P. 188 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mgmcl 18701	Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
Theorem	isnmgm 18702	A condition for a structure not to be a magma. (Contributed by AV, 30-Jan-2020.) (Proof shortened by NM, 5-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)
Theorem	mgmsscl 18703	If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. Formerly part of proof of grpissubg 19213. (Contributed by AV, 17-Feb-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐻)    ⇒   ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)
Theorem	plusffval 18704*	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
Theorem	plusfval 18705	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⨣ 𝑌) = (𝑋 + 𝑌))
Theorem	plusfeq 18706	If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ( + Fn (𝐵 × 𝐵) → ⨣ = + )
Theorem	plusffn 18707	The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ⨣ Fn (𝐵 × 𝐵)
Theorem	mgmplusf 18708	The group addition function of a magma is a function into its base set. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revisd by AV, 28-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⨣ = (+𝑓‘𝑀)    ⇒   ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
Theorem	mgmpropd 18709*	If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
Theorem	ismgmd 18710*	Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mgm)
Theorem	issstrmgm 18711*	Characterize a substructure as submagma by closure properties. (Contributed by AV, 30-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑆 ⊆ 𝐵) → (𝐻 ∈ Mgm ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))
Theorem	intopsn 18712	The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20858. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Theorem	mgmb1mgm1 18713	The only magma with a base set consisting of one element is the trivial magma (at least if its operation is an internal binary operation). (Contributed by AV, 23-Jan-2020.) (Revised by AV, 7-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Theorem	mgm0 18714	Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)
Theorem	mgm0b 18715	The structure with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
⊢ {⟨(Base‘ndx), ∅⟩, ⟨(+g‘ndx), 𝑂⟩} ∈ Mgm
Theorem	mgm1 18716	The structure with one element and the only closed internal operation for a singleton is a magma. (Contributed by AV, 10-Feb-2020.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mgm)
Theorem	opifismgm 18717*	A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝜓, 𝐶, 𝐷))    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑀 ∈ Mgm)
10.1.2  Identity elements According to Wikipedia ("Identity element", 7-Feb-2020, https://en.wikipedia.org/wiki/Identity_element): "In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it.". Or in more detail "... an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity." We concentrate on two-sided identities in the following. The existence of an identity (an identity is unique if it exists, see mgmidmo 18718) is an important property of monoids (see mndid 18802), and therefore also for groups (see grpid 19042), but also for magmas not required to be associative. Magmas with an identity element are called "unital magmas" (see Definition 2 in [BourbakiAlg1] p. 12) or, if the magmas are cancellative, "loops" (see definition in [Bruck] p. 15). In the context of extensible structures, the identity element (of any magma 𝑀) is defined as "group identity element" (0g‘𝑀), see df-0g 17494. Related theorems which are already valid for magmas are provided in the following.
Theorem	mgmidmo 18718*	A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)
⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)
Theorem	grpidval 18719*	The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
Theorem	grpidpropd 18720*	If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
Theorem	fn0g 18721	The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 0g Fn V
Theorem	0g0 18722	The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
⊢ ∅ = (0g‘∅)
Theorem	ismgmid 18723*	The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    ⇒   ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
Theorem	mgmidcl 18724*	The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    ⇒   ⊢ (𝜑 → 0 ∈ 𝐵)
Theorem	mgmlrid 18725*	The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
Theorem	ismgmid2 18726*	Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑈) = 𝑥)    ⇒   ⊢ (𝜑 → 𝑈 = 0 )
Theorem	lidrideqd 18727*	If there is a left and right identity element for any binary operation (group operation) +, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
⊢ (𝜑 → 𝐿 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    ⇒   ⊢ (𝜑 → 𝐿 = 𝑅)
Theorem	lidrididd 18728*	If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 18727) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
⊢ (𝜑 → 𝐿 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝜑 → 𝐿 = 0 )
Theorem	grpidd 18729*	Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)    ⇒   ⊢ (𝜑 → 0 = (0g‘𝐺))
Theorem	mgmidsssn0 18730*	Property of the set of identities of 𝐺. Either 𝐺 has no identities, and 𝑂 = ∅, or it has one and this identity is unique and identified by the 0g function. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝑂 ⊆ { 0 })
Theorem	grpinvalem 18731*	Lemma for grpinva 18732. (Contributed by NM, 9-Aug-2013.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ (𝜑 → 𝑂 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑋) = 𝑋)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝑋 = 𝑂)
Theorem	grpinva 18732*	Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ (𝜑 → 𝑂 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + 𝑋) = 𝑂)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) = 𝑂)
Theorem	grprida 18733*	Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ (𝜑 → 𝑂 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑂) = 𝑥)
10.1.3  Iterated sums in a magma The symbol Σg is mostly used in the context of abelian groups. Therefore, it is usually called "group sum". It can be defined, however, in arbitrary magmas (then it should be called "iterated sum"). If the magma is not required to be commutative or associative, then the order of the summands and the order in which summations are done become important. If the magma is not unital, then one cannot define a meaningful empty sum. See the comment for df-gsum 17495.
Theorem	gsumvalx 18734*	Expand out the substitutions in df-gsum 17495. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑂 = {𝑠 ∈ 𝐵 ∣ ∀𝑡 ∈ 𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)}    &   ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ 𝑂)))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ (𝜑 → dom 𝐹 = 𝐴)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))))
Theorem	gsumval 18735*	Expand out the substitutions in df-gsum 17495. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑂 = {𝑠 ∈ 𝐵 ∣ ∀𝑡 ∈ 𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)}    &   ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ 𝑂)))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))))
Theorem	gsumpropd 18736	The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 18817 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Theorem	gsumpropd2lem 18737*	Lemma for gsumpropd2 18738. (Contributed by Thierry Arnoux, 28-Jun-2017.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))    &   ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡))    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))    &   ⊢ 𝐴 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)}))    &   ⊢ 𝐵 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)}))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Theorem	gsumpropd2 18738*	A stronger version of gsumpropd 18736, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 18739. (Contributed by Thierry Arnoux, 28-Jun-2017.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))    &   ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡))    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Theorem	gsummgmpropd 18739*	A stronger version of gsumpropd 18736 if at least one of the involved structures is a magma, see gsumpropd2 18738. (Contributed by AV, 31-Jan-2020.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ Mgm)    &   ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡))    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Theorem	gsumress 18740*	The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither 𝐺 nor 𝐻 need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 0 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Theorem	gsumval1 18741*	Value of the group sum operation when every element being summed is an identity of 𝐺. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑂)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 )
Theorem	gsum0 18742	Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 Σg ∅) = 0
Theorem	gsumval2a 18743*	Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    &   ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}    &   ⊢ (𝜑 → ¬ ran 𝐹 ⊆ 𝑂)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Theorem	gsumval2 18744	Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Theorem	gsumsplit1r 18745	Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
Theorem	gsumprval 18746	Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 = (𝑀 + 1))    &   ⊢ (𝜑 → 𝐹:{𝑀, 𝑁}⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘𝑀) + (𝐹‘𝑁)))
Theorem	gsumpr12val 18747	Value of the group sum operation over the pair {1, 2}. (Contributed by AV, 14-Dec-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:{1, 2}⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘1) + (𝐹‘2)))
10.1.4  Magma homomorphisms and submagmas
Syntax	cmgmhm 18748	Hom-set generator class for magmas.
class MgmHom
Syntax	csubmgm 18749	Class function taking a magma to its lattice of submagmas.
class SubMgm
Definition	df-mgmhm 18750*	A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.)
⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))})
Definition	df-submgm 18751*	A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.)
⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡})
Theorem	mgmhmrcl 18752	Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.)
⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
Theorem	submgmrcl 18753	Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.)
⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
Theorem	ismgmhm 18754*	Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))))
Theorem	mgmhmf 18755	A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶)
Theorem	mgmhmpropd 18756*	Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)
⊢ (𝜑 → 𝐵 = (Base‘𝐽))    &   ⊢ (𝜑 → 𝐶 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐶 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ (𝜑 → 𝐶 ≠ ∅)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))    ⇒   ⊢ (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀))
Theorem	mgmhmlin 18757	A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))
Theorem	mgmhmf1o 18758	A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MgmHom 𝑅)))
Theorem	idmgmhm 18759	The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀))
Theorem	issubmgm 18760*	Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Theorem	issubmgm2 18761	Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝐻 = (𝑀 ↾s 𝑆)    ⇒   ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 𝐻 ∈ Mgm)))
Theorem	rabsubmgmd 18762*	Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ Mgm)    &   ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂)    &   ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏))    &   ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂))    ⇒   ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMgm‘𝑀))
Theorem	submgmss 18763	Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 ⊆ 𝐵)
Theorem	submgmid 18764	Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (𝑀 ∈ Mgm → 𝐵 ∈ (SubMgm‘𝑀))
Theorem	submgmcl 18765	Submagmas are closed under the magma operation. (Contributed by AV, 26-Feb-2020.)
⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Theorem	submgmmgm 18766	Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.)
⊢ 𝐻 = (𝑀 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝐻 ∈ Mgm)
Theorem	submgmbas 18767	The base set of a submagma. (Contributed by AV, 26-Feb-2020.)
⊢ 𝐻 = (𝑀 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 = (Base‘𝐻))
Theorem	subsubmgm 18768	A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)))
Theorem	resmgmhm 18769	Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
⊢ 𝑈 = (𝑆 ↾s 𝑋)    ⇒   ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MgmHom 𝑇))
Theorem	resmgmhm2 18770	One direction of resmgmhm2b 18771. (Contributed by AV, 26-Feb-2020.)
⊢ 𝑈 = (𝑇 ↾s 𝑋)    ⇒   ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
Theorem	resmgmhm2b 18771	Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
⊢ 𝑈 = (𝑇 ↾s 𝑋)    ⇒   ⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈)))
Theorem	mgmhmco 18772	The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MgmHom 𝑈))
Theorem	mgmhmima 18773	The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.)
⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMgm‘𝑁))
Theorem	mgmhmeql 18774	The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.)
⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMgm‘𝑆))
Theorem	submgmacs 18775	Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Mgm → (SubMgm‘𝐺) ∈ (ACS‘𝐵))
10.1.5  Semigroups A semigroup (Smgrp, see df-sgrp 18777) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup 18777). In other words, a semigroup is an associative magma. The notion of semigroup is a generalization of that of group where the existence of an identity or inverses is not required.
Syntax	csgrp 18776	Extend class notation with class of all semigroups.
class Smgrp
Definition	df-sgrp 18777*	A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 18698), whose operation is associative. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4 . (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
Theorem	issgrp 18778*	The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
Theorem	issgrpv 18779*	The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))
Theorem	issgrpn0 18780*	The predicate "is a semigroup" for a structure with a nonempty base set. (Contributed by AV, 1-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))
Theorem	isnsgrp 18781	A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍)) → 𝑀 ∉ Smgrp))
Theorem	sgrpmgm 18782	A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm)
Theorem	sgrpass 18783	A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⚬ = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Smgrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
Theorem	sgrpcl 18784	Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⚬ = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
Theorem	sgrp0 18785	Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp)
Theorem	sgrp0b 18786	The structure with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
⊢ {⟨(Base‘ndx), ∅⟩, ⟨(+g‘ndx), 𝑂⟩} ∈ Smgrp
Theorem	sgrp1 18787	The structure with one element and the only closed internal operation for a singleton is a semigroup. (Contributed by AV, 10-Feb-2020.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Smgrp)
Theorem	issgrpd 18788*	Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.)
⊢ (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 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Smgrp)
Theorem	sgrppropd 18789*	If two structures are sets, have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a semigroup iff the other one is. (Contributed by AV, 15-Feb-2025.)
⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))
Theorem	prdsplusgsgrpcl 18790	Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)
Theorem	prdssgrpd 18791	The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Smgrp)
10.1.6  Definition and basic properties of monoids According to Wikipedia ("Monoid", https://en.wikipedia.org/wiki/Monoid, 6-Feb-2020,) "In abstract algebra [...] a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are semigroups with identity.". In the following, monoids are defined in the second way (as semigroups with identity), see df-mnd 18793, whereas many authors define magmas in the first way (as algebraic structure with a single associative binary operation and an identity element, i.e. without the need of a definition for/knowledge about semigroups), see ismnd 18795. See, for example, the definition in [Lang] p. 3: "A monoid is a set G, with a law of composition which is associative, and having a unit element".
Syntax	cmnd 18792	Extend class notation with class of all monoids.
class Mnd
Definition	df-mnd 18793*	A monoid is a semigroup, which has a two-sided neutral element. Definition 2 in [BourbakiAlg1] p. 12. In other words (according to the definition in [Lang] p. 3), a monoid is a set equipped with an everywhere defined internal operation (see mndcl 18800), whose operation is associative (see mndass 18801) and has a two-sided neutral element (see mndid 18802), see also ismnd 18795. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
Theorem	ismnddef 18794*	The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
Theorem	ismnd 18795*	The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 18800), whose operation is associative (so, a semigroup, see also mndass 18801) and has a two-sided neutral element (see mndid 18802). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Mnd ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
Theorem	isnmnd 18796*	A condition for a structure not to be a monoid: every element of the base set is not a left identity for at least one element of the base set. (Contributed by AV, 4-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd)
Theorem	sgrpidmnd 18797*	A semigroup with an identity element which is not the empty set is a monoid. Of course there could be monoids with the empty set as identity element (see, for example, the monoid of the power set of a class under union, pwmnd 18999 and pwmndid 18998), but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)
Theorem	mndsgrp 18798	A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
Theorem	mndmgm 18799	A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm)
Theorem	mndcl 18800	Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Proof shortened by AV, 8-Feb-2020.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)