P. 192 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	idresefmnd 19101	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 19102	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 19103*	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 19104*	The constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.) Avoid ax-rep 5231 and shorten proof. (Revised by GG, 2-Apr-2026.)
⊢ 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	smndex1gbasOLD 19105*	Obsolete version of smndex1gbas 19104 as of 2-Apr-2026. (Contributed by AV, 12-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 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 19106*	The composition of a constant function (𝐺‘𝐾) with another endofunction on ℕ0 results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.) Avoid ax-rep 5231. (Revised by GG, 2-Apr-2026.)
⊢ 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	smndex1gidOLD 19107*	Obsolete version of smndex1gid 19106 as of 2-Apr-2026. (Contributed by AV, 14-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 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 19108*	The composition of the modulo function 𝐼 and a constant function (𝐺‘𝐾) results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.) Avoid ax-rep 5231. (Revised by GG, 2-Apr-2026.)
⊢ 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	smndex1igidOLD 19109*	Obsolete version of smndex1igid 19108 as of 2-Apr-2026. (Contributed by AV, 14-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 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 19110*	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 19111*	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 19112*	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 19113*	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 19114*	Lemma for smndex1mnd 19115 and smndex1id 19116. (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 19115*	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 19116*	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 19117*	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 19118*	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 19119	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 19120	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 19121	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 19122*	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 19123*	Lemma 1 for mgm2nsgrp 19127: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18839). (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 19124*	Lemma 2 for mgm2nsgrp 19127. (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 19125*	Lemma 3 for mgm2nsgrp 19127. (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 19126*	Lemma 4 for mgm2nsgrp 19127: 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 19127*	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 19128*	Lemma 1 for sgrp2nmnd 19135: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18839). (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 19129*	Lemma 2 for sgrp2nmnd 19135. (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 19130*	Lemma 3 for sgrp2nmnd 19135. (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 19131*	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 19132*	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 19133*	Lemma 4 for sgrp2nmnd 19135: 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 19134*	Lemma 5 for sgrp2nmnd 19135: 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 19135*	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 19136	There is a magma which is not a semigroup. (Contributed by AV, 29-Jan-2020.)
⊢ ∃𝑚 ∈ Mgm 𝑚 ∉ Smgrp
Theorem	sgrpnmndex 19137	There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.)
⊢ ∃𝑚 ∈ Smgrp 𝑚 ∉ Mnd
Theorem	sgrpssmgm 19138	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 19139	The class of all monoids is a proper subclass of the class of all semigroups. (Contributed by AV, 29-Jan-2020.)
⊢ Mnd ⊊ Smgrp
Theorem	degenmgmopdm 19140	The domain of the operation of a degenerate magma. (Contributed by AV, 18-Aug-2026.)
⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}    ⇒   ⊢ dom (+g‘𝑀) = ({1o} × {∅, 1o, 2o})
Theorem	degenmgmbas 19141	The base set of a degenerate magma. (Contributed by AV, 21-Aug-2026.)
⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}    &   ⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ 𝐵 = {∅, 1o}
Theorem	degenmgmnfn 19142	The operation of a degenerate magma is not a function on its base set. (Contributed by AV, 21-Aug-2026.)
⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}    &   ⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ¬ (+g‘𝑀) Fn (𝐵 × 𝐵)
Theorem	degenmgm 19143	A degenerate magma: although the operation is not defined for all pairs of elements of the base set ((∅(+g‘𝑀)1o) and (∅(+g‘𝑀)∅) are not defined, and therefore are ∅ by definition, which is contained in the base set), and its domain is not (a subset of) the base set (2o is in the domain of the operation, but not in the base set) , the structure 𝑀 is still a magma according to our definition. (Contributed by AV, 18-Aug-2026.)
⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}    ⇒   ⊢ 𝑀 ∈ Mgm
Theorem	degenmgm2opdm 19144	The domain of the operation of a second degenerate magma. (Contributed by AV, 21-Aug-2026.)
⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}⟩}    ⇒   ⊢ dom (+g‘𝑀) = ({1o} × {1o, 2o})
Theorem	degenmgm2nfun 19145	The operation of a second degenerate magma is not a function. (Contributed by AV, 21-Aug-2026.)
⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}⟩}    ⇒   ⊢ ¬ Fun (+g‘𝑀)
Theorem	degenmgm2 19146	A degenerate magma: although the operation is not defined for all pairs of elements of the base set, and its domain is not (a subset of) the base set, and the operation is not a function (see degenmgm2nfun 19145), the structure 𝑀 is still a magma according to our definition. (Contributed by AV, 21-Aug-2026.)
⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}⟩}    ⇒   ⊢ 𝑀 ∈ Mgm
Theorem	pwmndgplus 19147*	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 19148*	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 19149*	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 19150	Extend class notation with class of all groups.
class Grp
Syntax	cminusg 19151	Extend class notation with inverse of group element.
class invg
Syntax	csg 19152	Extend class notation with group subtraction (or division) operation.
class -g
Definition	df-grp 19153*	Define class of all groups. A group is a monoid (df-mnd 18930) 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 17394) and an internal group operation (notated (+g‘𝐺) per df-plusg 17447). 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 19158), associativity (so ((𝑎+g𝑏)+g𝑐) = (𝑎+g(𝑏+g𝑐)) for any a, b, c, see grpass 19159), 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 19161). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 20003). Subgroups can often be formed from groups, see df-subg 19339. An example of an (Abelian) group is the set of complex numbers ℂ over the group operation + (addition), as proven in cnaddablx 20088; an Abelian group is a group as proven in ablgrp 20005. Other structures include groups, including unital rings (df-ring 20467) and fields (df-field 20989). (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 19154*	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 19155*	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 19156*	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 19157	A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
Theorem	grpcl 19158	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 19159	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 19160*	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 19161*	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 19162	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 19163	A group is a monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mnd)
Theorem	grpcld 19164	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 19165	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 19166	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 19167	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 19168*	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 19169	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 19170	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 19171	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 20470, 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 19172*	Deduce a group from its properties. In this version of isgrpd2 19173, 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 19173*	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 2760, but we make an exception for theorems such as isgrpd2 19173, ismndd 18952, and islmodd 21147 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 19174*	Deduce a group from its properties. In this version of isgrpd 19175, we don't assume there is an expression for the inverse of 𝑥. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised 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 19175*	Deduce a group from its properties. Unlike isgrpd2 19173, 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 19176*	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 19177	A group is a semigroup. (Contributed by AV, 28-Aug-2021.)
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
Theorem	grpmgmd 19178	A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mgm)
Theorem	dfgrp2 19179*	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 19153, 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 19180*	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 19181*	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 19182	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 19183	The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
Theorem	grplid 19184	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 19185	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 19186	The identity element of a group is a left identity. Deduction associated with grplid 19184. (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 19187	The identity element of a group is a right identity. Deduction associated with grprid 19185. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 0 ) = 𝑋)
Theorem	grpn0 19188	A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
⊢ (𝐺 ∈ Grp → 𝐺 ≠ ∅)
Theorem	hashfingrpnn 19189	A finite group has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
Theorem	grprcan 19190	Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
Theorem	grpinveu 19191*	The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
Theorem	grpid 19192	Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋))
Theorem	isgrpid2 19193	Properties showing that an element 𝑍 is the identity element of a group. (Contributed by NM, 7-Aug-2013.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))
Theorem	grpidd2 19194*	Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 19175. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → 0 = (0g‘𝐺))
Theorem	grpinvfval 19195*	The inverse function of a group. For a shorter proof using ax-rep 5231, see grpinvfvalALT 19196. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5231. (Revised by Rohan Ridenour, 13-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
Theorem	grpinvfvalALT 19196*	Shorter proof of grpinvfval 19195 using ax-rep 5231. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
Theorem	grpinvval 19197*	The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
Theorem	grpinvfn 19198	Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ 𝑁 Fn 𝐵
Theorem	grpinvfvi 19199	The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)
⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ 𝑁 = (invg‘( I ‘𝐺))
Theorem	grpsubfval 19200*	Group subtraction (division) operation. For a shorter proof using ax-rep 5231, see grpsubfvalALT 19201. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5231. (Revised by Rohan Ridenour, 17-Aug-2023.) (Proof shortened by AV, 19-Feb-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))