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Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmgmb1mgm1 17301 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)    &    + = (+g𝑀)       ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Theoremmgm0 17302 Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)

Theoremmgm0b 17303 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

Theoremmgm1 17304 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⟩}⟩}       (𝐼𝑉𝑀 ∈ Mgm)

Theoremopifismgm 17305* 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)    &   (+g𝑀) = (𝑥𝐵, 𝑦𝐵 ↦ if(𝜓, 𝐶, 𝐷))    &   (𝜑𝐵 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷𝐵)       (𝜑𝑀 ∈ 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 17306) is an important property of monoids (see mndid 17350), and therefore also for groups (see grpid 17504), 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" (0g𝑀), see df-0g 16149. Related theorems which are already valid for magmas are provided in the following.

Theoremmgmidmo 17306* 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.)
∃*𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)

Theoremgrpidval 17307* 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)    &    + = (+g𝐺)    &    0 = (0g𝐺)        0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))

Theoremgrpidpropd 17308* 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (0g𝐾) = (0g𝐿))

Theoremfn0g 17309 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
0g Fn V

Theorem0g0 17310 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
∅ = (0g‘∅)

Theoremismgmid 17311* 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 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))       (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))

Theoremmgmidcl 17312* 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 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))       (𝜑0𝐵)

Theoremmgmlrid 17313* 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 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))       ((𝜑𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Theoremismgmid2 17314* Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝑈𝐵)    &   ((𝜑𝑥𝐵) → (𝑈 + 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 + 𝑈) = 𝑥)       (𝜑𝑈 = 0 )

Theoremgrpidd 17315* Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑0𝐵)    &   ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)       (𝜑0 = (0g𝐺))

Theoremmgmidsssn0 17316* 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 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}       (𝐺𝑉𝑂 ⊆ { 0 })

10.1.3  Ordered sums in a magma

The symbol Σg is mostly used in the context of abelian groups. Therefore, it is called "group sum". It can be defined, however, in arbitrary magmas. 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 Remark 2. in the comment for df-gsum 16150.

Theoremgsumvalx 17317* Expand out the substitutions in df-gsum 16150. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑠𝐵 ∣ ∀𝑡𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)}    &   (𝜑𝑊 = (𝐹 “ (V ∖ 𝑂)))    &   (𝜑𝐺𝑉)    &   (𝜑𝐹𝑋)    &   (𝜑 → dom 𝐹 = 𝐴)       (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)))))))

Theoremgsumval 17318* Expand out the substitutions in df-gsum 16150. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑠𝐵 ∣ ∀𝑡𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)}    &   (𝜑𝑊 = (𝐹 “ (V ∖ 𝑂)))    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)))))))

Theoremgsumpropd 17319 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 17363 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   (𝜑 → (+g𝐺) = (+g𝐻))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Theoremgsumpropd2lem 17320* Lemma for gsumpropd2 17321. (Contributed by Thierry Arnoux, 28-Jun-2017.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))    &   (𝜑 → Fun 𝐹)    &   (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))    &   𝐴 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)}))    &   𝐵 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Theoremgsumpropd2 17321* A stronger version of gsumpropd 17319, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 17322. (Contributed by Thierry Arnoux, 28-Jun-2017.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))    &   (𝜑 → Fun 𝐹)    &   (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Theoremgsummgmpropd 17322* A stronger version of gsumpropd 17319 if at least one of the involved structures is a magma, see gsumpropd2 17321. (Contributed by AV, 31-Jan-2020.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   (𝜑𝐺 ∈ Mgm)    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))    &   (𝜑 → Fun 𝐹)    &   (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Theoremgsumress 17323* 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)    &    + = (+g𝐺)    &   𝐻 = (𝐺s 𝑆)    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑆𝐵)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑0𝑆)    &   ((𝜑𝑥𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Theoremgsumval1 17324* 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 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝑊)    &   (𝜑𝐹:𝐴𝑂)       (𝜑 → (𝐺 Σg 𝐹) = 0 )

Theoremgsum0 17325 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
0 = (0g𝐺)       (𝐺 Σg ∅) = 0

Theoremgsumval2a 17326* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)⟶𝐵)    &   𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}    &   (𝜑 → ¬ ran 𝐹𝑂)       (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Theoremgsumval2 17327 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)⟶𝐵)       (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Theoremgsumprval 17328 Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 = (𝑀 + 1))    &   (𝜑𝐹:{𝑀, 𝑁}⟶𝐵)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐹𝑀) + (𝐹𝑁)))

Theoremgsumpr12val 17329 Value of the group sum operation over the pair {1, 2}. (Contributed by AV, 14-Dec-2018.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐹:{1, 2}⟶𝐵)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘1) + (𝐹‘2)))

10.1.4  Semigroups

A semigroup (SGrp, see df-sgrp 17331) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup). 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.

Syntaxcsgrp 17330 Extend class notation with class of all semigroups.
class SGrp

Definitiondf-sgrp 17331* A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 17289), 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.)
SGrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}

Theoremissgrp 17332* The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by by AV, 6-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))

Theoremissgrpv 17333* The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀𝑉 → (𝑀 ∈ SGrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))

Theoremissgrpn0 17334* The predicate "is a semigroup" for a structure with a nonempty base set. (Contributed by AV, 1-Feb-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝐴𝐵 → (𝑀 ∈ SGrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))

Theoremisnsgrp 17335 A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       ((𝑋𝐵𝑌𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → 𝑀 ∉ SGrp))

Theoremsgrpmgm 17336 A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
(𝑀 ∈ SGrp → 𝑀 ∈ Mgm)

Theoremsgrpass 17337 A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
𝐵 = (Base‘𝐺)    &    = (+g𝐺)       ((𝐺 ∈ SGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))

Theoremsgrp0 17338 Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ SGrp)

Theoremsgrp0b 17339 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⟩} ∈ SGrp

Theoremsgrp1 17340 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⟩}⟩}       (𝐼𝑉𝑀 ∈ SGrp)

10.1.5  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 17342, 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 17344. 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".

Syntaxcmnd 17341 Extend class notation with class of all monoids.
class Mnd

Definitiondf-mnd 17342* 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 17348), whose operation is associative (see mndass 17349) and has a two-sided neutral element (see mndid 17350), see also ismnd 17344. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
Mnd = {𝑔 ∈ SGrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑥𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}

Theoremismnddef 17343* 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)    &    + = (+g𝐺)       (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Theoremismnd 17344* The predicate "is a monoid". This is the definig 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 17348), whose operation is associative (so, a semigroup, see also mndass 17349) and has a two-sided neutral element (see mndid 17350). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Mnd ↔ (∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Theoremisnmnd 17345* 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)    &    = (+g𝑀)       (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥𝑀 ∉ Mnd)

Theoremmndsgrp 17346 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 ∈ SGrp)

Theoremmndmgm 17347 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)

Theoremmndcl 17348 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)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Theoremmndass 17349 A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Proof shortened by AV, 8-Feb-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theoremmndid 17350* A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Mnd → ∃𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

Theoremmndideu 17351* The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Mnd → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

Theoremmnd32g 17352 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))       (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Theoremmnd12g 17353 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))       (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

Theoremmnd4g 17354 Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝐵)    &   (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))       (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Theoremmndidcl 17355 The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Mnd → 0𝐵)

Theoremmndplusf 17356 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 3-Feb-2020.)
𝐵 = (Base‘𝐺)    &    = (+𝑓𝐺)       (𝐺 ∈ Mnd → :(𝐵 × 𝐵)⟶𝐵)

Theoremmndlrid 17357 A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Theoremmndlid 17358 The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)

Theoremmndrid 17359 The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (𝑋 + 0 ) = 𝑋)

Theoremismndd 17360* Deduce a monoid from its properties. (Contributed 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) → (𝑥 + 0 ) = 𝑥)       (𝜑𝐺 ∈ Mnd)

Theoremmndpfo 17361 The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.)
𝐵 = (Base‘𝐺)    &    = (+𝑓𝐺)       (𝐺 ∈ Mnd → :(𝐵 × 𝐵)–onto𝐵)

Theoremmndfo 17362 The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto𝐵)

Theoremmndpropd 17363* 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, one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))

Theoremmndprop 17364 If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)       (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)

Theoremissubmnd 17365* Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ Mnd ∧ 𝑆𝐵0𝑆) → (𝐻 ∈ Mnd ↔ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))

Theoremress0g 17366 0g is unaffected by restriction. This is a bit more generic than submnd0 17367. (Contributed by Thierry Arnoux, 23-Oct-2017.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Mnd ∧ 0𝐴𝐴𝐵) → 0 = (0g𝑆))

Theoremsubmnd0 17367 The zero of a submonoid is the same as the zero in the parent monoid. (Note that we must add the condition that the zero of the parent monoid is actually contained in the submonoid, because it is possible to have "subsets that are monoids" which are not submonoids because they have a different identity element.) (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐻 = (𝐺s 𝑆)       (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆)) → 0 = (0g𝐻))

Theoremprdsplusgcl 17368 Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Mnd)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)

Theoremprdsidlem 17369* Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Mnd)    &    0 = (0g𝑅)       (𝜑 → ( 0𝐵 ∧ ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))

Theoremprdsmndd 17370 The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)       (𝜑𝑌 ∈ Mnd)

Theoremprds0g 17371 Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)       (𝜑 → (0g𝑅) = (0g𝑌))

Theorempwsmnd 17372 The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Mnd ∧ 𝐼𝑉) → 𝑌 ∈ Mnd)

Theorempws0g 17373 Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &    0 = (0g𝑅)       ((𝑅 ∈ Mnd ∧ 𝐼𝑉) → (𝐼 × { 0 }) = (0g𝑌))

Theoremimasmnd2 17374* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+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) → (𝐹‘(𝑥 + 0 )) = (𝐹𝑥))       (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹0 ) = (0g𝑈)))

Theoremimasmnd 17375* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅 ∈ Mnd)    &    0 = (0g𝑅)       (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹0 ) = (0g𝑈)))

Theoremimasmndf1 17376 The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
𝑈 = (𝐹s 𝑅)    &   𝑉 = (Base‘𝑅)       ((𝐹:𝑉1-1𝐵𝑅 ∈ Mnd) → 𝑈 ∈ Mnd)

Theoremxpsmnd 17377 The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)       ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑇 ∈ Mnd)

Theoremmnd1 17378 The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Mnd)

Theoremmnd1id 17379 The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉 → (0g𝑀) = 𝐼)

10.1.6  Monoid homomorphisms and submonoids

Syntaxcmhm 17380 Hom-set generator class for monoids.
class MndHom

Syntaxcsubmnd 17381 Class function taking a monoid to its lattice of submonoids.
class SubMnd

Definitiondf-mhm 17382* A monoid homomorphism is a function on the base sets which preserves the binary operation and the identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡))})

Definitiondf-submnd 17383* A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g𝑠) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡)})

Theoremismhm 17384* Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)    &    0 = (0g𝑆)    &   𝑌 = (0g𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))

Theoremmhmrcl1 17385 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)

Theoremmhmrcl2 17386 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)

Theoremmhmf 17387 A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵𝐶)

Theoremmhmpropd 17388* Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))       (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))

Theoremmhmlin 17389 A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (+g𝑇)       ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))

Theoremmhm0 17390 A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
0 = (0g𝑆)    &   𝑌 = (0g𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹0 ) = 𝑌)

Theoremidmhm 17391 The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mnd → ( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀))

Theoremmhmf1o 17392 A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MndHom 𝑅)))

Theoremsubmrcl 17393 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)

Theoremissubm 17394* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &    + = (+g𝑀)       (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Theoremissubm2 17395 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐻 = (𝑀s 𝑆)       (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆𝐻 ∈ Mnd)))

Theoremissubmd 17396* Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝜒)    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)    &   (𝑧 = 0 → (𝜓𝜒))    &   (𝑧 = 𝑥 → (𝜓𝜃))    &   (𝑧 = 𝑦 → (𝜓𝜏))    &   (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))       (𝜑 → {𝑧𝐵𝜓} ∈ (SubMnd‘𝑀))

Theoremsubmss 17397 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 𝑆𝐵)

Theoremsubmid 17398 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mnd → 𝐵 ∈ (SubMnd‘𝑀))

Theoremsubm0cl 17399 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
0 = (0g𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 0𝑆)

Theoremsubmcl 17400 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
+ = (+g𝑀)       ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

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