P. 181 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gsumspl 18001	The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝑆 ∈ Word 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))    &   ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆)))    &   ⊢ (𝜑 → 𝑋 ∈ Word 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ Word 𝐵)    &   ⊢ (𝜑 → (𝑀 Σg 𝑋) = (𝑀 Σg 𝑌))    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝑆 splice ⟨𝐹, 𝑇, 𝑋⟩)) = (𝑀 Σg (𝑆 splice ⟨𝐹, 𝑇, 𝑌⟩)))
Theorem	gsumwmhm 18002	Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝐻 ∈ (𝑀 MndHom 𝑁) ∧ 𝑊 ∈ Word 𝐵) → (𝐻‘(𝑀 Σg 𝑊)) = (𝑁 Σg (𝐻 ∘ 𝑊)))
Theorem	gsumwspan 18003*	The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝐾 = (mrCls‘(SubMnd‘𝑀))    ⇒   ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))
10.1.8  Free monoids
Syntax	cfrmd 18004	Extend class definition with the free monoid construction.
class freeMnd
Syntax	cvrmd 18005	Extend class notation with free monoid injection.
class varFMnd
Definition	df-frmd 18006	Define a free monoid over a set 𝑖 of generators, defined as the set of finite strings on 𝐼 with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})
Definition	df-vrmd 18007*	Define a free monoid over a set 𝑖 of generators, defined as the set of finite strings on 𝐼 with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
Theorem	frmdval 18008	Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼)    &   ⊢ + = ( ++ ↾ (𝐵 × 𝐵))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑀 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩})
Theorem	frmdbas 18009	The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼)
Theorem	frmdelbas 18010	An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝐼)
Theorem	frmdplusg 18011	The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ + = ( ++ ↾ (𝐵 × 𝐵))
Theorem	frmdadd 18012	Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ++ 𝑌))
Theorem	vrmdfval 18013*	The canonical injection from the generating set 𝐼 to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑈 = (varFMnd‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ ⟨“𝑗”⟩))
Theorem	vrmdval 18014	The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑈 = (varFMnd‘𝐼)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = ⟨“𝐴”⟩)
Theorem	vrmdf 18015	The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑈 = (varFMnd‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶Word 𝐼)
Theorem	frmdmnd 18016	A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑀 = (freeMnd‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)
Theorem	frmd0 18017	The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    ⇒   ⊢ ∅ = (0g‘𝑀)
Theorem	frmdsssubm 18018	The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑀 = (freeMnd‘𝐼)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → Word 𝐽 ∈ (SubMnd‘𝑀))
Theorem	frmdgsum 18019	Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝑈 = (varFMnd‘𝐼)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐼) → (𝑀 Σg (𝑈 ∘ 𝑊)) = 𝑊)
Theorem	frmdss2 18020	A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of 𝐽 is Word 𝐽". (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    &   ⊢ 𝑈 = (varFMnd‘𝐼)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 ↔ Word 𝐽 ⊆ 𝐴))
Theorem	frmdup1 18021*	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 18022*	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 18023*	Lemma for frmdup3 18024. (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 18024*	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)
10.1.8.1  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 18025	Extend class notation to include the class of monoids of endofunctions.
class EndoFMnd
Definition	df-efmnd 18026*	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 18488 and symgvalstruct 18517. (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 18027*	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 18028	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 18029*	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 18030	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 18031	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 18032	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 18033	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 18034	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 18035	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 18036	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 18037*	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 18038	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 18039	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 18040	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 18041*	Lemma for symggrp 18520 and efmndsgrp 18043. Conditions for an operation to be associative. Formerly part of proof for symggrp 18520. (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 18042	The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 ∈ Mgm
Theorem	efmndsgrp 18043	The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 ∈ Smgrp
Theorem	ielefmnd 18044	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 18045	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 18046	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 18047	Even the monoid of endofunctions on the empty set is actually a monoid. (Contributed by AV, 31-Jan-2024.)
⊢ (EndoFMnd‘∅) ∈ Mnd
Theorem	efmndbas0 18048	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 18049	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 18050	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 18051	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 18052*	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 18529. (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 18053*	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 18054*	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 18055	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 18056	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 18057	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 18058*	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 18059*	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 18060*	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 18061*	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 18062*	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 18063*	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 18064*	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 18065*	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 18066*	Lemma for smndex1mnd 18067 and smndex1id 18068. (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 18067*	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 18068*	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 18069*	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 18070*	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 18071	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 18072	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 18073	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 18074*	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
10.1.9  Examples and counterexamples for magmas, semigroups and monoids
Theorem	mgm2nsgrplem1 18075*	Lemma 1 for mgm2nsgrp 18079: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17857). (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 18076*	Lemma 2 for mgm2nsgrp 18079. (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 18077*	Lemma 3 for mgm2nsgrp 18079. (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 18078*	Lemma 4 for mgm2nsgrp 18079: 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 18079*	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 18080*	Lemma 1 for sgrp2nmnd 18087: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17857). (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 18081*	Lemma 2 for sgrp2nmnd 18087. (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 18082*	Lemma 3 for sgrp2nmnd 18087. (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 18083*	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 18084*	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 18085*	Lemma 4 for sgrp2nmnd 18087: 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 18086*	Lemma 5 for sgrp2nmnd 18087: 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 18087*	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 18088	There is a magma which is not a semigroup. (Contributed by AV, 29-Jan-2020.)
⊢ ∃𝑚 ∈ Mgm 𝑚 ∉ Smgrp
Theorem	sgrpnmndex 18089	There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.)
⊢ ∃𝑚 ∈ Smgrp 𝑚 ∉ Mnd
Theorem	sgrpssmgm 18090	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 18091	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 18092*	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 18093*	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 18094*	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 18095	Extend class notation with class of all groups.
class Grp
Syntax	cminusg 18096	Extend class notation with inverse of group element.
class invg
Syntax	csg 18097	Extend class notation with group subtraction (or division) operation.
class -g
Definition	df-grp 18098*	Define class of all groups. A group is a monoid (df-mnd 17904) 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 16481) and an internal group operation (notated (+g‘𝐺) per df-plusg 16570). 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 18103), associativity (so ((𝑎+g𝑏)+g𝑐) = (𝑎+g(𝑏+g𝑐)) for any a, b, c, see grpass 18104), 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 18106). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 18901). Subgroups can often be formed from groups, see df-subg 18268. An example of an (Abelian) group is the set of complex numbers ℂ over the group operation + (addition), as proven in cnaddablx 18981; an Abelian group is a group as proven in ablgrp 18903. Other structures include groups, including unital rings (df-ring 19292) and fields (df-field 19498). (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 18099*	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 18100*	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))))