P. 488 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48701-48800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	upgrwlkupwlkb 48701	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 48702*	Alternate proof of upgriswlk 29776 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))})))
21.48.15.11  Edges of graphs expressed as sets of unordered pairs
Theorem	upgredgssspr 48703	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 48704*	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 48714. (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 48705*	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 48711. (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 48706*	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 48707*	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 48708*	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 48709*	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 48714. (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 48710*	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 48711*	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 48712*	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 48713*	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 48714. (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 48714*	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 48710) 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 48715*	Alternate proof of uspgrbisymrel 48714 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)
21.48.16  Monoids (extension)
21.48.16.1  Auxiliary theorems
Theorem	ovn0dmfun 48716	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 6892. (Contributed by AV, 27-Jan-2020.)
⊢ ((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))
Theorem	xpsnopab 48717*	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 48718*	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 48719*	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 6892. (Contributed by AV, 27-Jan-2020.)
⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
Theorem	fnxpdmdm 48720	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 48721	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 48722	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 48723	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))
21.48.16.2  Magmas, Semigroups and Monoids (extension)
Theorem	plusfreseq 48724	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 48725	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 48726	A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.)
⊢ (Base‘𝑀) = ∅    ⇒   ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm)
21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)
Theorem	opmpoismgm 48727*	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 48728*	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 48729*	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 48730*	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 48731*	1 is an odd integer. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    ⇒   ⊢ 1 ∈ 𝑂
Theorem	2nodd 48732*	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 48733*	Lemma 1 for oddinmgm 48735: 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 48734*	Lemma 2 for oddinmgm 48735: 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 48735*	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 48810, and even a non-unital ring, see 2zrng 48801. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   ⊢ 𝑀 = (ℂfld ↾s 𝑂)    ⇒   ⊢ 𝑀 ∉ Mgm
Theorem	nnsgrpmgm 48736	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 48737	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 48738	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 48739	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
21.48.16.4  Group sum operation (extension 1)
Theorem	gsumsplit2f 48740*	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 48741*	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 48742	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))
21.48.17  Magmas and internal binary operations (alternate approach) With df-mpo 7386, binary operations are defined by a rule, and with df-ov 7384, 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 7384 (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 48759. If, in addition, the result is also contained in the set 𝑆, the operation is called closed internal binary operation, see df-clintop 48760. 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 48760 ). 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 48760 and df-assintop 48761), 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 48746 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 48754, or for an alternate definition df-mgm2 48779 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups).
21.48.17.1  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 48743	Extend class notation for the closure law.
class clLaw
Syntax	casslaw 48744	Extend class notation for the associative law.
class assLaw
Syntax	ccomlaw 48745	Extend class notation for the commutative law.
class comLaw
Definition	df-cllaw 48746*	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 48747*	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 48748*	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 48749*	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 48750*	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 48751	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 48752*	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 48753*	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 48754	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 48755	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))
21.48.17.2  Internal binary operations In this subsection, "internal binary operations" obeying different laws are defined.
Syntax	cintop 48756	Extend class notation with class of internal (binary) operations for a set.
class intOp
Syntax	cclintop 48757	Extend class notation with class of closed operations for a set.
class clIntOp
Syntax	cassintop 48758	Extend class notation with class of associative operations for a set.
class assIntOp
Definition	df-intop 48759*	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 48760	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 48761*	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 48762	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 48763	An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → ⚬ :(𝑀 × 𝑀)⟶𝑁)
Theorem	clintopval 48764	The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
Theorem	assintopval 48765*	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 48766*	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 48767	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 48768	A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
Theorem	assintop 48769	An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))
Theorem	isassintop 48770*	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 48771	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 48772	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 48773	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 48774*	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)))
21.48.17.3  Alternative definitions for magmas and semigroups
Syntax	cmgm2 48775	Extend class notation with class of all magmas.
class MgmALT
Syntax	ccmgm2 48776	Extend class notation with class of all commutative magmas.
class CMgmALT
Syntax	csgrp2 48777	Extend class notation with class of all semigroups.
class SGrpALT
Syntax	ccsgrp2 48778	Extend class notation with class of all commutative semigroups.
class CSGrpALT
Definition	df-mgm2 48779	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 48780	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 48781	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 48782	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 48783	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 48784	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 48785	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 48786	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 48787	Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
Theorem	sgrp2sgrp 48788	Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp)
21.48.18  Rings (extension)
21.48.18.1  Nonzero rings (extension)
Theorem	lmod0rng 48789	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 48790	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))
21.48.18.2  Ideals as non-unital rings
Theorem	lidldomn1 48791*	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.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 }) ∧ 𝐼 ∈ 𝑈) → (∀𝑥 ∈ 𝑈 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥) → 𝐼 = 1 ))
Theorem	lidlabl 48792	A (left) ideal of a ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Abel)
Theorem	lidlrng 48793	A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) (Proof shortened by AV, 11-Mar-2025.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
Theorem	zlidlring 48794	The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
Theorem	uzlidlring 48795	Only the zero (left) ideal or the unit (left) ideal of a domain is a unital ring. (Contributed by AV, 18-Feb-2020.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
Theorem	lidldomnnring 48796	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.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → 𝐼 ∉ Ring)
21.48.18.3  The non-unital ring of even integers
Theorem	0even 48797*	0 is an even integer. (Contributed by AV, 11-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    ⇒   ⊢ 0 ∈ 𝐸
Theorem	1neven 48798*	1 is not an even integer. (Contributed by AV, 12-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    ⇒   ⊢ 1 ∉ 𝐸
Theorem	2even 48799*	2 is an even integer. (Contributed by AV, 12-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    ⇒   ⊢ 2 ∈ 𝐸
Theorem	2zlidl 48800*	The even integers are a (left) ideal of the ring of integers. (Contributed by AV, 20-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑈 = (LIdeal‘ℤring)    ⇒   ⊢ 𝐸 ∈ 𝑈