P. 496 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49501-49600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gpg5edgnedg 49501	Two consecutive (according to the numbering) inside vertices of the Petersen graph G(5,2) are not connected by an edge, but are connected by an edge in a 5-prism G(5,1). (Contributed by AV, 29-Dec-2025.)
⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))
Theorem	grlimedgnedg 49502*	In general, the image of an edge of a graph by a local isomprphism is not an edge of the other graph, proven by an example (see gpg5edgnedg 49501). This theorem proves that the analogon (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ 𝐾 ∈ 𝐼)) → (𝐹 “ 𝐾) ∈ 𝐸) of grimedgi 49307 for ordinarily isomorphic graphs does not hold in general. (Contributed by AV, 30-Dec-2025.)
⊢ ∃𝑔 ∈ USGraph ∃ℎ ∈ USGraph ∃𝑓 ∈ (𝑔 GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘𝑔)∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))
21.52.15.9  Loop-free graphs - extension
Theorem	1hegrlfgr 49503*	A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)    ⇒   ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
21.52.15.10  Walks - extension
Syntax	cupwlks 49504	Extend class notation with walks (of a pseudograph).
class UPWalks
Definition	df-upwlks 49505*	Define the set of all walks (in a pseudograph), called "simple walks" in the following. According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)." According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudographs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)
⊢ UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
Theorem	upwlksfval 49506*	The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
Theorem	isupwlk 49507*	Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
Theorem	isupwlkg 49508*	Generalization of isupwlk 49507: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
Theorem	upwlkbprop 49509	Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
Theorem	upwlkwlk 49510	A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.)
⊢ (𝐹(UPWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	upgrwlkupwlk 49511	In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.)
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃)
Theorem	upgrwlkupwlkb 49512	In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.)
⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ 𝐹(UPWalks‘𝐺)𝑃))
Theorem	upgrisupwlkALT 49513*	Alternate proof of upgriswlk 30278 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
21.52.15.11  Edges of graphs expressed as sets of unordered pairs
Theorem	upgredgssspr 49514	The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.)
⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ (Pairs‘(Vtx‘𝐺)))
Theorem	uspgropssxp 49515*	The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of a Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 49525. (Contributed by AV, 24-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝐺 ⊆ (𝑊 × 𝑃))
Theorem	uspgrsprfv 49516*	The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 49522. (Contributed by AV, 24-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))    ⇒   ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋))
Theorem	uspgrsprf 49517*	The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))    ⇒   ⊢ 𝐹:𝐺⟶𝑃
Theorem	uspgrsprf1 49518*	The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))    ⇒   ⊢ 𝐹:𝐺⟶1-1𝑃
Theorem	uspgrsprfo 49519*	The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺⟶onto𝑃)
Theorem	uspgrsprf1o 49520*	The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 49525. (Contributed by AV, 25-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺⟶1-1onto𝑃)
Theorem	uspgrex 49521*	The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V)
Theorem	uspgrbispr 49522*	There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    ⇒   ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺⟶1-1onto𝑃)
Theorem	uspgrspren 49523*	The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
⊢ 𝑃 = 𝒫 (Pairs‘𝑉)    &   ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑃)
Theorem	uspgrymrelen 49524*	The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 49525. (Contributed by AV, 27-Nov-2021.)
⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    ⇒   ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅)
Theorem	uspgrbisymrel 49525*	There is a bijection between the "simple pseudographs" for a fixed set 𝑉 of vertices and the class 𝑅 of the symmetric relations on the fixed set 𝑉. 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 𝐺 of such simple pseudographs is a set (if 𝑉 is a set, see uspgrex 49521) 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 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtx‘𝑔) = 𝑉} and even not for 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ⟨𝑣, 𝑒⟩ ∈ USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.)
⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    ⇒   ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺⟶1-1onto𝑅)
Theorem	uspgrbisymrelALT 49526*	Alternate proof of uspgrbisymrel 49525 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}    ⇒   ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺⟶1-1onto𝑅)
21.52.16  Monoids (extension)
21.52.16.1  Auxiliary theorems
Theorem	ovn0dmfun 49527	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 6956. (Contributed by AV, 27-Jan-2020.)
⊢ ((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))
Theorem	xpsnopab 49528*	A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.)
⊢ ({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)}
Theorem	xpiun 49529*	A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)
⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)}
Theorem	fnxpdmdm 49530	The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.)
⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)
Theorem	cnfldsrngbas 49531	The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
⊢ 𝑅 = (ℂfld ↾s 𝑆)    ⇒   ⊢ (𝑆 ⊆ ℂ → 𝑆 = (Base‘𝑅))
Theorem	cnfldsrngadd 49532	The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
⊢ 𝑅 = (ℂfld ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ 𝑉 → + = (+g‘𝑅))
Theorem	cnfldsrngmul 49533	The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
⊢ 𝑅 = (ℂfld ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ 𝑉 → · = (.r‘𝑅))
21.52.16.2  Magmas, Semigroups and Monoids (extension)
Theorem	plusfreseq 49534	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.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ ⨣ = (+𝑓‘𝑀)    ⇒   ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
Theorem	mgmplusfreseq 49535	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.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ ⨣ = (+𝑓‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ )
Theorem	0mgm 49536	A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.)
⊢ (Base‘𝑀) = ∅    ⇒   ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm)
21.52.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)
Theorem	opmpoismgm 49537*	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.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑀 ∈ Mgm)
Theorem	copissgrp 49538*	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.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑀 ∈ Smgrp)
Theorem	copisnmnd 49539*	A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 1 < (♯‘𝐵))    ⇒   ⊢ (𝜑 → 𝑀 ∉ Mnd)
Theorem	0nodd 49540*	0 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    ⇒   ⊢ 0 ∉ 𝑂
Theorem	1odd 49541*	1 is an odd integer. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    ⇒   ⊢ 1 ∈ 𝑂
Theorem	2nodd 49542*	2 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    ⇒   ⊢ 2 ∉ 𝑂
Theorem	oddibas 49543*	Lemma 1 for oddinmgm 49545: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   ⊢ 𝑀 = (ℂfld ↾s 𝑂)    ⇒   ⊢ 𝑂 = (Base‘𝑀)
Theorem	oddiadd 49544*	Lemma 2 for oddinmgm 49545: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   ⊢ 𝑀 = (ℂfld ↾s 𝑂)    ⇒   ⊢ + = (+g‘𝑀)
Theorem	oddinmgm 49545*	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 49620, and even a non-unital ring, see 2zrng 49611. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   ⊢ 𝑀 = (ℂfld ↾s 𝑂)    ⇒   ⊢ 𝑀 ∉ Mgm
Theorem	nnsgrpmgm 49546	The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.)
⊢ 𝑀 = (ℂfld ↾s ℕ)    ⇒   ⊢ 𝑀 ∈ Mgm
Theorem	nnsgrp 49547	The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.)
⊢ 𝑀 = (ℂfld ↾s ℕ)    ⇒   ⊢ 𝑀 ∈ Smgrp
Theorem	nnsgrpnmnd 49548	The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.)
⊢ 𝑀 = (ℂfld ↾s ℕ)    ⇒   ⊢ 𝑀 ∉ Mnd
Theorem	nn0mnd 49549	The set of nonnegative integers under (complex) addition is a monoid. Example in [Lang] p. 6. Remark: 𝑀 could have also been written as (ℂfld ↾s ℕ0). (Contributed by AV, 27-Dec-2023.)
⊢ 𝑀 = {⟨(Base‘ndx), ℕ0⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ 𝑀 ∈ Mnd
21.52.16.4  Group sum operation (extension 1)
Theorem	gsumsplit2f 49550*	Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋))))
Theorem	gsumdifsndf 49551*	Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑘𝑌    &   ⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))
Theorem	gsumfsupp 49552	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.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐼 = (𝐹 supp 0 )    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐼)) = (𝐺 Σg 𝐹))
21.52.17  Magmas and internal binary operations (alternate approach) With df-mpo 7456, binary operations are defined by a rule, and with df-ov 7454, 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 7454 (19-Jan-2020), "... a binary operation on a set 𝑆 is a mapping of the elements of the Cartesian product 𝑆 × 𝑆 to S: 𝑓:𝑆 × 𝑆⟶𝑆. 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 𝑆 × 𝑆 internal binary operations, see df-intop 49569. If, in addition, the result is also contained in the set 𝑆, the operation is called closed internal binary operation, see df-clintop 49570. Therefore, a "binary operation on a set 𝑆 " 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 49570 ). 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 49570 and df-assintop 49571), 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 49556 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 49564, or for an alternate definition df-mgm2 49589 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups).
21.52.17.1  Laws for internal binary operations 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 49553	Extend class notation for the closure law.
class clLaw
Syntax	casslaw 49554	Extend class notation for the associative law.
class assLaw
Syntax	ccomlaw 49555	Extend class notation for the commutative law.
class comLaw
Definition	df-cllaw 49556*	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 need not to be a function. (Contributed by AV, 7-Jan-2020.)
⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
Definition	df-comlaw 49557*	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.)
⊢ comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
Definition	df-asslaw 49558*	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.)
⊢ assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
Theorem	iscllaw 49559*	The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.)
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
Theorem	iscomlaw 49560*	The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.)
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ comLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥)))
Theorem	clcllaw 49561	Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.)
⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)
Theorem	isasslaw 49562*	The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
Theorem	asslawass 49563*	Associativity of an associative operation. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 21-Jan-2020.)
⊢ ( ⚬ assLaw 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
Theorem	mgmplusgiopALT 49564	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.)
⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))
Theorem	sgrpplusgaopALT 49565	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.)
⊢ (𝐺 ∈ Smgrp → (+g‘𝐺) assLaw (Base‘𝐺))
21.52.17.2  Internal binary operations In this subsection, "internal binary operations" obeying different laws are defined.
Syntax	cintop 49566	Extend class notation with class of internal (binary) operations for a set.
class intOp
Syntax	cclintop 49567	Extend class notation with class of closed operations for a set.
class clIntOp
Syntax	cassintop 49568	Extend class notation with class of associative operations for a set.
class assIntOp
Definition	df-intop 49569*	Function mapping a set to the class of all internal (binary) operations for this set. (Contributed by AV, 20-Jan-2020.)
⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚)))
Definition	df-clintop 49570	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.)
⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
Definition	df-assintop 49571*	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.)
⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
Theorem	intopval 49572	The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))
Theorem	intop 49573	An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → ⚬ :(𝑀 × 𝑀)⟶𝑁)
Theorem	clintopval 49574	The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
Theorem	assintopval 49575*	The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
Theorem	assintopmap 49576*	The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
Theorem	isclintop 49577	The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
Theorem	clintop 49578	A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
Theorem	assintop 49579	An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))
Theorem	isassintop 49580*	The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))
Theorem	clintopcllaw 49581	The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀)
Theorem	assintopcllaw 49582	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.)
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀)
Theorem	assintopasslaw 49583	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.)
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ assLaw 𝑀)
Theorem	assintopass 49584*	An associative (closed internal binary) operation for a set is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
21.52.17.3  Alternative definitions for magmas and semigroups
Syntax	cmgm2 49585	Extend class notation with class of all magmas.
class MgmALT
Syntax	ccmgm2 49586	Extend class notation with class of all commutative magmas.
class CMgmALT
Syntax	csgrp2 49587	Extend class notation with class of all semigroups.
class SGrpALT
Syntax	ccsgrp2 49588	Extend class notation with class of all commutative semigroups.
class CSGrpALT
Definition	df-mgm2 49589	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.)
⊢ MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)}
Definition	df-cmgm2 49590	A commutative magma is a magma with a commutative operation. Definition 8 of [BourbakiAlg1] p. 7. (Contributed by AV, 20-Jan-2020.)
⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)}
Definition	df-sgrp2 49591	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.)
⊢ SGrpALT = {𝑔 ∈ MgmALT ∣ (+g‘𝑔) assLaw (Base‘𝑔)}
Definition	df-csgrp2 49592	A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.)
⊢ CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+g‘𝑔) comLaw (Base‘𝑔)}
Theorem	ismgmALT 49593	The predicate "is a magma". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵))
Theorem	iscmgmALT 49594	The predicate "is a commutative magma". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵))
Theorem	issgrpALT 49595	The predicate "is a semigroup". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ assLaw 𝐵))
Theorem	iscsgrpALT 49596	The predicate "is a commutative semigroup". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ ⚬ comLaw 𝐵))
Theorem	mgm2mgm 49597	Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
Theorem	sgrp2sgrp 49598	Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp)
21.52.18  Rings (extension)
21.52.18.1  Nonzero rings (extension)
Theorem	lmod0rng 49599	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.)
⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing) → (Base‘𝑀) = {(0g‘𝑀)})
Theorem	nzrneg1ne0 49600	The additive inverse of the 1 in a nonzero ring is not zero ( -1 =/= 0 ). (Contributed by AV, 29-Apr-2019.)
⊢ (𝑅 ∈ NzRing → ((invg‘𝑅)‘(1r‘𝑅)) ≠ (0g‘𝑅))