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Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgrpsubpropd 17501 Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
(𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   (𝜑 → (+g𝐺) = (+g𝐻))       (𝜑 → (-g𝐺) = (-g𝐻))

Theoremgrpsubpropd2 17502* Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐵 = (Base‘𝐻))    &   (𝜑𝐺 ∈ Grp)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))       (𝜑 → (-g𝐺) = (-g𝐻))

Theoremgrp1 17503 The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Grp)

Theoremgrp1inv 17504 The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Revised by AV, 26-Aug-2021.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉 → (invg𝑀) = ( I ↾ {𝐼}))

Theoremprdsinvlem 17505* Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Grp)    &   (𝜑𝐹𝐵)    &    0 = (0g𝑅)    &   𝑁 = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝐹𝑦)))       (𝜑 → (𝑁𝐵 ∧ (𝑁 + 𝐹) = 0 ))

Theoremprdsgrpd 17506 The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)       (𝜑𝑌 ∈ Grp)

Theoremprdsinvgd 17507* Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)    &   𝐵 = (Base‘𝑌)    &   𝑁 = (invg𝑌)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = (𝑥𝐼 ↦ ((invg‘(𝑅𝑥))‘(𝑋𝑥))))

Theorempwsgrp 17508 The product of a family of groups is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Grp ∧ 𝐼𝑉) → 𝑌 ∈ Grp)

Theorempwsinvg 17509 Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (invg𝑅)    &   𝑁 = (invg𝑌)       ((𝑅 ∈ Grp ∧ 𝐼𝑉𝑋𝐵) → (𝑁𝑋) = (𝑀𝑋))

Theorempwssub 17510 Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (-g𝑅)    &    = (-g𝑌)       (((𝑅 ∈ Grp ∧ 𝐼𝑉) ∧ (𝐹𝐵𝐺𝐵)) → (𝐹 𝐺) = (𝐹𝑓 𝑀𝐺))

Theoremimasgrp2 17511* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅𝑊)    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))    &   (𝜑0𝑉)    &   ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))    &   ((𝜑𝑥𝑉) → 𝑁𝑉)    &   ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))       (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))

Theoremimasgrp 17512* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅 ∈ Grp)    &    0 = (0g𝑅)       (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))

Theoremimasgrpf1 17513 The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑈 = (𝐹s 𝑅)    &   𝑉 = (Base‘𝑅)       ((𝐹:𝑉1-1𝐵𝑅 ∈ Grp) → 𝑈 ∈ Grp)

Theoremqusgrp2 17514* Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑 Er 𝑉)    &   (𝜑𝑅𝑋)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 + 𝑏) (𝑝 + 𝑞)))    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) (𝑥 + (𝑦 + 𝑧)))    &   (𝜑0𝑉)    &   ((𝜑𝑥𝑉) → ( 0 + 𝑥) 𝑥)    &   ((𝜑𝑥𝑉) → 𝑁𝑉)    &   ((𝜑𝑥𝑉) → (𝑁 + 𝑥) 0 )       (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] = (0g𝑈)))

Theoremxpsgrp 17515 The binary product of groups is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)       ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp)

Theoremmhmlem 17516* Lemma for mhmmnd 17518 and ghmgrp 17520. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵)))

Theoremmhmid 17517* A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Mnd)    &    0 = (0g𝐺)       (𝜑 → (𝐹0 ) = (0g𝐻))

Theoremmhmmnd 17518* The image of a monoid 𝐺 under a monoid homomorphism 𝐹 is a monoid. (Contributed by Thierry Arnoux, 25-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Mnd)       (𝜑𝐻 ∈ Mnd)

Theoremmhmfmhm 17519* The function fulfilling the conditions of mhmmnd 17518 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Mnd)       (𝜑𝐹 ∈ (𝐺 MndHom 𝐻))

Theoremghmgrp 17520* The image of a group 𝐺 under a group homomorphism 𝐹 is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator 𝑂 in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Grp)       (𝜑𝐻 ∈ Grp)

10.2.2  Group multiple operation

The "group multiple" operation (if the group is multiplicative, also called "group power" or "group exponentiation" operation), can be defined for arbitrary magmas, if the multiplier/exponent is a nonnegative integer. See also the definition in [Lang] p. 6, where an element 𝑥(of a monoid) to the power of a nonnegative integer 𝑛 is defined and denoted by 𝑥𝑛. df-mulg 17522, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires the definition df-minusg 17407 of the inverse operation invg.

Syntaxcmg 17521 Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group.
class .g

Definitiondf-mulg 17522* Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g𝑔), seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛))))))

Theoremmulgfval 17523* Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐼 = (invg𝐺)    &    · = (.g𝐺)        · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))

Theoremmulgval 17524 Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐼 = (invg𝐺)    &    · = (.g𝐺)    &   𝑆 = seq1( + , (ℕ × {𝑋}))       ((𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))

Theoremmulgfn 17525 Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)        · Fn (ℤ × 𝐵)

Theoremmulgfvi 17526 The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
· = (.g𝐺)        · = (.g‘( I ‘𝐺))

Theoremmulg0 17527 Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)       (𝑋𝐵 → (0 · 𝑋) = 0 )

Theoremmulgnn 17528 Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    · = (.g𝐺)    &   𝑆 = seq1( + , (ℕ × {𝑋}))       ((𝑁 ∈ ℕ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = (𝑆𝑁))

Theoremmulg1 17529 Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       (𝑋𝐵 → (1 · 𝑋) = 𝑋)

Theoremmulgnnp1 17530 Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝑁 ∈ ℕ ∧ 𝑋𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))

Theoremmulg2 17531 Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       (𝑋𝐵 → (2 · 𝑋) = (𝑋 + 𝑋))

Theoremmulgnegnn 17532 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝑁 ∈ ℕ ∧ 𝑋𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋)))

Theoremmulgnn0p1 17533 Group multiple (exponentiation) operation at a successor, extended to 0. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0𝑋𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))

Theoremmulgnnsubcl 17534* Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑆𝐵)    &   ((𝜑𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)       ((𝜑𝑁 ∈ ℕ ∧ 𝑋𝑆) → (𝑁 · 𝑋) ∈ 𝑆)

Theoremmulgnn0subcl 17535* Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑆𝐵)    &   ((𝜑𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &    0 = (0g𝐺)    &   (𝜑0𝑆)       ((𝜑𝑁 ∈ ℕ0𝑋𝑆) → (𝑁 · 𝑋) ∈ 𝑆)

Theoremmulgsubcl 17536* Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑆𝐵)    &   ((𝜑𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &    0 = (0g𝐺)    &   (𝜑0𝑆)    &   𝐼 = (invg𝐺)    &   ((𝜑𝑥𝑆) → (𝐼𝑥) ∈ 𝑆)       ((𝜑𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) ∈ 𝑆)

Theoremmulgnncl 17537 Closure of the group multiple (exponentiation) operation for a positive multiplier in a magma. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵) → (𝑁 · 𝑋) ∈ 𝐵)

TheoremmulgnnclOLD 17538 Obsolete proof of mulgnncl 17537 as of 29-Aug-2021. Closure of the group multiple (exponentiation) operation. TODO: This can be generalized to a magma if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵) → (𝑁 · 𝑋) ∈ 𝐵)

Theoremmulgnn0cl 17539 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝑁 · 𝑋) ∈ 𝐵)

Theoremmulgcl 17540 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · 𝑋) ∈ 𝐵)

Theoremmulgneg 17541 Group multiple (exponentiation) operation at a negative integer. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋)))

Theoremmulgnegneg 17542 The inverse of a negative group multiple is the positive group multiple. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝐼‘(-𝑁 · 𝑋)) = (𝑁 · 𝑋))

Theoremmulgm1 17543 Group multiple (exponentiation) operation at negative one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 20-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (-1 · 𝑋) = (𝐼𝑋))

Theoremmulgaddcomlem 17544 Lemma for mulgaddcom 17545. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋𝐵) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋)))

Theoremmulgaddcom 17545 The group multiple operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋)))

Theoremmulginvcom 17546 The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · (𝐼𝑋)) = (𝐼‘(𝑁 · 𝑋)))

Theoremmulginvinv 17547 The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝐼‘(𝑁 · (𝐼𝑋))) = (𝑁 · 𝑋))

Theoremmulgnn0z 17548 A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 )

Theoremmulgz 17549 A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 · 0 ) = 0 )

Theoremmulgnndir 17550 Sum of group multiples, for positive multiples. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ SGrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))

TheoremmulgnndirOLD 17551 Obsolete proof of mulgnndir 17550 as of 29-Aug-2021. Sum of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))

Theoremmulgnn0dir 17552 Sum of group multiples, generalized to 0. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))

Theoremmulgdirlem 17553 Lemma for mulgdir 17554. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ (𝑀 + 𝑁) ∈ ℕ0) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))

Theoremmulgdir 17554 Sum of group multiples, generalized to . (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))

Theoremmulgp1 17555 Group multiple (exponentiation) operation at a successor, extended to . (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))

Theoremmulgneg2 17556 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (-𝑁 · 𝑋) = (𝑁 · (𝐼𝑋)))

Theoremmulgnnass 17557 Product of group multiples, for positive multiples in a semigroup. (Contributed by Mario Carneiro, 13-Dec-2014.) (Revised by AV, 29-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ SGrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))

TheoremmulgnnassOLD 17558 Obsolete proof of mulgnnass 17557 as of 29-Aug-2021. Product of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 13-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))

Theoremmulgnn0ass 17559 Product of group multiples, generalized to 0. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))

Theoremmulgass 17560 Product of group multiples, generalized to . (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))

Theoremmulgassr 17561 Reversed product of group multiples. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑁 · 𝑀) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))

Theoremmulgmodid 17562 Casting out multiples of the identity element leaves the group multiple unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ (𝑋𝐵 ∧ (𝑀 · 𝑋) = 0 )) → ((𝑁 mod 𝑀) · 𝑋) = (𝑁 · 𝑋))

Theoremmulgsubdir 17563 Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑀𝑁) · 𝑋) = ((𝑀 · 𝑋) (𝑁 · 𝑋)))

Theoremmhmmulg 17564 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    × = (.g𝐻)       ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))

Theoremmulgpropd 17565* Two structures with the same group-nature have the same group multiple function. 𝐾 is expected to either be V (when strong equality is available) or 𝐵 (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
· = (.g𝐺)    &    × = (.g𝐻)    &   (𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐵 = (Base‘𝐻))    &   (𝜑𝐵𝐾)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))       (𝜑· = × )

Theoremsubmmulgcl 17566 Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
= (.g𝐺)       ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0𝑋𝑆) → (𝑁 𝑋) ∈ 𝑆)

Theoremsubmmulg 17567 A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
= (.g𝐺)    &   𝐻 = (𝐺s 𝑆)    &    · = (.g𝐻)       ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0𝑋𝑆) → (𝑁 𝑋) = (𝑁 · 𝑋))

Theorempwsmulg 17568 Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &    = (.g𝑌)    &    · = (.g𝑅)       (((𝑅 ∈ Mnd ∧ 𝐼𝑉) ∧ (𝑁 ∈ ℕ0𝑋𝐵𝐴𝐼)) → ((𝑁 𝑋)‘𝐴) = (𝑁 · (𝑋𝐴)))

10.2.3  Subgroups and Quotient groups

Syntaxcsubg 17569 Extend class notation with all subgroups of a group.
class SubGrp

Syntaxcnsg 17570 Extend class notation with all normal subgroups of a group.
class NrmSGrp

Syntaxcqg 17571 Quotient group equivalence class.
class ~QG

Definitiondf-subg 17572* Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17590), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17585), contains the neutral element of the group (see subg0 17581) and contains the inverses for all of its elements (see subginvcl 17584). (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})

Definitiondf-nsg 17573* Define the equivalence relation in a quotient ring or quotient group (where 𝑖 is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g𝑤) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})

Definitiondf-eqg 17574* Define the equivalence relation in a quotient ring or quotient group (where 𝑖 is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)})

Theoremissubg 17575 The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)       (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))

Theoremsubgss 17576 A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)       (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝐵)

Theoremsubgid 17577 A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))

Theoremsubggrp 17578 A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)

Theoremsubgbas 17579 The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))

Theoremsubgrcl 17580 Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
(𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)

Theoremsubg0 17581 A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐻 = (𝐺s 𝑆)    &    0 = (0g𝐺)       (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g𝐻))

Theoremsubginv 17582 The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐻 = (𝐺s 𝑆)    &   𝐼 = (invg𝐺)    &   𝐽 = (invg𝐻)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) = (𝐽𝑋))

Theoremsubg0cl 17583 The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
0 = (0g𝐺)       (𝑆 ∈ (SubGrp‘𝐺) → 0𝑆)

Theoremsubginvcl 17584 The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐼 = (invg𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) ∈ 𝑆)

Theoremsubgcl 17585 A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
+ = (+g𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Theoremsubgsubcl 17586 A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
= (-g𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 𝑌) ∈ 𝑆)

Theoremsubgsub 17587 The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
= (-g𝐺)    &   𝐻 = (𝐺s 𝑆)    &   𝑁 = (-g𝐻)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 𝑌) = (𝑋𝑁𝑌))

Theoremsubgmulgcl 17588 Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
· = (.g𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) ∈ 𝑆)

Theoremsubgmulg 17589 A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
· = (.g𝐺)    &   𝐻 = (𝐺s 𝑆)    &    = (.g𝐻)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) = (𝑁 𝑋))

Theoremissubg2 17590* Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝐵𝑆 ≠ ∅ ∧ ∀𝑥𝑆 (∀𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼𝑥) ∈ 𝑆))))

Theoremissubgrpd2 17591* Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑆 = (𝐼s 𝐷))    &   (𝜑0 = (0g𝐼))    &   (𝜑+ = (+g𝐼))    &   (𝜑𝐷 ⊆ (Base‘𝐼))    &   (𝜑0𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ((𝜑𝑥𝐷) → ((invg𝐼)‘𝑥) ∈ 𝐷)    &   (𝜑𝐼 ∈ Grp)       (𝜑𝐷 ∈ (SubGrp‘𝐼))

Theoremissubgrpd 17592* Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑆 = (𝐼s 𝐷))    &   (𝜑0 = (0g𝐼))    &   (𝜑+ = (+g𝐼))    &   (𝜑𝐷 ⊆ (Base‘𝐼))    &   (𝜑0𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ((𝜑𝑥𝐷) → ((invg𝐼)‘𝑥) ∈ 𝐷)    &   (𝜑𝐼 ∈ Grp)       (𝜑𝑆 ∈ Grp)

Theoremissubg3 17593* A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐼 = (invg𝐺)       (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ∈ (SubMnd‘𝐺) ∧ ∀𝑥𝑆 (𝐼𝑥) ∈ 𝑆)))

Theoremissubg4 17594* A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝐵𝑆 ≠ ∅ ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 𝑦) ∈ 𝑆)))

Theoremgrpissubg 17595 If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the (base set of the) group is subgroup of the other group. (Contributed by AV, 14-Mar-2019.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)       ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubGrp‘𝐺)))

Theoremresgrpisgrp 17596 If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the other group restricted to the base set of the group is a group. (Contributed by AV, 14-Mar-2019.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)       ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → (𝐺s 𝑆) ∈ Grp))

Theoremsubgsubm 17597 A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))

Theoremsubsubg 17598 A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)))

Theoremsubgint 17599 The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
((𝑆 ⊆ (SubGrp‘𝐺) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubGrp‘𝐺))

Theorem0subg 17600 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.)
0 = (0g𝐺)       (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))

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