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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremefgcpbl2 17901* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 𝑋𝐵 𝑌) → (𝐴 ++ 𝐵) (𝑋 ++ 𝑌))

Theoremfrgpval 17902 Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)    &   𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &    = ( ~FG𝐼)       (𝐼𝑉𝐺 = (𝑀 /s ))

Theoremfrgpcpbl 17903 Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &    + = (+g𝑀)       ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷))

Theoremfrgp0 17904 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)       (𝐼𝑉 → (𝐺 ∈ Grp ∧ [∅] = (0g𝐺)))

Theoremfrgpeccl 17905 Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐵 = (Base‘𝐺)       (𝑋𝑊 → [𝑋] 𝐵)

Theoremfrgpgrp 17906 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)       (𝐼𝑉𝐺 ∈ Grp)

Theoremfrgpadd 17907 Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &    + = (+g𝐺)       ((𝐴𝑊𝐵𝑊) → ([𝐴] + [𝐵] ) = [(𝐴 ++ 𝐵)] )

Theoremfrgpinv 17908* The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝑁 = (invg𝐺)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       (𝐴𝑊 → (𝑁‘[𝐴] ) = [(𝑀 ∘ (reverse‘𝐴))] )

Theoremfrgpmhm 17909* The "natural map" from words of the free monoid to their cosets in the free group is a surjective monoid homomorphism. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &   𝑊 = (Base‘𝑀)    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝐹 = (𝑥𝑊 ↦ [𝑥] )       (𝐼𝑉𝐹 ∈ (𝑀 MndHom 𝐺))

Theoremvrgpfval 17910* The canonical injection from the generating set 𝐼 to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)       (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))

Theoremvrgpval 17911 The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)       ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )

Theoremvrgpf 17912 The mapping from the index set to the generators is a function into the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)       (𝐼𝑉𝑈:𝐼𝑋)

Theoremvrgpinv 17913 The inverse of a generating element is represented by 𝐴, 1⟩ instead of 𝐴, 0⟩. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑁 = (invg𝐺)       ((𝐼𝑉𝐴𝐼) → (𝑁‘(𝑈𝐴)) = [⟨“⟨𝐴, 1𝑜⟩”⟩] )

Theoremfrgpuptf 17914* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)       (𝜑𝑇:(𝐼 × 2𝑜)⟶𝐵)

Theoremfrgpuptinv 17915* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       ((𝜑𝐴 ∈ (𝐼 × 2𝑜)) → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴)))

Theoremfrgpuplem 17916* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)       ((𝜑𝐴 𝐶) → (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))

Theoremfrgpupf 17917* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       (𝜑𝐸:𝑋𝐵)

Theoremfrgpupval 17918* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       ((𝜑𝐴𝑊) → (𝐸‘[𝐴] ) = (𝐻 Σg (𝑇𝐴)))

Theoremfrgpup1 17919* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       (𝜑𝐸 ∈ (𝐺 GrpHom 𝐻))

Theoremfrgpup2 17920* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐴𝐼)       (𝜑 → (𝐸‘(𝑈𝐴)) = (𝐹𝐴))

Theoremfrgpup3lem 17921* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐾 ∈ (𝐺 GrpHom 𝐻))    &   (𝜑 → (𝐾𝑈) = 𝐹)       (𝜑𝐾 = 𝐸)

Theoremfrgpup3 17922* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝐵 = (Base‘𝐻)    &   𝑈 = (varFGrp𝐼)       ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚𝑈) = 𝐹)

Theorem0frgp 17923 The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘∅)    &   𝐵 = (Base‘𝐺)       𝐵 ≈ 1𝑜

10.3  Abelian groups

10.3.1  Definition and basic properties

Syntaxccmn 17924 Extend class notation with class of all commutative monoids.
class CMnd

Syntaxcabl 17925 Extend class notation with class of all Abelian groups.
class Abel

Definitiondf-cmn 17926* Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g𝑔)𝑏) = (𝑏(+g𝑔)𝑎)}

Definitiondf-abl 17927 Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Abel = (Grp ∩ CMnd)

Theoremisabl 17928 The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
(𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Theoremablgrp 17929 An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
(𝐺 ∈ Abel → 𝐺 ∈ Grp)

Theoremablcmn 17930 An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Theoremiscmn 17931* The predicate "is a commutative monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Theoremisabl2 17932* The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Theoremcmnpropd 17933* 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))

Theoremablpropd 17934* 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))

Theoremablprop 17935 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)    &   (+g𝐾) = (+g𝐿)       (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Theoremiscmnd 17936* Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑𝐺 ∈ Mnd)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       (𝜑𝐺 ∈ CMnd)

Theoremisabld 17937* Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑𝐺 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       (𝜑𝐺 ∈ Abel)

Theoremisabli 17938* Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
𝐺 ∈ Grp    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       𝐺 ∈ Abel

Theoremcmnmnd 17939 A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝐺 ∈ CMnd → 𝐺 ∈ Mnd)

Theoremcmncom 17940 A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremablcom 17941 An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremcmn32 17942 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Theoremcmn4 17943 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Theoremcmn12 17944 Commutative/associative law for Abelian monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

Theoremabl32 17945 Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Theoremablinvadd 17946 The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑋) + (𝑁𝑌)))

Theoremablsub2inv 17947 Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))

Theoremablsubadd 17948 Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋))

Theoremablsub4 17949 Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 𝑍) + (𝑌 𝑊)))

𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 𝑍) = (𝑊 𝑌)))

Theoremabladdsub 17951 Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 𝑍) + 𝑌))

Theoremablpncan2 17952 Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑋) = 𝑌)

Theoremablpncan3 17953 A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = 𝑌)

Theoremablsubsub 17954 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) + 𝑍))

Theoremablsubsub4 17955 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))

Theoremablpnpcan 17956 Cancellation law for mixed addition and subtraction. (pnpcan 10071 analog.) (Contributed by NM, 29-May-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))

Theoremablnncan 17957 Cancellation law for group subtraction. (nncan 10061 analog.) (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 (𝑋 𝑌)) = 𝑌)

Theoremablsub32 17958 Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Theoremablnnncan 17959 Cancellation law for group subtraction. (nnncan 10067 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 𝑌))

Theoremablnnncan1 17960 Cancellation law for group subtraction. (nnncan1 10068 analog.) (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑍 𝑌))

Theoremmulgnn0di 17961 Group multiple of a sum, for nonnegative multiples. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑀 ∈ ℕ0𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌)))

Theoremmulgdi 17962 Group multiple of a sum. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌)))

Theoremmulgmhm 17963* 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)    &    · = (.g𝐺)       ((𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0) → (𝑥𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 MndHom 𝐺))

Theoremmulgghm 17964* 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)    &    · = (.g𝐺)       ((𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ) → (𝑥𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 GrpHom 𝐺))

Theoremmulgsubdi 17965 Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 𝑌)) = ((𝑀 · 𝑋) (𝑀 · 𝑌)))

Theoremghmfghm 17966* The function fulfilling the conditions of ghmgrp 17254 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Grp)       (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))

Theoremghmcmn 17967* 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)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ CMnd)       (𝜑𝐻 ∈ CMnd)

Theoremghmabl 17968* 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)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Abel)       (𝜑𝐻 ∈ Abel)

Theoreminvghm 17969 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 = (invg𝐺)       (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))

Theoremeqgabl 17970 Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    = (𝐺 ~QG 𝑆)       ((𝐺 ∈ Abel ∧ 𝑆𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐵 𝐴) ∈ 𝑆)))

Theoremsubgabl 17971 A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Theoremsubcmn 17972 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd)

Theoremsubmcmn 17973 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ CMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ CMnd)

Theoremsubmcmn2 17974 A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐻 = (𝐺s 𝑆)    &   𝑍 = (Cntz‘𝐺)       (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ 𝑆 ⊆ (𝑍𝑆)))

Theoremcntzcmn 17975 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 ∧ 𝑆𝐵) → (𝑍𝑆) = 𝐵)

Theoremcntzcmnss 17976 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 ∧ 𝑆𝐵) → 𝑆 ⊆ (𝑍𝑆))

Theoremcntzspan 17977 If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   𝐾 = (mrCls‘(SubMnd‘𝐺))    &   𝐻 = (𝐺s (𝐾𝑆))       ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍𝑆)) → 𝐻 ∈ CMnd)

Theoremcntzcmnf 17978 Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Theoremghmplusg 17979 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
+ = (+g𝑁)       ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝑓 + 𝐺) ∈ (𝑀 GrpHom 𝑁))

Theoremablnsg 17980 Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))

Theoremodadd1 17981 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)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂𝐴) · (𝑂𝐵)))

Theoremodadd2 17982 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)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))

Theoremodadd 17983 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)    &    + = (+g𝐺)       (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂𝐴) · (𝑂𝐵)))

Theoremgex2abl 17984 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)

Theoremgexexlem 17985* Lemma for gexex 17986. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐸 ∈ ℕ)    &   (𝜑𝐴𝑋)    &   ((𝜑𝑦𝑋) → (𝑂𝑦) ≤ (𝑂𝐴))       (𝜑 → (𝑂𝐴) = 𝐸)

Theoremgexex 17986* 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) → ∃𝑥𝑋 (𝑂𝑥) = 𝐸)

Theoremtorsubg 17987 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))

Theoremoddvdssubg 17988* 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) → {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁} ∈ (SubGrp‘𝐺))

Theoremlsmcomx 17989 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 ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))

Theoremablcntzd 17990 All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))       (𝜑𝑇 ⊆ (𝑍𝑈))

Theoremlsmcom 17991 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))

Theoremlsmsubg2 17992 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))

Theoremlsm4 17993 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)))

Theoremprdscmnd 17994 The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶CMnd)       (𝜑𝑌 ∈ CMnd)

Theoremprdsabld 17995 The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Abel)       (𝜑𝑌 ∈ Abel)

Theorempwscmn 17996 The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ CMnd ∧ 𝐼𝑉) → 𝑌 ∈ CMnd)

Theorempwsabl 17997 The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Abel ∧ 𝐼𝑉) → 𝑌 ∈ Abel)

Theoremqusabl 17998 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)

Theoremabl1 17999 The (smallest) structure representing a trivial abelian group. (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Abel)

Theoremabln0 18000 Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.)
Abel ≠ ∅

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