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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-abl 18901 | Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
⊢ Abel = (Grp ∩ CMnd) | ||
Theorem | isabl 18902 | The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) |
⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | ||
Theorem | ablgrp 18903 | An Abelian group is a group. (Contributed by NM, 26-Aug-2011.) |
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | ||
Theorem | ablgrpd 18904 | An Abelian group is a group, deduction form of ablgrp 18903. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝜑 → 𝐺 ∈ Abel) ⇒ ⊢ (𝜑 → 𝐺 ∈ Grp) | ||
Theorem | ablcmn 18905 | An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | ||
Theorem | iscmn 18906* | The predicate "is a commutative monoid." (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) | ||
Theorem | isabl2 18907* | The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) | ||
Theorem | cmnpropd 18908* | If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd)) | ||
Theorem | ablpropd 18909* | If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) | ||
Theorem | ablprop 18910 | If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.) |
⊢ (Base‘𝐾) = (Base‘𝐿) & ⊢ (+g‘𝐾) = (+g‘𝐿) ⇒ ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel) | ||
Theorem | iscmnd 18911* | Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → + = (+g‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) ⇒ ⊢ (𝜑 → 𝐺 ∈ CMnd) | ||
Theorem | isabld 18912* | Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → + = (+g‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) ⇒ ⊢ (𝜑 → 𝐺 ∈ Abel) | ||
Theorem | isabli 18913* | Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.) |
⊢ 𝐺 ∈ Grp & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | cmnmnd 18914 | A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | ||
Theorem | cmncom 18915 | A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | ablcom 18916 | An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | cmn32 18917 | Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) | ||
Theorem | cmn4 18918 | Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) | ||
Theorem | cmn12 18919 | Commutative/associative law for Abelian monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) | ||
Theorem | abl32 18920 | Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) | ||
Theorem | rinvmod 18921* | Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovmo 7377. (Contributed by AV, 31-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 ) | ||
Theorem | ablinvadd 18922 | The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑋) + (𝑁‘𝑌))) | ||
Theorem | ablsub2inv 18923 | Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋)) | ||
Theorem | ablsubadd 18924 | Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋)) | ||
Theorem | ablsub4 18925 | Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) − (𝑍 + 𝑊)) = ((𝑋 − 𝑍) + (𝑌 − 𝑊))) | ||
Theorem | abladdsub4 18926 | Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 − 𝑍) = (𝑊 − 𝑌))) | ||
Theorem | abladdsub 18927 | Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = ((𝑋 − 𝑍) + 𝑌)) | ||
Theorem | ablpncan2 18928 | Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑋) = 𝑌) | ||
Theorem | ablpncan3 18929 | A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 + (𝑌 − 𝑋)) = 𝑌) | ||
Theorem | ablsubsub 18930 | Law for double subtraction. (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − (𝑌 − 𝑍)) = ((𝑋 − 𝑌) + 𝑍)) | ||
Theorem | ablsubsub4 18931 | Law for double subtraction. (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌 + 𝑍))) | ||
Theorem | ablpnpcan 18932 | Cancellation law for mixed addition and subtraction. (pnpcan 10917 analog.) (Contributed by NM, 29-May-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍)) | ||
Theorem | ablnncan 18933 | Cancellation law for group subtraction. (nncan 10907 analog.) (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌) | ||
Theorem | ablsub32 18934 | Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌)) | ||
Theorem | ablnnncan 18935 | Cancellation law for group subtraction. (nnncan 10913 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌)) | ||
Theorem | ablnnncan1 18936 | Cancellation law for group subtraction. (nnncan1 10914 analog.) (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) | ||
Theorem | ablsubsub23 18937 | Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵)) | ||
Theorem | mulgnn0di 18938 | Group multiple of a sum, for nonnegative multiples. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ (𝑀 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌))) | ||
Theorem | mulgdi 18939 | Group multiple of a sum. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌))) | ||
Theorem | mulgmhm 18940* | The map from 𝑥 to 𝑛𝑥 for a fixed positive integer 𝑛 is a monoid homomorphism if the monoid is commutative. (Contributed by Mario Carneiro, 4-May-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 MndHom 𝐺)) | ||
Theorem | mulgghm 18941* | The map from 𝑥 to 𝑛𝑥 for a fixed integer 𝑛 is a group homomorphism if the group is commutative. (Contributed by Mario Carneiro, 4-May-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 GrpHom 𝐺)) | ||
Theorem | mulgsubdi 18942 | Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑀 · (𝑋 − 𝑌)) = ((𝑀 · 𝑋) − (𝑀 · 𝑌))) | ||
Theorem | ghmfghm 18943* | The function fulfilling the conditions of ghmgrp 18215 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | ||
Theorem | ghmcmn 18944* | The image of a commutative monoid 𝐺 under a group homomorphism 𝐹 is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ CMnd) ⇒ ⊢ (𝜑 → 𝐻 ∈ CMnd) | ||
Theorem | ghmabl 18945* | The image of an abelian group 𝐺 under a group homomorphism 𝐹 is an abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Abel) ⇒ ⊢ (𝜑 → 𝐻 ∈ Abel) | ||
Theorem | invghm 18946 | The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺)) | ||
Theorem | eqgabl 18947 | Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑆) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐵 − 𝐴) ∈ 𝑆))) | ||
Theorem | subgabl 18948 | A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel) | ||
Theorem | subcmn 18949 | A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd) | ||
Theorem | submcmn 18950 | A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ CMnd) | ||
Theorem | submcmn2 18951 | A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ 𝑆 ⊆ (𝑍‘𝑆))) | ||
Theorem | cntzcmn 18952 | The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = 𝐵) | ||
Theorem | cntzcmnss 18953 | Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ (𝑍‘𝑆)) | ||
Theorem | cntrcmnd 18954 | The center of a monoid is a commutative submonoid. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ 𝑍 = (𝑀 ↾s (Cntr‘𝑀)) ⇒ ⊢ (𝑀 ∈ Mnd → 𝑍 ∈ CMnd) | ||
Theorem | cntrabl 18955 | The center of a group is an abelian group. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ 𝑍 = (𝑀 ↾s (Cntr‘𝑀)) ⇒ ⊢ (𝑀 ∈ Grp → 𝑍 ∈ Abel) | ||
Theorem | cntzspan 18956 | If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺)) & ⊢ 𝐻 = (𝐺 ↾s (𝐾‘𝑆)) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝐻 ∈ CMnd) | ||
Theorem | cntzcmnf 18957 | Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | ||
Theorem | ghmplusg 18958 | The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ + = (+g‘𝑁) ⇒ ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁)) | ||
Theorem | ablnsg 18959 | Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺)) | ||
Theorem | odadd1 18960 | The order of a product in an abelian group divides the LCM of the orders of the factors. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵))) | ||
Theorem | odadd2 18961 | The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2))) | ||
Theorem | odadd 18962 | The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂‘𝐴) · (𝑂‘𝐵))) | ||
Theorem | gex2abl 18963 | A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) → 𝐺 ∈ Abel) | ||
Theorem | gexexlem 18964* | Lemma for gexex 18965. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐸 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ (𝑂‘𝐴)) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) | ||
Theorem | gexex 18965* | In an abelian group with finite exponent, there is an element in the group with order equal to the exponent. In other words, all orders of elements divide the largest order of an element of the group. This fails if 𝐸 = 0, for example in an infinite p-group, where there are elements of arbitrarily large orders (so 𝐸 is zero) but no elements of infinite order. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 𝑂 = (od‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸) | ||
Theorem | torsubg 18966 | The set of all elements of finite order forms a subgroup of any abelian group, called the torsion subgroup. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑂 = (od‘𝐺) ⇒ ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺)) | ||
Theorem | oddvdssubg 18967* | The set of all elements whose order divides a fixed integer is a subgroup of any abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁} ∈ (SubGrp‘𝐺)) | ||
Theorem | lsmcomx 18968 | Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) | ||
Theorem | ablcntzd 18969 | All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | ||
Theorem | lsmcom 18970 | Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) | ||
Theorem | lsmsubg2 18971 | The sum of two subgroups is a subgroup. (Contributed by NM, 4-Feb-2014.) (Proof shortened by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺)) | ||
Theorem | lsm4 18972 | Commutative/associative law for subgroup sum. (Contributed by NM, 26-Sep-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑄 ∈ (SubGrp‘𝐺) ∧ 𝑅 ∈ (SubGrp‘𝐺)) ∧ (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺))) → ((𝑄 ⊕ 𝑅) ⊕ (𝑇 ⊕ 𝑈)) = ((𝑄 ⊕ 𝑇) ⊕ (𝑅 ⊕ 𝑈))) | ||
Theorem | prdscmnd 18973 | The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶CMnd) ⇒ ⊢ (𝜑 → 𝑌 ∈ CMnd) | ||
Theorem | prdsabld 18974 | The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Abel) ⇒ ⊢ (𝜑 → 𝑌 ∈ Abel) | ||
Theorem | pwscmn 18975 | The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ CMnd) | ||
Theorem | pwsabl 18976 | The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Abel) | ||
Theorem | qusabl 18977 | If 𝑌 is a subgroup of the abelian group 𝐺, then 𝐻 = 𝐺 / 𝑌 is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel) | ||
Theorem | abl1 18978 | The (smallest) structure representing a trivial abelian group. (Contributed by AV, 28-Apr-2019.) |
⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Abel) | ||
Theorem | abln0 18979 | Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.) |
⊢ Abel ≠ ∅ | ||
Theorem | cnaddablx 18980 | The complex numbers are an Abelian group under addition. This version of cnaddabl 18981 shows the explicit structure "scaffold" we chose for the definition for Abelian groups. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use cnaddabl 18981 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.) |
⊢ 𝐺 = {〈1, ℂ〉, 〈2, + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | cnaddabl 18981 | The complex numbers are an Abelian group under addition. This version of cnaddablx 18980 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how Base and +g is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnring 20559. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.) |
⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | cnaddid 18982 | The group identity element of complex number addition is zero. See also cnfld0 20561. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.) |
⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ (0g‘𝐺) = 0 | ||
Theorem | cnaddinv 18983 | Value of the group inverse of complex number addition. See also cnfldneg 20563. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.) |
⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ (𝐴 ∈ ℂ → ((invg‘𝐺)‘𝐴) = -𝐴) | ||
Theorem | zaddablx 18984 | The integers are an Abelian group under addition. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. Use zsubrg 20590 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.) |
⊢ 𝐺 = {〈1, ℤ〉, 〈2, + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | frgpnabllem1 18985* | Lemma for frgpnabl 18987. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ + = (+g‘𝐺) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑈 = (varFGrp‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ (𝜑 → 𝐵 ∈ 𝐼) ⇒ ⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵)))) | ||
Theorem | frgpnabllem2 18986* | Lemma for frgpnabl 18987. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ + = (+g‘𝐺) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑈 = (varFGrp‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ (𝜑 → 𝐵 ∈ 𝐼) & ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ((𝑈‘𝐵) + (𝑈‘𝐴))) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | frgpnabl 18987 | The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) ⇒ ⊢ (1o ≺ 𝐼 → ¬ 𝐺 ∈ Abel) | ||
Syntax | ccyg 18988 | Cyclic group. |
class CycGrp | ||
Definition | df-cyg 18989* | Define a cyclic group, which is a group with an element 𝑥, called the generator of the group, such that all elements in the group are multiples of 𝑥. A generator is usually not unique. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)} | ||
Theorem | iscyg 18990* | Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) | ||
Theorem | iscyggen 18991* | The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ⇒ ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) | ||
Theorem | iscyggen2 18992* | The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ⇒ ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)))) | ||
Theorem | iscyg2 18993* | A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ⇒ ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅)) | ||
Theorem | cyggeninv 18994* | The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → (𝑁‘𝑋) ∈ 𝐸) | ||
Theorem | cyggenod 18995* | An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ 𝑂 = (od‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) | ||
Theorem | cyggenod2 18996* | In an infinite cyclic group, the generator must have infinite order, but this property no longer characterizes the generators. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ 𝑂 = (od‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → (𝑂‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0)) | ||
Theorem | iscyg3 18997* | Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑥))) | ||
Theorem | iscygd 18998* | Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)) ⇒ ⊢ (𝜑 → 𝐺 ∈ CycGrp) | ||
Theorem | iscygodd 18999 | Show that a group with an element the same order as the group is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑂‘𝑋) = (♯‘𝐵)) ⇒ ⊢ (𝜑 → 𝐺 ∈ CycGrp) | ||
Theorem | cycsubmcmn 19000* | The set of nonnegative integer powers of an element 𝐴 of a monoid forms a commutative monoid. (Contributed by AV, 20-Jan-2024.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) & ⊢ 𝐶 = ran 𝐹 ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s 𝐶) ∈ CMnd) |
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