P. 469 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46801-46900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isomgrref 46801	The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.)
⊢ (𝐺 ∈ UHGraph → 𝐺 IsomGr 𝐺)
Theorem	isomgrsym 46802	The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.)
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 𝐵 → 𝐵 IsomGr 𝐴))
Theorem	isomgrsymb 46803	The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.)
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 IsomGr 𝐵 ↔ 𝐵 IsomGr 𝐴))
Theorem	isomgrtrlem 46804*	Lemma for isomgrtr 46805. (Contributed by AV, 5-Dec-2022.)
⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))
Theorem	isomgrtr 46805	The isomorphy relation is transitive for hypergraphs. (Contributed by AV, 5-Dec-2022.)
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝐴 IsomGr 𝐵 ∧ 𝐵 IsomGr 𝐶) → 𝐴 IsomGr 𝐶))
Theorem	strisomgrop 46806	A graph represented as an extensible structure with vertices as base set and indexed edges is isomorphic to a hypergraph represented as ordered pair with the same vertices and edges. (Contributed by AV, 11-Nov-2022.)
⊢ 𝐺 = ⟨𝑉, 𝐸⟩    &   ⊢ 𝐻 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 IsomGr 𝐻)
Theorem	ushrisomgr 46807	A simple hypergraph (with arbitrarily indexed edges) is isomorphic to a graph with the same vertices and the same edges, indexed by the edges themselves. (Contributed by AV, 11-Nov-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩    ⇒   ⊢ (𝐺 ∈ USHGraph → 𝐺 IsomGr 𝐻)
21.45.15.2  Loop-free graphs - extension
Theorem	1hegrlfgr 46808*	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)})
21.45.15.3  Walks - extension
Syntax	cupwlks 46809	Extend class notation with walks (of a pseudograph).
class UPWalks
Definition	df-upwlks 46810*	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 46811*	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 46812*	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 46813*	Generalization of isupwlk 46812: 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 46814	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 46815	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 46816	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 46817	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 46818*	Alternate proof of upgriswlk 29165 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))})))
21.45.15.4  Edges of graphs expressed as sets of unordered pairs
Theorem	upgredgssspr 46819	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 46820*	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 46830. (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 46821*	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 46827. (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 46822*	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 46823*	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–1-1→𝑃
Theorem	uspgrsprfo 46824*	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–onto→𝑃)
Theorem	uspgrsprf1o 46825*	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 46830. (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–1-1-onto→𝑃)
Theorem	uspgrex 46826*	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 46827*	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–1-1-onto→𝑃)
Theorem	uspgrspren 46828*	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 46829*	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 46830. (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 46830*	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 46826) 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–1-1-onto→𝑅)
Theorem	uspgrbisymrelALT 46831*	Alternate proof of uspgrbisymrel 46830 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–1-1-onto→𝑅)
21.45.16  Monoids (extension)
21.45.16.1  Auxiliary theorems
Theorem	ovn0dmfun 46832	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 6933. (Contributed by AV, 27-Jan-2020.)
⊢ ((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))
Theorem	xpsnopab 46833*	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 46834*	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	ovn0ssdmfun 46835*	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 6933. (Contributed by AV, 27-Jan-2020.)
⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
Theorem	fnxpdmdm 46836	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 46837	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 46838	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 46839	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))
21.45.16.2  Magmas, Semigroups and Monoids (extension)
Theorem	plusfreseq 46840	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 46841	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 46842	A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.)
⊢ (Base‘𝑀) = ∅    ⇒   ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm)
21.45.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)
Theorem	opmpoismgm 46843*	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 46844*	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 46845*	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 46846*	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 46847*	1 is an odd integer. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    ⇒   ⊢ 1 ∈ 𝑂
Theorem	2nodd 46848*	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 46849*	Lemma 1 for oddinmgm 46851: 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 46850*	Lemma 2 for oddinmgm 46851: 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 46851*	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 46930, and even a non-unital ring, see 2zrng 46921. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   ⊢ 𝑀 = (ℂfld ↾s 𝑂)    ⇒   ⊢ 𝑀 ∉ Mgm
Theorem	nnsgrpmgm 46852	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 46853	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 46854	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 46855	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
21.45.16.4  Group sum operation (extension 1)
Theorem	gsumsplit2f 46856*	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 46857*	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 46858	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))
21.45.17  Magmas and internal binary operations (alternate approach) With df-mpo 7416, binary operations are defined by a rule, and with df-ov 7414, 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 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 46875. If, in addition, the result is also contained in the set 𝑆, the operation is called closed internal binary operation, see df-clintop 46876. 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 46876 ). 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 46876 and df-assintop 46877), 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 46862 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 46870, or for an alternate definition df-mgm2 46895 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups).
21.45.17.1  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 46859	Extend class notation for the closure law.
class clLaw
Syntax	casslaw 46860	Extend class notation for the associative law.
class assLaw
Syntax	ccomlaw 46861	Extend class notation for the commutative law.
class comLaw
Definition	df-cllaw 46862*	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.)
⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
Definition	df-comlaw 46863*	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 46864*	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 46865*	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 46866*	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 46867	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 46868*	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 46869*	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 46870	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 46871	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))
21.45.17.2  Internal binary operations In this subsection, "internal binary operations" obeying different laws are defined.
Syntax	cintop 46872	Extend class notation with class of internal (binary) operations for a set.
class intOp
Syntax	cclintop 46873	Extend class notation with class of closed operations for a set.
class clIntOp
Syntax	cassintop 46874	Extend class notation with class of associative operations for a set.
class assIntOp
Definition	df-intop 46875*	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 46876	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 46877*	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 46878	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 46879	An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → ⚬ :(𝑀 × 𝑀)⟶𝑁)
Theorem	clintopval 46880	The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
Theorem	assintopval 46881*	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 46882*	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 46883	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 46884	A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
Theorem	assintop 46885	An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))
Theorem	isassintop 46886*	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 46887	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 46888	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 46889	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 46890*	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)))
21.45.17.3  Alternative definitions for magmas and semigroups
Syntax	cmgm2 46891	Extend class notation with class of all magmas.
class MgmALT
Syntax	ccmgm2 46892	Extend class notation with class of all commutative magmas.
class CMgmALT
Syntax	csgrp2 46893	Extend class notation with class of all semigroups.
class SGrpALT
Syntax	ccsgrp2 46894	Extend class notation with class of all commutative semigroups.
class CSGrpALT
Definition	df-mgm2 46895	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 46896	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 46897	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 46898	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 46899	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 46900	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))