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Type | Label | Description |
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Statement | ||
Theorem | xpsgrpsub 19101 | Value of the subtraction operation in a binary structure product of groups. (Contributed by AV, 24-Mar-2025.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ Grp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ · = (-g‘𝑅) & ⊢ × = (-g‘𝑆) & ⊢ − = (-g‘𝑇) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 − 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉) | ||
Theorem | mhmlem 19102* | Lemma for mhmmnd 19104 and ghmgrp 19106. (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 19103* | 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 19104* | 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 19105* | The function fulfilling the conditions of mhmmnd 19104 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 19106* | 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 19108, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires Definition df-minusg 18977 of the inverse operation invg. | ||
Syntax | cmg 19107 | Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group. |
class .g | ||
Definition | df-mulg 19108* | 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 19109* | Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5303, see mulgfvalALT 19110. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5303. (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 19110* | Shorter proof of mulgfval 19109 using ax-rep 5303. (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 19111 | 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 19112 | 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 19113 | The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ · = (.g‘𝐺) ⇒ ⊢ · = (.g‘( I ‘𝐺)) | ||
Theorem | mulg0 19114 | 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 19115 | 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 | ressmulgnn 19116 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝐴 ⊆ (Base‘𝐺) & ⊢ ∗ = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) | ||
Theorem | ressmulgnn0 19117 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝐴 ⊆ (Base‘𝐺) & ⊢ ∗ = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ (0g‘𝐺) = (0g‘𝐻) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) | ||
Theorem | ressmulgnnd 19118 | Values for the group multiple function in a restricted structure, a deduction version. (Contributed by metakunt, 14-May-2025.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋)) | ||
Theorem | mulgnngsum 19119* | 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 19120* | 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 19121 | 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 19122 | 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 19123 | 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 19124 | 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 19125 | 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 19126* | Closure of the group multiple (exponentiation) operation in a submagma. (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 19127* | 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 19128* | 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 19129 | 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 19130 | 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 19131 | 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 19132 | 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 19133 | 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 19134 | 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 | mulgnn0cld 19135 | Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 19130. (Contributed by SN, 1-Feb-2025.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgcld 19136 | Deduction associated with mulgcl 19131. (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 19137 | Lemma for mulgaddcom 19138. (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 19138 | 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 19139 | 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 19140 | 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 19141 | 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 19142 | 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 19143 | 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 19144 | 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 19145 | Lemma for mulgdir 19146. (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 19146 | 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 19147 | 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 19148 | 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 19149 | 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 19150 | 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 19151 | 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 19152 | 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 19153 | 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 19154 | Distribution of group multiples over subtraction for group elements, subdir 11724 analog. (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 19155 | 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 19156* | 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 19157 | 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 19158 | 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 19159 | 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 19160 | Extend class notation with all subgroups of a group. |
class SubGrp | ||
Syntax | cnsg 19161 | Extend class notation with all normal subgroups of a group. |
class NrmSGrp | ||
Syntax | cqg 19162 | Quotient group equivalence class. |
class ~QG | ||
Definition | df-subg 19163* | Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19181), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19176), contains the neutral element of the group (see subg0 19172) and contains the inverses for all of its elements (see subginvcl 19175). (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) | ||
Definition | df-nsg 19164* | 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 19165* | 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 19166 | 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 19167 | A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵) | ||
Theorem | subgid 19168 | A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) | ||
Theorem | subggrp 19169 | A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) | ||
Theorem | subgbas 19170 | 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 19171 | Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | ||
Theorem | subg0 19172 | 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 19173 | 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 19174 | 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 19175 | 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 19176 | 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 19177 | 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 19178 | 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 19179 | 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 19180 | 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 19181* | 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 19182* | 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 19183* | 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 19184* | 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 19185* | 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 19186 | 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 19187 | 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 19188 | A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺)) | ||
Theorem | subsubg 19189 | 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 19190 | 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 19191 | The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) | ||
Theorem | 0subgOLD 19192 | Obsolete version of 0subg 19191 as of 31-Jan-2025. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) | ||
Theorem | trivsubgd 19193 | 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 19194 | 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}) | ||
Theorem | isnsg 19195* | Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))) | ||
Theorem | isnsg2 19196* | Weaken the condition of isnsg 19195 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))) | ||
Theorem | nsgbi 19197 | Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)) | ||
Theorem | nsgsubg 19198 | A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | ||
Theorem | nsgconj 19199 | The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐵) − 𝐴) ∈ 𝑆) | ||
Theorem | isnsg3 19200* | A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)) |
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