P. 190 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18901-19000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	efmndhash 18901	The monoid of endofunctions on 𝑛 objects has cardinality 𝑛↑𝑛. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ Fin → (♯‘𝐵) = ((♯‘𝐴)↑(♯‘𝐴)))
Theorem	efmndbasfi 18902	The monoid of endofunctions on a finite set 𝐴 is finite. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ Fin → 𝐵 ∈ Fin)
Theorem	efmndfv 18903	The function value of an endofunction. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐴)
Theorem	efmndtset 18904	The topology of the monoid of endofunctions on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just endofunctions - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by AV, 25-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))
Theorem	efmndplusg 18905*	The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
Theorem	efmndov 18906	The value of the group operation of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌))
Theorem	efmndcl 18907	The group operation of the monoid of endofunctions on 𝐴 is closed. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	efmndtopn 18908	The topology of the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺))
Theorem	symggrplem 18909*	Lemma for symggrp 19432 and efmndsgrp 18911. Conditions for an operation to be associative. Formerly part of proof for symggrp 19432. (Contributed by AV, 28-Jan-2024.)
⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ∘ 𝑦))    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	efmndmgm 18910	The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 ∈ Mgm
Theorem	efmndsgrp 18911	The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 ∈ Smgrp
Theorem	ielefmnd 18912	The identity function restricted to a set 𝐴 is an element of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))
Theorem	efmndid 18913	The identity function restricted to a set 𝐴 is the identity element of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝐺))
Theorem	efmndmnd 18914	The monoid of endofunctions on a set 𝐴 is actually a monoid. (Contributed by AV, 31-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd)
Theorem	efmnd0nmnd 18915	Even the monoid of endofunctions on the empty set is actually a monoid. (Contributed by AV, 31-Jan-2024.)
⊢ (EndoFMnd‘∅) ∈ Mnd
Theorem	efmndbas0 18916	The base set of the monoid of endofunctions on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 31-Mar-2024.)
⊢ (Base‘(EndoFMnd‘∅)) = {∅}
Theorem	efmnd1hash 18917	The monoid of endofunctions on a singleton has cardinality 1. (Contributed by AV, 27-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (♯‘𝐵) = 1)
Theorem	efmnd1bas 18918	The monoid of endofunctions on a singleton consists of the identity only. (Contributed by AV, 31-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{⟨𝐼, 𝐼⟩}})
Theorem	efmnd2hash 18919	The monoid of endofunctions on a (proper) pair has cardinality 4. (Contributed by AV, 18-Feb-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼, 𝐽}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐵) = 4)
Theorem	submefmnd 18920*	If the base set of a monoid is contained in the base set of the monoid of endofunctions on a set 𝐴, contains the identity function and has the function composition as group operation, then its base set is a submonoid of the monoid of endofunctions on set 𝐴. Analogous to pgrpsubgsymg 19441. (Contributed by AV, 17-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐹 = (Base‘𝑆)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (((𝑆 ∈ Mnd ∧ 𝐹 ⊆ 𝐵 ∧ 0 ∈ 𝐹) ∧ (+g‘𝑆) = (𝑓 ∈ 𝐹, 𝑔 ∈ 𝐹 ↦ (𝑓 ∘ 𝑔))) → 𝐹 ∈ (SubMnd‘𝑀)))
Theorem	sursubmefmnd 18921*	The set of surjective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–onto→𝐴} ∈ (SubMnd‘𝑀))
Theorem	injsubmefmnd 18922*	The set of injective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀))
Theorem	idressubmefmnd 18923	The singleton containing only the identity function restricted to a set is a submonoid of the monoid of endofunctions on this set. (Contributed by AV, 17-Feb-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {( I ↾ 𝐴)} ∈ (SubMnd‘𝐺))
Theorem	idresefmnd 18924	The structure with the singleton containing only the identity function restricted to a set 𝐴 as base set and the function composition as group operation, constructed by (structure) restricting the monoid of endofunctions on 𝐴 to that singleton, is a monoid whose base set is a subset of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 17-Feb-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    &   ⊢ 𝐸 = (𝐺 ↾s {( I ↾ 𝐴)})    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐸 ∈ Mnd ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))
Theorem	smndex1ibas 18925	The modulo function 𝐼 is an endofunction on ℕ0. (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    ⇒   ⊢ 𝐼 ∈ (Base‘𝑀)
Theorem	smndex1iidm 18926*	The modulo function 𝐼 is idempotent. (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    ⇒   ⊢ (𝐼 ∘ 𝐼) = 𝐼
Theorem	smndex1gbas 18927*	The constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   ⊢ (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾) ∈ (Base‘𝑀))
Theorem	smndex1gid 18928*	The composition of a constant function (𝐺‘𝐾) with another endofunction on ℕ0 results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺‘𝐾) ∘ 𝐹) = (𝐺‘𝐾))
Theorem	smndex1igid 18929*	The composition of the modulo function 𝐼 and a constant function (𝐺‘𝐾) results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   ⊢ (𝐾 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝐾)) = (𝐺‘𝐾))
Theorem	smndex1basss 18930*	The modulo function 𝐼 and the constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    ⇒   ⊢ 𝐵 ⊆ (Base‘𝑀)
Theorem	smndex1bas 18931*	The base set of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 12-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ (Base‘𝑆) = 𝐵
Theorem	smndex1mgm 18932*	The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a magma. (Contributed by AV, 14-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝑆 ∈ Mgm
Theorem	smndex1sgrp 18933*	The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a semigroup. (Contributed by AV, 14-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝑆 ∈ Smgrp
Theorem	smndex1mndlem 18934*	Lemma for smndex1mnd 18935 and smndex1id 18936. (Contributed by AV, 16-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ (𝑋 ∈ 𝐵 → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋))
Theorem	smndex1mnd 18935*	The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a monoid. (Contributed by AV, 16-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝑆 ∈ Mnd
Theorem	smndex1id 18936*	The modulo function 𝐼 is the identity of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 16-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝐼 = (0g‘𝑆)
Theorem	smndex1n0mnd 18937*	The identity of the monoid 𝑀 of endofunctions on set ℕ0 is not contained in the base set of the constructed monoid 𝑆. (Contributed by AV, 17-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ (0g‘𝑀) ∉ 𝐵
Theorem	nsmndex1 18938*	The base set 𝐵 of the constructed monoid 𝑆 is not a submonoid of the monoid 𝑀 of endofunctions on set ℕ0, although 𝑀 ∈ Mnd and 𝑆 ∈ Mnd and 𝐵 ⊆ (Base‘𝑀) hold. (Contributed by AV, 17-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   ⊢ 𝑆 = (𝑀 ↾s 𝐵)    ⇒   ⊢ 𝐵 ∉ (SubMnd‘𝑀)
Theorem	smndex2dbas 18939	The doubling function 𝐷 is an endofunction on ℕ0. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    ⇒   ⊢ 𝐷 ∈ 𝐵
Theorem	smndex2dnrinv 18940	The doubling function 𝐷 has no right inverse in the monoid of endofunctions on ℕ0. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    ⇒   ⊢ ∀𝑓 ∈ 𝐵 (𝐷 ∘ 𝑓) ≠ 0
Theorem	smndex2hbas 18941	The halving functions 𝐻 are endofunctions on ℕ0. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))    ⇒   ⊢ 𝐻 ∈ 𝐵
Theorem	smndex2dlinvh 18942*	The halving functions 𝐻 are left inverses of the doubling function 𝐷. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘ℕ0)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))    ⇒   ⊢ (𝐻 ∘ 𝐷) = 0
10.1.10  Examples and counterexamples for magmas, semigroups and monoids
Theorem	mgm2nsgrplem1 18943*	Lemma 1 for mgm2nsgrp 18947: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18680). (Contributed by AV, 27-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm)
Theorem	mgm2nsgrplem2 18944*	Lemma 2 for mgm2nsgrp 18947. (Contributed by AV, 27-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ⚬ 𝐴) ⚬ 𝐵) = 𝐴)
Theorem	mgm2nsgrplem3 18945*	Lemma 3 for mgm2nsgrp 18947. (Contributed by AV, 28-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⚬ (𝐴 ⚬ 𝐵)) = 𝐵)
Theorem	mgm2nsgrplem4 18946*	Lemma 4 for mgm2nsgrp 18947: M is not a semigroup. (Contributed by AV, 28-Jan-2020.) (Proof shortened by AV, 31-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    ⇒   ⊢ ((♯‘𝑆) = 2 → 𝑀 ∉ Smgrp)
Theorem	mgm2nsgrp 18947*	A small magma (with two elements) which is not a semigroup. (Contributed by AV, 28-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))    ⇒   ⊢ ((♯‘𝑆) = 2 → (𝑀 ∈ Mgm ∧ 𝑀 ∉ Smgrp))
Theorem	sgrp2nmndlem1 18948*	Lemma 1 for sgrp2nmnd 18955: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18680). (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm)
Theorem	sgrp2nmndlem2 18949*	Lemma 2 for sgrp2nmnd 18955. (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ⚬ 𝐶) = 𝐴)
Theorem	sgrp2nmndlem3 18950*	Lemma 3 for sgrp2nmnd 18955. (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵)
Theorem	sgrp2rid2 18951*	A small semigroup (with two elements) with two right identities which are different if 𝐴 ≠ 𝐵. (Contributed by AV, 10-Feb-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦)
Theorem	sgrp2rid2ex 18952*	A small semigroup (with two elements) with two right identities which are different. (Contributed by AV, 10-Feb-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((♯‘𝑆) = 2 → ∃𝑥 ∈ 𝑆 ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ≠ 𝑧 ∧ (𝑦 ⚬ 𝑥) = 𝑦 ∧ (𝑦 ⚬ 𝑧) = 𝑦))
Theorem	sgrp2nmndlem4 18953*	Lemma 4 for sgrp2nmnd 18955: M is a semigroup. (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    ⇒   ⊢ ((♯‘𝑆) = 2 → 𝑀 ∈ Smgrp)
Theorem	sgrp2nmndlem5 18954*	Lemma 5 for sgrp2nmnd 18955: M is not a monoid. (Contributed by AV, 29-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    ⇒   ⊢ ((♯‘𝑆) = 2 → 𝑀 ∉ Mnd)
Theorem	sgrp2nmnd 18955*	A small semigroup (with two elements) which is not a monoid. (Contributed by AV, 26-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    &   ⊢ (Base‘𝑀) = 𝑆    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    ⇒   ⊢ ((♯‘𝑆) = 2 → (𝑀 ∈ Smgrp ∧ 𝑀 ∉ Mnd))
Theorem	mgmnsgrpex 18956	There is a magma which is not a semigroup. (Contributed by AV, 29-Jan-2020.)
⊢ ∃𝑚 ∈ Mgm 𝑚 ∉ Smgrp
Theorem	sgrpnmndex 18957	There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.)
⊢ ∃𝑚 ∈ Smgrp 𝑚 ∉ Mnd
Theorem	sgrpssmgm 18958	The class of all semigroups is a proper subclass of the class of all magmas. (Contributed by AV, 29-Jan-2020.)
⊢ Smgrp ⊊ Mgm
Theorem	mndsssgrp 18959	The class of all monoids is a proper subclass of the class of all semigroups. (Contributed by AV, 29-Jan-2020.)
⊢ Mnd ⊊ Smgrp
Theorem	pwmndgplus 18960*	The operation of the monoid of the power set of a class 𝐴 under union. (Contributed by AV, 27-Feb-2024.)
⊢ (Base‘𝑀) = 𝒫 𝐴    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋(+g‘𝑀)𝑌) = (𝑋 ∪ 𝑌))
Theorem	pwmndid 18961*	The identity of the monoid of the power set of a class 𝐴 under union is the empty set. (Contributed by AV, 27-Feb-2024.)
⊢ (Base‘𝑀) = 𝒫 𝐴    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   ⊢ (0g‘𝑀) = ∅
Theorem	pwmnd 18962*	The power set of a class 𝐴 is a monoid under union. (Contributed by AV, 27-Feb-2024.)
⊢ (Base‘𝑀) = 𝒫 𝐴    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   ⊢ 𝑀 ∈ Mnd
10.2  Groups
10.2.1  Definition and basic properties
Syntax	cgrp 18963	Extend class notation with class of all groups.
class Grp
Syntax	cminusg 18964	Extend class notation with inverse of group element.
class invg
Syntax	csg 18965	Extend class notation with group subtraction (or division) operation.
class -g
Definition	df-grp 18966*	Define class of all groups. A group is a monoid (df-mnd 18760) whose internal operation is such that every element admits a left inverse (which can be proven to be a two-sided inverse). Thus, a group 𝐺 is an algebraic structure formed from a base set of elements (notated (Base‘𝐺) per df-base 17245) and an internal group operation (notated (+g‘𝐺) per df-plusg 17310). The operation combines any two elements of the group base set and must satisfy the 4 group axioms: closure (the result of the group operation must always be a member of the base set, see grpcl 18971), associativity (so ((𝑎+g𝑏)+g𝑐) = (𝑎+g(𝑏+g𝑐)) for any a, b, c, see grpass 18972), identity (there must be an element 𝑒 = (0g‘𝐺) such that 𝑒+g𝑎 = 𝑎+g𝑒 = 𝑎 for any a), and inverse (for each element a in the base set, there must be an element 𝑏 = invg𝑎 in the base set such that 𝑎+g𝑏 = 𝑏+g𝑎 = 𝑒). It can be proven that the identity element is unique (grpideu 18974). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 19815). Subgroups can often be formed from groups, see df-subg 19153. An example of an (Abelian) group is the set of complex numbers ℂ over the group operation + (addition), as proven in cnaddablx 19900; an Abelian group is a group as proven in ablgrp 19817. Other structures include groups, including unital rings (df-ring 20252) and fields (df-field 20748). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)}
Definition	df-minusg 18967*	Define inverse of group element. (Contributed by NM, 24-Aug-2011.)
⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+g‘𝑔)𝑥) = (0g‘𝑔))))
Definition	df-sbg 18968*	Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
Theorem	isgrp 18969*	The predicate "is a group". (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))
Theorem	grpmnd 18970	A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
Theorem	grpcl 18971	Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	grpass 18972	A group operation is associative. (Contributed by NM, 14-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	grpinvex 18973*	Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
Theorem	grpideu 18974*	The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
Theorem	grpassd 18975	A group operation is associative. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	grpmndd 18976	A group is a monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mnd)
Theorem	grpcld 18977	Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	grpplusf 18978	The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = (+𝑓‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)⟶𝐵)
Theorem	grpplusfo 18979	The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = (+𝑓‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)–onto→𝐵)
Theorem	resgrpplusfrn 18980	The underlying set of a group operation which is a restriction of a structure. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by AV, 30-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ 𝐹 = (+𝑓‘𝐻)    ⇒   ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = ran 𝐹)
Theorem	grppropd 18981*	If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
Theorem	grpprop 18982	If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
⊢ (Base‘𝐾) = (Base‘𝐿)    &   ⊢ (+g‘𝐾) = (+g‘𝐿)    ⇒   ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)
Theorem	grppropstr 18983	Generalize a specific 2-element group 𝐿 to show that any set 𝐾 with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ (Base‘𝐾) = 𝐵    &   ⊢ (+g‘𝐾) = +    &   ⊢ 𝐿 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)
Theorem	grpss 18984	Show that a structure extending a constructed group (e.g., a ring) is also a group. This allows to prove that a constructed potential ring 𝑅 is a group before we know that it is also a ring. (Theorem ringgrp 20255, on the other hand, requires that we know in advance that 𝑅 is a ring.) (Contributed by NM, 11-Oct-2013.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    &   ⊢ 𝑅 ∈ V    &   ⊢ 𝐺 ⊆ 𝑅    &   ⊢ Fun 𝑅    ⇒   ⊢ (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)
Theorem	isgrpd2e 18985*	Deduce a group from its properties. In this version of isgrpd2 18986, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 10-Aug-2013.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 0 = (0g‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	isgrpd2 18986*	Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2734, but we make an exception for theorems such as isgrpd2 18986, ismndd 18781, and islmodd 20880 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 0 = (0g‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	isgrpde 18987*	Deduce a group from its properties. In this version of isgrpd 18988, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 6-Jan-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	isgrpd 18988*	Deduce a group from its properties. Unlike isgrpd2 18986, this one goes straight from the base properties rather than going through Mnd. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	isgrpi 18989*	Properties that determine a group. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 3-Sep-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ 0 ∈ 𝐵    &   ⊢ (𝑥 ∈ 𝐵 → ( 0 + 𝑥) = 𝑥)    &   ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 )    ⇒   ⊢ 𝐺 ∈ Grp
Theorem	grpsgrp 18990	A group is a semigroup. (Contributed by AV, 28-Aug-2021.)
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
Theorem	grpmgmd 18991	A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mgm)
Theorem	dfgrp2 18992*	Alternate definition of a group as semigroup with a left identity and a left inverse for each element. This "definition" is weaker than df-grp 18966, based on the definition of a monoid which provides a left and a right identity. (Contributed by AV, 28-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))
Theorem	dfgrp2e 18993*	Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))
Theorem	isgrpix 18994*	Properties that determine a group. Read 𝑁 as 𝑁(𝑥). Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ 0 ∈ 𝐵    &   ⊢ (𝑥 ∈ 𝐵 → ( 0 + 𝑥) = 𝑥)    &   ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 )    ⇒   ⊢ 𝐺 ∈ Grp
Theorem	grpidcl 18995	The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
Theorem	grpbn0 18996	The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
Theorem	grplid 18997	The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
Theorem	grprid 18998	The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋)
Theorem	grplidd 18999	The identity element of a group is a left identity. Deduction associated with grplid 18997. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ( 0 + 𝑋) = 𝑋)
Theorem	grpridd 19000	The identity element of a group is a right identity. Deduction associated with grprid 18998. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 0 ) = 𝑋)