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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | upwlkbprop 47001 | Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΉ(UPWalksβπΊ)π β (πΊ β V β§ πΉ β V β§ π β V)) | ||
Theorem | upwlkwlk 47002 | A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.) |
β’ (πΉ(UPWalksβπΊ)π β πΉ(WalksβπΊ)π) | ||
Theorem | upgrwlkupwlk 47003 | In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.) |
β’ ((πΊ β UPGraph β§ πΉ(WalksβπΊ)π) β πΉ(UPWalksβπΊ)π) | ||
Theorem | upgrwlkupwlkb 47004 | In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.) |
β’ (πΊ β UPGraph β (πΉ(WalksβπΊ)π β πΉ(UPWalksβπΊ)π)) | ||
Theorem | upgrisupwlkALT 47005* | Alternate proof of upgriswlk 29367 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ β π β§ π β π) β (πΉ(WalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) | ||
Theorem | upgredgssspr 47006 | The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.) |
β’ (πΊ β UPGraph β (EdgβπΊ) β (Pairsβ(VtxβπΊ))) | ||
Theorem | uspgropssxp 47007* | The set πΊ of "simple pseudographs" for a fixed set π of vertices is a subset of a Cartesian product. For more details about the class πΊ of all "simple pseudographs" see comments on uspgrbisymrel 47017. (Contributed by AV, 24-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} β β’ (π β π β πΊ β (π Γ π)) | ||
Theorem | uspgrsprfv 47008* | The value of the function πΉ which maps a "simple pseudograph" for a fixed set π of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for πΊ as defined here, the function πΉ is a bijection between the "simple pseudographs" and the subsets of the set of pairs π over the fixed set π of vertices, see uspgrbispr 47014. (Contributed by AV, 24-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ (π β πΊ β (πΉβπ) = (2nd βπ)) | ||
Theorem | uspgrsprf 47009* | The mapping πΉ is a function from the "simple pseudographs" with a fixed set of vertices π into the subsets of the set of pairs over the set π. (Contributed by AV, 24-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ πΉ:πΊβΆπ | ||
Theorem | uspgrsprf1 47010* | The mapping πΉ is a one-to-one function from the "simple pseudographs" with a fixed set of vertices π into the subsets of the set of pairs over the set π. (Contributed by AV, 25-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ πΉ:πΊβ1-1βπ | ||
Theorem | uspgrsprfo 47011* | The mapping πΉ is a function from the "simple pseudographs" with a fixed set of vertices π onto the subsets of the set of pairs over the set π. (Contributed by AV, 25-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ (π β π β πΉ:πΊβontoβπ) | ||
Theorem | uspgrsprf1o 47012* | The mapping πΉ is a bijection between the "simple pseudographs" with a fixed set of vertices π and the subsets of the set of pairs over the set π. See also the comments on uspgrbisymrel 47017. (Contributed by AV, 25-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ (π β π β πΉ:πΊβ1-1-ontoβπ) | ||
Theorem | uspgrex 47013* | The class πΊ of all "simple pseudographs" with a fixed set of vertices π is a set. (Contributed by AV, 26-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} β β’ (π β π β πΊ β V) | ||
Theorem | uspgrbispr 47014* | There is a bijection between the "simple pseudographs" with a fixed set of vertices π and the subsets of the set of pairs over the set π. (Contributed by AV, 26-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} β β’ (π β π β βπ π:πΊβ1-1-ontoβπ) | ||
Theorem | uspgrspren 47015* | The set πΊ of the "simple pseudographs" with a fixed set of vertices π and the class π of subsets of the set of pairs over the fixed set π are equinumerous. (Contributed by AV, 27-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} β β’ (π β π β πΊ β π) | ||
Theorem | uspgrymrelen 47016* | The set πΊ of the "simple pseudographs" with a fixed set of vertices π and the class π of the symmetric relations on the fixed set π are equinumerous. For more details about the class πΊ of all "simple pseudographs" see comments on uspgrbisymrel 47017. (Contributed by AV, 27-Nov-2021.) |
β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ π = {π β π« (π Γ π) β£ βπ₯ β π βπ¦ β π (π₯ππ¦ β π¦ππ₯)} β β’ (π β π β πΊ β π ) | ||
Theorem | uspgrbisymrel 47017* |
There is a bijection between the "simple pseudographs" for a fixed
set
π of vertices and the class π
of the
symmetric relations on the
fixed set π. The simple pseudographs, which are
graphs without
hyper- or multiedges, but which may contain loops, are expressed as
ordered pairs of the vertices and the edges (as proper or improper
unordered pairs of vertices, not as indexed edges!) in this theorem.
That class πΊ of such simple pseudographs is a set
(if π is a
set, see uspgrex 47013) of equivalence classes of graphs
abstracting from
the index sets of their edge functions.
Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of πΊ and the symmetric relations π on the fixed set π of vertices. This theorem would not hold for πΊ = {π β USPGraph β£ (Vtxβπ) = π} and even not for πΊ = {β¨π£, πβ© β£ (π£ = π β§ β¨π£, πβ© β USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.) |
β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ π = {π β π« (π Γ π) β£ βπ₯ β π βπ¦ β π (π₯ππ¦ β π¦ππ₯)} β β’ (π β π β βπ π:πΊβ1-1-ontoβπ ) | ||
Theorem | uspgrbisymrelALT 47018* | Alternate proof of uspgrbisymrel 47017 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ π = {π β π« (π Γ π) β£ βπ₯ β π βπ¦ β π (π₯ππ¦ β π¦ππ₯)} β β’ (π β π β βπ π:πΊβ1-1-ontoβπ ) | ||
Theorem | ovn0dmfun 47019 | If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6924. (Contributed by AV, 27-Jan-2020.) |
β’ ((π΄πΉπ΅) β β β (β¨π΄, π΅β© β dom πΉ β§ Fun (πΉ βΎ {β¨π΄, π΅β©}))) | ||
Theorem | xpsnopab 47020* | A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.) |
β’ ({π} Γ πΆ) = {β¨π, πβ© β£ (π = π β§ π β πΆ)} | ||
Theorem | xpiun 47021* | A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.) |
β’ (π΅ Γ πΆ) = βͺ π₯ β π΅ {β¨π, πβ© β£ (π = π₯ β§ π β πΆ)} | ||
Theorem | ovn0ssdmfun 47022* | If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6924. (Contributed by AV, 27-Jan-2020.) |
β’ (βπ β π· βπ β πΈ (ππΉπ) β β β ((π· Γ πΈ) β dom πΉ β§ Fun (πΉ βΎ (π· Γ πΈ)))) | ||
Theorem | fnxpdmdm 47023 | The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.) |
β’ (πΉ Fn (π΄ Γ π΄) β dom dom πΉ = π΄) | ||
Theorem | cnfldsrngbas 47024 | The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
β’ π = (βfld βΎs π) β β’ (π β β β π = (Baseβπ )) | ||
Theorem | cnfldsrngadd 47025 | The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
β’ π = (βfld βΎs π) β β’ (π β π β + = (+gβπ )) | ||
Theorem | cnfldsrngmul 47026 | The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
β’ π = (βfld βΎs π) β β’ (π β π β Β· = (.rβπ )) | ||
Theorem | plusfreseq 47027 | If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+πβπ) β β’ (β β ran ⨣ β ( + βΎ (π΅ Γ π΅)) = ⨣ ) | ||
Theorem | mgmplusfreseq 47028 | If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+πβπ) β β’ ((π β Mgm β§ β β π΅) β ( + βΎ (π΅ Γ π΅)) = ⨣ ) | ||
Theorem | 0mgm 47029 | A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.) |
β’ (Baseβπ) = β β β’ (π β π β π β Mgm) | ||
Theorem | opmpoismgm 47030* | A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ (+gβπ) = (π₯ β π΅, π¦ β π΅ β¦ πΆ) & β’ (π β π΅ β β ) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β πΆ β π΅) β β’ (π β π β Mgm) | ||
Theorem | copissgrp 47031* | A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ (+gβπ) = (π₯ β π΅, π¦ β π΅ β¦ πΆ) & β’ (π β π΅ β β ) & β’ (π β πΆ β π΅) β β’ (π β π β Smgrp) | ||
Theorem | copisnmnd 47032* | A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ (+gβπ) = (π₯ β π΅, π¦ β π΅ β¦ πΆ) & β’ (π β πΆ β π΅) & β’ (π β 1 < (β―βπ΅)) β β’ (π β π β Mnd) | ||
Theorem | 0nodd 47033* | 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
β’ π = {π§ β β€ β£ βπ₯ β β€ π§ = ((2 Β· π₯) + 1)} β β’ 0 β π | ||
Theorem | 1odd 47034* | 1 is an odd integer. (Contributed by AV, 3-Feb-2020.) |
β’ π = {π§ β β€ β£ βπ₯ β β€ π§ = ((2 Β· π₯) + 1)} β β’ 1 β π | ||
Theorem | 2nodd 47035* | 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
β’ π = {π§ β β€ β£ βπ₯ β β€ π§ = ((2 Β· π₯) + 1)} β β’ 2 β π | ||
Theorem | oddibas 47036* | Lemma 1 for oddinmgm 47038: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.) |
β’ π = {π§ β β€ β£ βπ₯ β β€ π§ = ((2 Β· π₯) + 1)} & β’ π = (βfld βΎs π) β β’ π = (Baseβπ) | ||
Theorem | oddiadd 47037* | Lemma 2 for oddinmgm 47038: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.) |
β’ π = {π§ β β€ β£ βπ₯ β β€ π§ = ((2 Β· π₯) + 1)} & β’ π = (βfld βΎs π) β β’ + = (+gβπ) | ||
Theorem | oddinmgm 47038* | The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 47113, and even a non-unital ring, see 2zrng 47104. (Contributed by AV, 3-Feb-2020.) |
β’ π = {π§ β β€ β£ βπ₯ β β€ π§ = ((2 Β· π₯) + 1)} & β’ π = (βfld βΎs π) β β’ π β Mgm | ||
Theorem | nnsgrpmgm 47039 | The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.) |
β’ π = (βfld βΎs β) β β’ π β Mgm | ||
Theorem | nnsgrp 47040 | The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.) |
β’ π = (βfld βΎs β) β β’ π β Smgrp | ||
Theorem | nnsgrpnmnd 47041 | The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.) |
β’ π = (βfld βΎs β) β β’ π β Mnd | ||
Theorem | nn0mnd 47042 | The set of nonnegative integers under (complex) addition is a monoid. Example in [Lang] p. 6. Remark: π could have also been written as (βfld βΎs β0). (Contributed by AV, 27-Dec-2023.) |
β’ π = {β¨(Baseβndx), β0β©, β¨(+gβndx), + β©} β β’ π β Mnd | ||
Theorem | gsumsplit2f 47043* | Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.) |
β’ β²ππ & β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ + = (+gβπΊ) & β’ (π β πΊ β CMnd) & β’ (π β π΄ β π) & β’ ((π β§ π β π΄) β π β π΅) & β’ (π β (π β π΄ β¦ π) finSupp 0 ) & β’ (π β (πΆ β© π·) = β ) & β’ (π β π΄ = (πΆ βͺ π·)) β β’ (π β (πΊ Ξ£g (π β π΄ β¦ π)) = ((πΊ Ξ£g (π β πΆ β¦ π)) + (πΊ Ξ£g (π β π· β¦ π)))) | ||
Theorem | gsumdifsndf 47044* | Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.) |
β’ β²ππ & β’ β²ππ & β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) & β’ (π β πΊ β CMnd) & β’ (π β π΄ β π) & β’ (π β (π β π΄ β¦ π) finSupp (0gβπΊ)) & β’ ((π β§ π β π΄) β π β π΅) & β’ (π β π β π΄) & β’ (π β π β π΅) & β’ ((π β§ π = π) β π = π) β β’ (π β (πΊ Ξ£g (π β π΄ β¦ π)) = ((πΊ Ξ£g (π β (π΄ β {π}) β¦ π)) + π)) | ||
Theorem | gsumfsupp 47045 | A group sum of a family can be restricted to the support of that family without changing its value, provided that that support is finite. This corresponds to the definition of an (infinite) product in [Lang] p. 5, last two formulas. (Contributed by AV, 27-Dec-2023.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ πΌ = (πΉ supp 0 ) & β’ (π β πΊ β CMnd) & β’ (π β π΄ β π) & β’ (π β πΉ:π΄βΆπ΅) & β’ (π β πΉ finSupp 0 ) β β’ (π β (πΊ Ξ£g (πΉ βΎ πΌ)) = (πΊ Ξ£g πΉ)) | ||
With df-mpo 7406, binary operations are defined by a rule, and with df-ov 7404, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation (19-Jan-2020), "... a binary operation on a set π is a mapping of the elements of the Cartesian product π Γ π to S: π:π Γ πβΆπ. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, we call binary operations mapping the elements of the Cartesian product π Γ π internal binary operations, see df-intop 47062. If, in addition, the result is also contained in the set π, the operation is called closed internal binary operation, see df-clintop 47063. Therefore, a "binary operation on a set π " according to Wikipedia is a "closed internal binary operation" in our terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations 47063 ). Taking a step back, we define "laws" applicable for "binary operations" (which even need not to be functions), according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7. These laws are used, on the one hand, to specialize internal binary operations (see df-clintop 47063 and df-assintop 47064), and on the other hand to define the common algebraic structures like magmas, groups, rings, etc. Internal binary operations, which obey these laws, are defined afterwards. Notice that in [BourbakiAlg1] p. 1, p. 4 and p. 7, these operations are called "laws" by themselves. In the following, an alternate definition df-cllaw 47049 for an internal binary operation is provided, which does not require function-ness, but only closure. Therefore, this definition could be used as binary operation (Slot 2) defined for a magma as extensible structure, see mgmplusgiopALT 47057, or for an alternate definition df-mgm2 47082 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups). | ||
In this subsection, the "laws" applicable for "binary operations" according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7 are defined. These laws are called "internal laws" in [BourbakiAlg1] p. xxi. | ||
Syntax | ccllaw 47046 | Extend class notation for the closure law. |
class clLaw | ||
Syntax | casslaw 47047 | Extend class notation for the associative law. |
class assLaw | ||
Syntax | ccomlaw 47048 | Extend class notation for the commutative law. |
class comLaw | ||
Definition | df-cllaw 47049* | The closure law for binary operations, see definitions of laws A0. and M0. in section 1.1 of [Hall] p. 1, or definition 1 in [BourbakiAlg1] p. 1: the value of a binary operation applied to two operands of a given sets is an element of this set. By this definition, the closure law is expressed as binary relation: a binary operation is related to a set by clLaw if the closure law holds for this binary operation regarding this set. Note that the binary operation needs not to be a function. (Contributed by AV, 7-Jan-2020.) |
β’ clLaw = {β¨π, πβ© β£ βπ₯ β π βπ¦ β π (π₯ππ¦) β π} | ||
Definition | df-comlaw 47050* | The commutative law for binary operations, see definitions of laws A2. and M2. in section 1.1 of [Hall] p. 1, or definition 8 in [BourbakiAlg1] p. 7: the value of a binary operation applied to two operands equals the value of a binary operation applied to the two operands in reversed order. By this definition, the commutative law is expressed as binary relation: a binary operation is related to a set by comLaw if the commutative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by AV, 7-Jan-2020.) |
β’ comLaw = {β¨π, πβ© β£ βπ₯ β π βπ¦ β π (π₯ππ¦) = (π¦ππ₯)} | ||
Definition | df-asslaw 47051* | The associative law for binary operations, see definitions of laws A1. and M1. in section 1.1 of [Hall] p. 1, or definition 5 in [BourbakiAlg1] p. 4: the value of a binary operation applied the value of the binary operation applied to two operands and a third operand equals the value of the binary operation applied to the first operand and the value of the binary operation applied to the second and third operand. By this definition, the associative law is expressed as binary relation: a binary operation is related to a set by assLaw if the associative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.) |
β’ assLaw = {β¨π, πβ© β£ βπ₯ β π βπ¦ β π βπ§ β π ((π₯ππ¦)ππ§) = (π₯π(π¦ππ§))} | ||
Theorem | iscllaw 47052* | The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.) |
β’ (( β¬ β π β§ π β π) β ( β¬ clLaw π β βπ₯ β π βπ¦ β π (π₯ β¬ π¦) β π)) | ||
Theorem | iscomlaw 47053* | The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.) |
β’ (( β¬ β π β§ π β π) β ( β¬ comLaw π β βπ₯ β π βπ¦ β π (π₯ β¬ π¦) = (π¦ β¬ π₯))) | ||
Theorem | clcllaw 47054 | Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.) |
β’ (( β¬ clLaw π β§ π β π β§ π β π) β (π β¬ π) β π) | ||
Theorem | isasslaw 47055* | The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.) |
β’ (( β¬ β π β§ π β π) β ( β¬ assLaw π β βπ₯ β π βπ¦ β π βπ§ β π ((π₯ β¬ π¦) β¬ π§) = (π₯ β¬ (π¦ β¬ π§)))) | ||
Theorem | asslawass 47056* | Associativity of an associative operation. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 21-Jan-2020.) |
β’ ( β¬ assLaw π β βπ₯ β π βπ¦ β π βπ§ β π ((π₯ β¬ π¦) β¬ π§) = (π₯ β¬ (π¦ β¬ π§))) | ||
Theorem | mgmplusgiopALT 47057 | Slot 2 (group operation) of a magma as extensible structure is a closed operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ (π β Mgm β (+gβπ) clLaw (Baseβπ)) | ||
Theorem | sgrpplusgaopALT 47058 | Slot 2 (group operation) of a semigroup as extensible structure is an associative operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ (πΊ β Smgrp β (+gβπΊ) assLaw (BaseβπΊ)) | ||
In this subsection, "internal binary operations" obeying different laws are defined. | ||
Syntax | cintop 47059 | Extend class notation with class of internal (binary) operations for a set. |
class intOp | ||
Syntax | cclintop 47060 | Extend class notation with class of closed operations for a set. |
class clIntOp | ||
Syntax | cassintop 47061 | Extend class notation with class of associative operations for a set. |
class assIntOp | ||
Definition | df-intop 47062* | Function mapping a set to the class of all internal (binary) operations for this set. (Contributed by AV, 20-Jan-2020.) |
β’ intOp = (π β V, π β V β¦ (π βm (π Γ π))) | ||
Definition | df-clintop 47063 | Function mapping a set to the class of all closed (internal binary) operations for this set, see definition in section 1.2 of [Hall] p. 2, definition in section I.1 of [Bruck] p. 1, or definition 1 in [BourbakiAlg1] p. 1, where it is called "a law of composition". (Contributed by AV, 20-Jan-2020.) |
β’ clIntOp = (π β V β¦ (π intOp π)) | ||
Definition | df-assintop 47064* | Function mapping a set to the class of all associative (closed internal binary) operations for this set, see definition 5 in [BourbakiAlg1] p. 4, where it is called "an associative law of composition". (Contributed by AV, 20-Jan-2020.) |
β’ assIntOp = (π β V β¦ {π β ( clIntOp βπ) β£ π assLaw π}) | ||
Theorem | intopval 47065 | The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
β’ ((π β π β§ π β π) β (π intOp π) = (π βm (π Γ π))) | ||
Theorem | intop 47066 | An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
β’ ( β¬ β (π intOp π) β β¬ :(π Γ π)βΆπ) | ||
Theorem | clintopval 47067 | The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
β’ (π β π β ( clIntOp βπ) = (π βm (π Γ π))) | ||
Theorem | assintopval 47068* | The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
β’ (π β π β ( assIntOp βπ) = {π β ( clIntOp βπ) β£ π assLaw π}) | ||
Theorem | assintopmap 47069* | The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.) |
β’ (π β π β ( assIntOp βπ) = {π β (π βm (π Γ π)) β£ π assLaw π}) | ||
Theorem | isclintop 47070 | The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
β’ (π β π β ( β¬ β ( clIntOp βπ) β β¬ :(π Γ π)βΆπ)) | ||
Theorem | clintop 47071 | A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
β’ ( β¬ β ( clIntOp βπ) β β¬ :(π Γ π)βΆπ) | ||
Theorem | assintop 47072 | An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
β’ ( β¬ β ( assIntOp βπ) β ( β¬ :(π Γ π)βΆπ β§ β¬ assLaw π)) | ||
Theorem | isassintop 47073* | The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
β’ (π β π β ( β¬ β ( assIntOp βπ) β ( β¬ :(π Γ π)βΆπ β§ βπ₯ β π βπ¦ β π βπ§ β π ((π₯ β¬ π¦) β¬ π§) = (π₯ β¬ (π¦ β¬ π§))))) | ||
Theorem | clintopcllaw 47074 | The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
β’ ( β¬ β ( clIntOp βπ) β β¬ clLaw π) | ||
Theorem | assintopcllaw 47075 | The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
β’ ( β¬ β ( assIntOp βπ) β β¬ clLaw π) | ||
Theorem | assintopasslaw 47076 | The associative low holds for a associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
β’ ( β¬ β ( assIntOp βπ) β β¬ assLaw π) | ||
Theorem | assintopass 47077* | An associative (closed internal binary) operation for a set is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
β’ ( β¬ β ( assIntOp βπ) β βπ₯ β π βπ¦ β π βπ§ β π ((π₯ β¬ π¦) β¬ π§) = (π₯ β¬ (π¦ β¬ π§))) | ||
Syntax | cmgm2 47078 | Extend class notation with class of all magmas. |
class MgmALT | ||
Syntax | ccmgm2 47079 | Extend class notation with class of all commutative magmas. |
class CMgmALT | ||
Syntax | csgrp2 47080 | Extend class notation with class of all semigroups. |
class SGrpALT | ||
Syntax | ccsgrp2 47081 | Extend class notation with class of all commutative semigroups. |
class CSGrpALT | ||
Definition | df-mgm2 47082 | A magma is a set equipped with a closed operation. Definition 1 of [BourbakiAlg1] p. 1, or definition of a groupoid in section I.1 of [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by AV, 6-Jan-2020.) |
β’ MgmALT = {π β£ (+gβπ) clLaw (Baseβπ)} | ||
Definition | df-cmgm2 47083 | A commutative magma is a magma with a commutative operation. Definition 8 of [BourbakiAlg1] p. 7. (Contributed by AV, 20-Jan-2020.) |
β’ CMgmALT = {π β MgmALT β£ (+gβπ) comLaw (Baseβπ)} | ||
Definition | df-sgrp2 47084 | A semigroup is a magma with an associative operation. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4, or of a semigroup in section 1.3 of [Hall] p. 7. (Contributed by AV, 6-Jan-2020.) |
β’ SGrpALT = {π β MgmALT β£ (+gβπ) assLaw (Baseβπ)} | ||
Definition | df-csgrp2 47085 | A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.) |
β’ CSGrpALT = {π β SGrpALT β£ (+gβπ) comLaw (Baseβπ)} | ||
Theorem | ismgmALT 47086 | The predicate "is a magma". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ π΅ = (Baseβπ) & β’ β¬ = (+gβπ) β β’ (π β π β (π β MgmALT β β¬ clLaw π΅)) | ||
Theorem | iscmgmALT 47087 | The predicate "is a commutative magma". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ π΅ = (Baseβπ) & β’ β¬ = (+gβπ) β β’ (π β CMgmALT β (π β MgmALT β§ β¬ comLaw π΅)) | ||
Theorem | issgrpALT 47088 | The predicate "is a semigroup". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ π΅ = (Baseβπ) & β’ β¬ = (+gβπ) β β’ (π β SGrpALT β (π β MgmALT β§ β¬ assLaw π΅)) | ||
Theorem | iscsgrpALT 47089 | The predicate "is a commutative semigroup". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ π΅ = (Baseβπ) & β’ β¬ = (+gβπ) β β’ (π β CSGrpALT β (π β SGrpALT β§ β¬ comLaw π΅)) | ||
Theorem | mgm2mgm 47090 | Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.) |
β’ (π β MgmALT β π β Mgm) | ||
Theorem | sgrp2sgrp 47091 | Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.) |
β’ (π β SGrpALT β π β Smgrp) | ||
Theorem | lmod0rng 47092 | If the scalar ring of a module is the zero ring, the module is the zero module, i.e. the base set of the module is the singleton consisting of the identity element only. (Contributed by AV, 17-Apr-2019.) |
β’ ((π β LMod β§ Β¬ (Scalarβπ) β NzRing) β (Baseβπ) = {(0gβπ)}) | ||
Theorem | nzrneg1ne0 47093 | The additive inverse of the 1 in a nonzero ring is not zero ( -1 =/= 0 ). (Contributed by AV, 29-Apr-2019.) |
β’ (π β NzRing β ((invgβπ )β(1rβπ )) β (0gβπ )) | ||
Theorem | lidldomn1 47094* | If a (left) ideal (which is not the zero ideal) of a domain has a multiplicative identity element, the identity element is the identity of the domain. (Contributed by AV, 17-Feb-2020.) |
β’ πΏ = (LIdealβπ ) & β’ Β· = (.rβπ ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ ((π β Domn β§ (π β πΏ β§ π β { 0 }) β§ πΌ β π) β (βπ₯ β π ((πΌ Β· π₯) = π₯ β§ (π₯ Β· πΌ) = π₯) β πΌ = 1 )) | ||
Theorem | lidlabl 47095 | A (left) ideal of a ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.) |
β’ πΏ = (LIdealβπ ) & β’ πΌ = (π βΎs π) β β’ ((π β Ring β§ π β πΏ) β πΌ β Abel) | ||
Theorem | lidlrng 47096 | A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) (Proof shortened by AV, 11-Mar-2025.) |
β’ πΏ = (LIdealβπ ) & β’ πΌ = (π βΎs π) β β’ ((π β Ring β§ π β πΏ) β πΌ β Rng) | ||
Theorem | zlidlring 47097 | The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.) |
β’ πΏ = (LIdealβπ ) & β’ πΌ = (π βΎs π) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β Ring β§ π = { 0 }) β πΌ β Ring) | ||
Theorem | uzlidlring 47098 | Only the zero (left) ideal or the unit (left) ideal of a domain is a unital ring. (Contributed by AV, 18-Feb-2020.) |
β’ πΏ = (LIdealβπ ) & β’ πΌ = (π βΎs π) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β Domn β§ π β πΏ) β (πΌ β Ring β (π = { 0 } β¨ π = π΅))) | ||
Theorem | lidldomnnring 47099 | A (left) ideal of a domain which is neither the zero ideal nor the unit ideal is not a unital ring. (Contributed by AV, 18-Feb-2020.) |
β’ πΏ = (LIdealβπ ) & β’ πΌ = (π βΎs π) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β Domn β§ (π β πΏ β§ π β { 0 } β§ π β π΅)) β πΌ β Ring) | ||
Theorem | 0even 47100* | 0 is an even integer. (Contributed by AV, 11-Feb-2020.) |
β’ πΈ = {π§ β β€ β£ βπ₯ β β€ π§ = (2 Β· π₯)} β β’ 0 β πΈ |
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