndex 18869	There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.)
⊢ ∃𝑚 ∈ Smgrp 𝑚 ∉ Mnd
Theorem	sgrpssmgm 18870	The class of all semigroups is a proper subclass of the class of all magmas. (Contributed by AV, 29-Jan-2020.)
⊢ Smgrp ⊊ Mgm
Theorem	mndsssgrp 18871	The class of all monoids is a proper subclass of the class of all semigroups. (Contributed by AV, 29-Jan-2020.)
⊢ Mnd ⊊ Smgrp
Theorem	pwmndgplus 18872*	The operation of the monoid of the power set of a class 𝐴 under union. (Contributed by AV, 27-Feb-2024.)
⊢ (Base‘𝑀) = 𝒫 𝐴    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋(+g‘𝑀)𝑌) = (𝑋 ∪ 𝑌))
Theorem	pwmndid 18873*	The identity of the monoid of the power set of a class 𝐴 under union is the empty set. (Contributed by AV, 27-Feb-2024.)
⊢ (Base‘𝑀) = 𝒫 𝐴    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   ⊢ (0g‘𝑀) = ∅
Theorem	pwmnd 18874*	The power set of a class 𝐴 is a monoid under union. (Contributed by AV, 27-Feb-2024.)
⊢ (Base‘𝑀) = 𝒫 𝐴    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   ⊢ 𝑀 ∈ Mnd
10.2  Groups
10.2.1  Definition and basic properties
Syntax	cgrp 18875	Extend class notation with class of all groups.
class Grp
Syntax	cminusg 18876	Extend class notation with inverse of group element.
class invg
Syntax	csg 18877	Extend class notation with group subtraction (or division) operation.
class -g
Definition	df-grp 18878*	Define class of all groups. A group is a monoid (df-mnd 18672) 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 17149) and an internal group operation (notated (+g‘𝐺) per df-plusg 17202). 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 18883), associativity (so ((𝑎+g𝑏)+g𝑐) = (𝑎+g(𝑏+g𝑐)) for any a, b, c, see grpass 18884), identity (there must be an element 𝑒 = (0g‘𝐺) 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 18886). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 19724). Subgroups can often be formed from groups, see df-subg 19065. An example of an (Abelian) group is the set of complex numbers ℂ over the group operation + (addition), as proven in cnaddablx 19809; an Abelian group is a group as proven in ablgrp 19726. Other structures include groups, including unital rings (df-ring 20182) and fields (df-field 20677). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)}
Definition	df-minusg 18879*	Define inverse of group element. (Contributed by NM, 24-Aug-2011.)
⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+g‘𝑔)𝑥) = (0g‘𝑔))))
Definition	df-sbg 18880*	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(+g‘𝑔)((invg‘𝑔)‘𝑦))))
Theorem	isgrp 18881*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))
Theorem	grpmnd 18882	A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
Theorem	grpcl 18883	Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	grpass 18884	A group operation is associative. (Contributed by NM, 14-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	grpinvex 18885*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
Theorem	grpideu 18886*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
Theorem	grpassd 18887	A group operation is associative. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	grpmndd 18888	A group is a monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mnd)
Theorem	grpcld 18889	Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	grpplusf 18890	The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = (+𝑓‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)⟶𝐵)
Theorem	grpplusfo 18891	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 = (+𝑓‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)–onto→𝐵)
Theorem	resgrpplusfrn 18892	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 = (𝐺 ↾s 𝑆)    &   ⊢ 𝐹 = (+𝑓‘𝐻)    ⇒   ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = ran 𝐹)
Theorem	grppropd 18893*	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.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
Theorem	grpprop 18894	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)    &   ⊢ (+g‘𝐾) = (+g‘𝐿)    ⇒   ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)
Theorem	grppropstr 18895	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    &   ⊢ (+g‘𝐾) = +    &   ⊢ 𝐿 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)
Theorem	grpss 18896	Show that a structure extending a constructed group (e.g., a ring) is also a group. This allows to prove that a constructed potential ring 𝑅 is a group before we know that it is also a ring. (Theorem ringgrp 20185, 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    &   ⊢ 𝐺 ⊆ 𝑅    &   ⊢ Fun 𝑅    ⇒   ⊢ (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)
Theorem	isgrpd2e 18897*	Deduce a group from its properties. In this version of isgrpd2 18898, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 10-Aug-2013.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 0 = (0g‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	isgrpd2 18898*	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 2737, but we make an exception for theorems such as isgrpd2 18898, ismndd 18693, and islmodd 20829 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 0 = (0g‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	isgrpde 18899*	Deduce a group from its properties. In this version of isgrpd 18900, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 6-Jan-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	isgrpd 18900*	Deduce a group from its properties. Unlike isgrpd2 18898, 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.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)