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Type | Label | Description |
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Statement | ||
Theorem | grp1inv 18201 | 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〉}〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀) = ( I ↾ {𝐼})) | ||
Theorem | prdsinvlem 18202* | Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 0 = (0g ∘ 𝑅) & ⊢ 𝑁 = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝐹‘𝑦))) ⇒ ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ (𝑁 + 𝐹) = 0 )) | ||
Theorem | prdsgrpd 18203 | The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) ⇒ ⊢ (𝜑 → 𝑌 ∈ Grp) | ||
Theorem | prdsinvgd 18204* | Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑁 = (invg‘𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))) | ||
Theorem | pwsgrp 18205 | A structure power of a group is a group. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Grp) | ||
Theorem | pwsinvg 18206 | Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑀 = (invg‘𝑅) & ⊢ 𝑁 = (invg‘𝑌) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑀 ∘ 𝑋)) | ||
Theorem | pwssub 18207 | Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑀 = (-g‘𝑅) & ⊢ − = (-g‘𝑌) ⇒ ⊢ (((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉) ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝐹 − 𝐺) = (𝐹 ∘f 𝑀𝐺)) | ||
Theorem | imasgrp2 18208* | The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧)))) & ⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘ 0 )) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) | ||
Theorem | imasgrp 18209* | The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) | ||
Theorem | imasgrpf1 18210 | The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑈 = (𝐹 “s 𝑅) & ⊢ 𝑉 = (Base‘𝑅) ⇒ ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝑈 ∈ Grp) | ||
Theorem | qusgrp2 18211* | Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) ∼ (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑁 + 𝑥) ∼ 0 ) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0g‘𝑈))) | ||
Theorem | xpsgrp 18212 | 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) | ||
Theorem | mhmlem 18213* | Lemma for mhmmnd 18215 and ghmgrp 18217. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))) | ||
Theorem | mhmid 18214* | A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻)) | ||
Theorem | mhmmnd 18215* | The image of a monoid 𝐺 under a monoid homomorphism 𝐹 is a monoid. (Contributed by Thierry Arnoux, 25-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐻 ∈ Mnd) | ||
Theorem | mhmfmhm 18216* | The function fulfilling the conditions of mhmmnd 18215 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) | ||
Theorem | ghmgrp 18217* | 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.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐻 ∈ Grp) | ||
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 𝑥↑𝑛. Definition df-mulg 18219, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires the definition df-minusg 18101 of the inverse operation invg. | ||
Syntax | cmg 18218 | Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group. |
class .g | ||
Definition | df-mulg 18219* | 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, (0g‘𝑔), ⦋seq1((+g‘𝑔), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑔)‘(𝑠‘-𝑛)))))) | ||
Theorem | mulgfval 18220* | Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5182, see mulgfvalALT 18221. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5182. (Revised by Rohan Ridenour, 17-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) | ||
Theorem | mulgfvalALT 18221* | Shorter proof of mulgfval 18220 using ax-rep 5182. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) | ||
Theorem | mulgval 18222 | Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) | ||
Theorem | mulgfn 18223 | Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · Fn (ℤ × 𝐵) | ||
Theorem | mulgfvi 18224 | The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ · = (.g‘𝐺) ⇒ ⊢ · = (.g‘( I ‘𝐺)) | ||
Theorem | mulg0 18225 | Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) | ||
Theorem | mulgnn 18226 | Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝑆‘𝑁)) | ||
Theorem | mulgnngsum 18227* | Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) | ||
Theorem | mulgnn0gsum 18228* | Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) | ||
Theorem | mulg1 18229 | Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) | ||
Theorem | mulgnnp1 18230 | Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) | ||
Theorem | mulg2 18231 | Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐵 → (2 · 𝑋) = (𝑋 + 𝑋)) | ||
Theorem | mulgnegnn 18232 | Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋))) | ||
Theorem | mulgnn0p1 18233 | Group multiple (exponentiation) operation at a successor, extended to ℕ0. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) | ||
Theorem | mulgnnsubcl 18234* | Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) | ||
Theorem | mulgnn0subcl 18235* | Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 0 ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) | ||
Theorem | mulgsubcl 18236* | Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 0 ∈ 𝑆) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐼‘𝑥) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) | ||
Theorem | mulgnncl 18237 | 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) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgnn0cl 18238 | Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgcl 18239 | Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgneg 18240 | 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋))) | ||
Theorem | mulgnegneg 18241 | 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(-𝑁 · 𝑋)) = (𝑁 · 𝑋)) | ||
Theorem | mulgm1 18242 | Group multiple (exponentiation) operation at negative one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 20-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (-1 · 𝑋) = (𝐼‘𝑋)) | ||
Theorem | mulgcld 18243 | Deduction associated with mulgcl 18239. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgaddcomlem 18244 | Lemma for mulgaddcom 18245. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋))) | ||
Theorem | mulgaddcom 18245 | 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) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋))) | ||
Theorem | mulginvcom 18246 | 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋))) | ||
Theorem | mulginvinv 18247 | 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑁 · (𝐼‘𝑋))) = (𝑁 · 𝑋)) | ||
Theorem | mulgnn0z 18248 | A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) | ||
Theorem | mulgz 18249 | A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 · 0 ) = 0 ) | ||
Theorem | mulgnndir 18250 | Sum of group multiples, for positive multiples. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
Theorem | mulgnn0dir 18251 | Sum of group multiples, generalized to ℕ0. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
Theorem | mulgdirlem 18252 | Lemma for mulgdir 18253. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑀 + 𝑁) ∈ ℕ0) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
Theorem | mulgdir 18253 | Sum of group multiples, generalized to ℤ. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
Theorem | mulgp1 18254 | Group multiple (exponentiation) operation at a successor, extended to ℤ. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) | ||
Theorem | mulgneg2 18255 | Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-𝑁 · 𝑋) = (𝑁 · (𝐼‘𝑋))) | ||
Theorem | mulgnnass 18256 | 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) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
Theorem | mulgnn0ass 18257 | Product of group multiples, generalized to ℕ0. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
Theorem | mulgass 18258 | Product of group multiples, generalized to ℤ. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
Theorem | mulgassr 18259 | Reversed product of group multiples. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑁 · 𝑀) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
Theorem | mulgmodid 18260 | 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 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ (𝑋 ∈ 𝐵 ∧ (𝑀 · 𝑋) = 0 )) → ((𝑁 mod 𝑀) · 𝑋) = (𝑁 · 𝑋)) | ||
Theorem | mulgsubdir 18261 | Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 − 𝑁) · 𝑋) = ((𝑀 · 𝑋) − (𝑁 · 𝑋))) | ||
Theorem | mhmmulg 18262 | A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ × = (.g‘𝐻) ⇒ ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋))) | ||
Theorem | mulgpropd 18263* | 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.) |
⊢ · = (.g‘𝐺) & ⊢ × = (.g‘𝐻) & ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐻)) & ⊢ (𝜑 → 𝐵 ⊆ 𝐾) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐾) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) ⇒ ⊢ (𝜑 → · = × ) | ||
Theorem | submmulgcl 18264 | Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ ∙ = (.g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) ∈ 𝑆) | ||
Theorem | submmulg 18265 | A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ ∙ = (.g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ · = (.g‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) = (𝑁 · 𝑋)) | ||
Theorem | pwsmulg 18266 | Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ ∙ = (.g‘𝑌) & ⊢ · = (.g‘𝑅) ⇒ ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴))) | ||
Syntax | csubg 18267 | Extend class notation with all subgroups of a group. |
class SubGrp | ||
Syntax | cnsg 18268 | Extend class notation with all normal subgroups of a group. |
class NrmSGrp | ||
Syntax | cqg 18269 | Quotient group equivalence class. |
class ~QG | ||
Definition | df-subg 18270* | Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18288), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18283), contains the neutral element of the group (see subg0 18279) and contains the inverses for all of its elements (see subginvcl 18282). (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) | ||
Definition | df-nsg 18271* | 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][(+g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)}) | ||
Definition | df-eqg 18272* | Define the equivalence relation in a group generated by a subgroup. More precisely, if 𝐺 is a group and 𝐻 is a subgroup, then 𝐺 ~QG 𝐻 is the equivalence relation on 𝐺 associated with the left cosets of 𝐻. A typical application of this definition is the construction of the quotient group (resp. ring) of a group (resp. ring) by a normal subgroup (resp. two-sided ideal). (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)}) | ||
Theorem | issubg 18273 | The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)) | ||
Theorem | subgss 18274 | A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵) | ||
Theorem | subgid 18275 | A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) | ||
Theorem | subggrp 18276 | A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) | ||
Theorem | subgbas 18277 | The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) | ||
Theorem | subgrcl 18278 | Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | ||
Theorem | subg0 18279 | 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 = (𝐺 ↾s 𝑆) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘𝐻)) | ||
Theorem | subginv 18280 | 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 = (𝐺 ↾s 𝑆) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ 𝐽 = (invg‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋)) | ||
Theorem | subg0cl 18281 | The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆) | ||
Theorem | subginvcl 18282 | The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) ∈ 𝑆) | ||
Theorem | subgcl 18283 | A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | ||
Theorem | subgsubcl 18284 | A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) ∈ 𝑆) | ||
Theorem | subgsub 18285 | The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ − = (-g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ 𝑁 = (-g‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋𝑁𝑌)) | ||
Theorem | subgmulgcl 18286 | Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.) |
⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) | ||
Theorem | subgmulg 18287 | A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ · = (.g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ ∙ = (.g‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = (𝑁 ∙ 𝑋)) | ||
Theorem | issubg2 18288* | Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆)))) | ||
Theorem | issubgrpd2 18289* | Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) & ⊢ (𝜑 → 0 = (0g‘𝐼)) & ⊢ (𝜑 → + = (+g‘𝐼)) & ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) & ⊢ (𝜑 → 0 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) | ||
Theorem | issubgrpd 18290* | Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) & ⊢ (𝜑 → 0 = (0g‘𝐼)) & ⊢ (𝜑 → + = (+g‘𝐼)) & ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) & ⊢ (𝜑 → 0 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ∈ Grp) ⇒ ⊢ (𝜑 → 𝑆 ∈ Grp) | ||
Theorem | issubg3 18291* | A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆))) | ||
Theorem | issubg4 18292* | A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 − 𝑦) ∈ 𝑆))) | ||
Theorem | grpissubg 18293 | 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) → ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubGrp‘𝐺))) | ||
Theorem | resgrpisgrp 18294 | 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) → ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝐺 ↾s 𝑆) ∈ Grp)) | ||
Theorem | subgsubm 18295 | A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺)) | ||
Theorem | subsubg 18296 | A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆))) | ||
Theorem | subgint 18297 | 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)) | ||
Theorem | 0subg 18298 | The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) | ||
Theorem | trivsubgd 18299 | The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = { 0 }) & ⊢ (𝜑 → 𝐴 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | trivsubgsnd 18300 | The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = { 0 }) ⇒ ⊢ (𝜑 → (SubGrp‘𝐺) = {𝐵}) |
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