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Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremodbezout 17701* If 𝑁 is coprime to the order of 𝐴, there is a modular inverse 𝑥 to cancel multiplication by 𝑁. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑁 gcd (𝑂𝐴)) = 1) → ∃𝑥 ∈ ℤ (𝑥 · (𝑁 · 𝐴)) = 𝐴)
 
Theoremod1 17702 The order of the group identity is one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
𝑂 = (od‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp → (𝑂0 ) = 1)
 
Theoremodeq1 17703 The group identity is the unique element of a group with order one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
𝑂 = (od‘𝐺)    &    0 = (0g𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑂𝐴) = 1 ↔ 𝐴 = 0 ))
 
Theoremodinv 17704 The order of the inverse of a group element. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)    &   𝐼 = (invg𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂‘(𝐼𝐴)) = (𝑂𝐴))
 
Theoremodf1 17705* The multiples of an element with infinite order form an infinite cyclic subgroup of 𝐺. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑂𝐴) = 0 ↔ 𝐹:ℤ–1-1𝑋))
 
Theoremodinf 17706* The multiples of an element with infinite order form an infinite cyclic subgroup of 𝐺. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
 
Theoremdfod2 17707* An alternative definition of the order of a group element is as the cardinality of the cyclic subgroup generated by the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (#‘ran 𝐹), 0))
 
Theoremodcl2 17708 The order of an element of a finite group is finite. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴𝑋) → (𝑂𝐴) ∈ ℕ)
 
Theoremoddvds2 17709 The order of an element of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴𝑋) → (𝑂𝐴) ∥ (#‘𝑋))
 
Theoremsubmod 17710 The order of an element is the same in a subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
𝐻 = (𝐺s 𝑌)    &   𝑂 = (od‘𝐺)    &   𝑃 = (od‘𝐻)       ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = (𝑃𝐴))
 
Theoremsubgod 17711 The order of an element is the same in a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.) (Proof shortened by Stefan O'Rear, 12-Sep-2015.)
𝐻 = (𝐺s 𝑌)    &   𝑂 = (od‘𝐺)    &   𝑃 = (od‘𝐻)       ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = (𝑃𝐴))
 
Theoremodsubdvds 17712 The order of an element of a subgroup divides the order of the subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑂 = (od‘𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴𝑆) → (𝑂𝐴) ∥ (#‘𝑆))
 
Theoremodf1o1 17713* An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)):ℤ–1-1-onto→(𝐾‘{𝐴}))
 
Theoremodf1o2 17714* An element with nonzero order has as many multiples as its order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) ∈ ℕ) → (𝑥 ∈ (0..^(𝑂𝐴)) ↦ (𝑥 · 𝐴)):(0..^(𝑂𝐴))–1-1-onto→(𝐾‘{𝐴}))
 
Theoremodhash 17715 An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (#‘(𝐾‘{𝐴})) = +∞)
 
Theoremodhash2 17716 If an element has nonzero order, it generates a subgroup with size equal to the order. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) ∈ ℕ) → (#‘(𝐾‘{𝐴})) = (𝑂𝐴))
 
Theoremodhash3 17717 An element which generates a finite subgroup has order the size of that subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝐾‘{𝐴}) ∈ Fin) → (𝑂𝐴) = (#‘(𝐾‘{𝐴})))
 
Theoremodngen 17718* A cyclic subgroup of size (𝑂𝐴) has (ϕ‘(𝑂𝐴)) generators. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) ∈ ℕ) → (#‘{𝑥 ∈ (𝐾‘{𝐴}) ∣ (𝑂𝑥) = (𝑂𝐴)}) = (ϕ‘(𝑂𝐴)))
 
Theoremgexval 17719* Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) (Revised by AV, 26-Sep-2020.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }       (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
 
Theoremgexlem1 17720* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.) (Proof shortened by AV, 26-Sep-2020.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }       (𝐺𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸𝐼))
 
TheoremgexvalOLD 17721* Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) Obsolete version of gexval 17719 as of 26-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }       (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, sup(𝐼, ℝ, < )))
 
Theoremgexlem1OLD 17722* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.) Obsolete version of gexlem1 17720 as of 26-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }       (𝐺𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸𝐼))
 
Theoremgexcl 17723 The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       (𝐺𝑉𝐸 ∈ ℕ0)
 
Theoremgexid 17724 Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (𝐴𝑋 → (𝐸 · 𝐴) = 0 )
 
Theoremgexlem2 17725* Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) (Proof shortened by AV, 26-Sep-2020.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺𝑉𝑁 ∈ ℕ ∧ ∀𝑥𝑋 (𝑁 · 𝑥) = 0 ) → 𝐸 ∈ (1...𝑁))
 
Theoremgexdvdsi 17726 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝐸𝑁) → (𝑁 · 𝐴) = 0 )
 
Theoremgexdvds 17727* The only 𝑁 that annihilate all the elements of the group are the multiples of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸𝑁 ↔ ∀𝑥𝑋 (𝑁 · 𝑥) = 0 ))
 
Theoremgexlem2OLD 17728* Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) Obsolete version of gexlem2 17725 as of 26-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺𝑉𝑁 ∈ ℕ ∧ ∀𝑥𝑋 (𝑁 · 𝑥) = 0 ) → 𝐸 ∈ (1...𝑁))
 
Theoremgexdvds2 17729* An integer divides the group exponent iff it divides all the group orders. In other words, the group exponent is the LCM of the orders of all the elements. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸𝑁 ↔ ∀𝑥𝑋 (𝑂𝑥) ∥ 𝑁))
 
Theoremgexod 17730 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) ∥ 𝐸)
 
Theoremgexcl3 17731* If the order of every group element is bounded by 𝑁, the group has finite exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ ∀𝑥𝑋 (𝑂𝑥) ∈ (1...𝑁)) → 𝐸 ∈ ℕ)
 
Theoremgexnnod 17732 Every group element has finite order if the exponent is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝐴𝑋) → (𝑂𝐴) ∈ ℕ)
 
Theoremgexcl2 17733 The exponent of a finite group is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → 𝐸 ∈ ℕ)
 
Theoremgexdvds3 17734 The exponent of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → 𝐸 ∥ (#‘𝑋))
 
Theoremgex1 17735 A group or monoid has exponent 1 iff it is trivial. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝑋 ≈ 1𝑜))
 
Theoremispgp 17736* A group is a 𝑃-group if every element has some power of 𝑃 as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
 
Theorempgpprm 17737 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝑃 pGrp 𝐺𝑃 ∈ ℙ)
 
Theorempgpgrp 17738 Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝑃 pGrp 𝐺𝐺 ∈ Grp)
 
Theorempgpfi1 17739 A finite group with order a power of a prime 𝑃 is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → ((#‘𝑋) = (𝑃𝑁) → 𝑃 pGrp 𝐺))
 
Theorempgp0 17740 The identity subgroup is a 𝑃-group for every prime 𝑃. (Contributed by Mario Carneiro, 19-Jan-2015.)
0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 pGrp (𝐺s { 0 }))
 
Theoremsubgpgp 17741 A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑆))
 
Theoremsylow1lem1 17742* Lemma for sylow1 17747. The p-adic valuation of the size of 𝑆 is equal to the number of excess powers of 𝑃 in (#‘𝑋) / (𝑃𝑁). (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}       (𝜑 → ((#‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
 
Theoremsylow1lem2 17743* Lemma for sylow1 17747. The function is a group action on 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))       (𝜑 ∈ (𝐺 GrpAct 𝑆))
 
Theoremsylow1lem3 17744* Lemma for sylow1 17747. One of the orbits of the group action has p-adic valuation less than the prime count of the set 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝜑 → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
 
Theoremsylow1lem4 17745* Lemma for sylow1 17747. The stabilizer subgroup of any element of 𝑆 is at most 𝑃𝑁 in size. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}    &   (𝜑𝐵𝑆)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐵) = 𝐵}       (𝜑 → (#‘𝐻) ≤ (𝑃𝑁))
 
Theoremsylow1lem5 17746* Lemma for sylow1 17747. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly 𝑃𝑁. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}    &   (𝜑𝐵𝑆)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐵) = 𝐵}    &   (𝜑 → (𝑃 pCnt (#‘[𝐵] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))       (𝜑 → ∃ ∈ (SubGrp‘𝐺)(#‘) = (𝑃𝑁))
 
Theoremsylow1 17747* Sylow's first theorem. If 𝑃𝑁 is a prime power that divides the cardinality of 𝐺, then 𝐺 has a supgroup with size 𝑃𝑁. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))       (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(#‘𝑔) = (𝑃𝑁))
 
Theoremodcau 17748* Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime 𝑃 contains an element of order 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ∃𝑔𝑋 (𝑂𝑔) = 𝑃)
 
Theorempgpfi 17749* The converse to pgpfi1 17739. A finite group is a 𝑃-group iff it has size some power of 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
 
Theorempgpfi2 17750 Alternate version of pgpfi 17749. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ (#‘𝑋) = (𝑃↑(𝑃 pCnt (#‘𝑋))))))
 
Theorempgphash 17751 The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝑋 = (Base‘𝐺)       ((𝑃 pGrp 𝐺𝑋 ∈ Fin) → (#‘𝑋) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
 
Theoremisslw 17752* The property of being a Sylow subgroup. A Sylow 𝑃-subgroup is a 𝑃-group which has no proper supersets that are also 𝑃-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
 
Theoremslwprm 17753 Reverse closure for the first argument of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
 
Theoremslwsubg 17754 A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
 
Theoremslwispgp 17755 Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐾)       ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
 
Theoremslwpss 17756 A proper superset of a Sylow subgroup is not a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐾)       ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → ¬ 𝑃 pGrp 𝑆)
 
Theoremslwpgp 17757 A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐻)       (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)
 
Theorempgpssslw 17758* Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑆 = (𝐺s 𝐻)    &   𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↦ (#‘𝑥))       ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻𝑘)
 
Theoremslwn0 17759 Every finite group contains a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝑃 pSyl 𝐺) ≠ ∅)
 
Theoremsubgslw 17760 A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse need not be true.) (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐻 = (𝐺s 𝑆)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))
 
Theoremsylow2alem1 17761* Lemma for sylow2a 17763. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       ((𝜑𝐴𝑍) → [𝐴] = {𝐴})
 
Theoremsylow2alem2 17762* Lemma for sylow2a 17763. All the orbits which are not for fixed points have size 𝐺 ∣ / ∣ 𝐺𝑥 (where 𝐺𝑥 is the stabilizer subgroup) and thus are powers of 𝑃. And since they are all nontrivial (because any orbit which is a singleton is a fixed point), they all divide 𝑃, and so does the sum of all of them. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(#‘𝑧))
 
Theoremsylow2a 17763* A named lemma of Sylow's second and third theorems. If 𝐺 is a finite 𝑃-group that acts on the finite set 𝑌, then the set 𝑍 of all points of 𝑌 fixed by every element of 𝐺 has cardinality equivalent to the cardinality of 𝑌, mod 𝑃. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝜑𝑃 ∥ ((#‘𝑌) − (#‘𝑍)))
 
Theoremsylow2blem1 17764* Lemma for sylow2b 17767. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))       ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 · [𝐶] ) = [(𝐵 + 𝐶)] )
 
Theoremsylow2blem2 17765* Lemma for sylow2b 17767. Left multiplication in a subgroup 𝐻 is a group action on the set of all left cosets of 𝐾. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))       (𝜑· ∈ ((𝐺s 𝐻) GrpAct (𝑋 / )))
 
Theoremsylow2blem3 17766* Sylow's second theorem. Putting together the results of sylow2a 17763 and the orbit-stabilizer theorem to show that 𝑃 does not divide the set of all fixed points under the group action, we get that there is a fixed point of the group action, so that there is some 𝑔𝑋 with 𝑔𝐾 = 𝑔𝐾 for all 𝐻. This implies that invg(𝑔)𝑔𝐾, so is in the conjugated subgroup 𝑔𝐾invg(𝑔). (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &   (𝜑𝑃 pGrp (𝐺s 𝐻))    &   (𝜑 → (#‘𝐾) = (𝑃↑(𝑃 pCnt (#‘𝑋))))    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 ⊆ ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))
 
Theoremsylow2b 17767* Sylow's second theorem. Any 𝑃-group 𝐻 is a subgroup of a conjugated 𝑃-group 𝐾 of order 𝑃𝑛 ∥ (#‘𝑋) with 𝑛 maximal. This is usually stated under the assumption that 𝐾 is a Sylow subgroup, but we use a slightly different definition, whose equivalence to this one requires this theorem. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &   (𝜑𝑃 pGrp (𝐺s 𝐻))    &   (𝜑 → (#‘𝐾) = (𝑃↑(𝑃 pCnt (#‘𝑋))))    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 ⊆ ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))
 
Theoremslwhash 17768 A sylow subgroup has cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (𝑃 pSyl 𝐺))       (𝜑 → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
 
Theoremfislw 17769 The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))))
 
Theoremsylow2 17770* Sylow's second theorem. See also sylow2b 17767 for the "hard" part of the proof. Any two Sylow 𝑃-subgroups are conjugate to one another, and hence the same size, namely 𝑃↑(𝑃 pCnt ∣ 𝑋 ∣ ) (see fislw 17769). This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (𝑃 pSyl 𝐺))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    + = (+g𝐺)    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 = ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))
 
Theoremsylow3lem1 17771* Lemma for sylow3 17777, first part. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))       (𝜑 ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)))
 
Theoremsylow3lem2 17772* Lemma for sylow3 17777, first part. The stabilizer of a given Sylow subgroup 𝐾 in the group action acting on all of 𝐺 is the normalizer NG(K). (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑𝐻 = 𝑁)
 
Theoremsylow3lem3 17773* Lemma for sylow3 17777, first part. The number of Sylow subgroups is the same as the index (number of cosets) of the normalizer of the Sylow subgroup 𝐾. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑 → (#‘(𝑃 pSyl 𝐺)) = (#‘(𝑋 / (𝐺 ~QG 𝑁))))
 
Theoremsylow3lem4 17774* Lemma for sylow3 17777, first part. The number of Sylow subgroups is a divisor of the size of 𝐺 reduced by the size of a Sylow subgroup of 𝐺. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑 → (#‘(𝑃 pSyl 𝐺)) ∥ ((#‘𝑋) / (𝑃↑(𝑃 pCnt (#‘𝑋)))))
 
Theoremsylow3lem5 17775* Lemma for sylow3 17777, second part. Reduce the group action of sylow3lem1 17771 to a given Sylow subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    = (𝑥𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))       (𝜑 ∈ ((𝐺s 𝐾) GrpAct (𝑃 pSyl 𝐺)))
 
Theoremsylow3lem6 17776* Lemma for sylow3 17777, second part. Using the lemma sylow2a 17763, show that the number of sylow subgroups is equivalent mod 𝑃 to the number of fixed points under the group action. But 𝐾 is the unique element of the set of Sylow subgroups that is fixed under the group action, so there is exactly one fixed point and so ((#‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    = (𝑥𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}       (𝜑 → ((#‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)
 
Theoremsylow3 17777 Sylow's third theorem. The number of Sylow subgroups is a divisor of 𝐺 ∣ / 𝑑, where 𝑑 is the common order of a Sylow subgroup, and is equivalent to 1 mod 𝑃. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   𝑁 = (#‘(𝑃 pSyl 𝐺))       (𝜑 → (𝑁 ∥ ((#‘𝑋) / (𝑃↑(𝑃 pCnt (#‘𝑋)))) ∧ (𝑁 mod 𝑃) = 1))
 
10.2.11  Direct products
 
Syntaxclsm 17778 Extend class notation with subgroup sum.
class LSSum
 
Syntaxcpj1 17779 Extend class notation with left projection.
class proj1
 
Definitiondf-lsm 17780* Define subgroup sum (inner direct product of subgroups). (Contributed by NM, 28-Jan-2014.)
LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))
 
Definitiondf-pj1 17781* Define the left projection function, which takes two subgroups 𝑡, 𝑢 with trivial intersection and returns a function mapping the elements of the subgroup sum 𝑡 + 𝑢 to their projections onto 𝑡. (The other projection function can be obtained by swapping the roles of 𝑡 and 𝑢.) (Contributed by Mario Carneiro, 15-Oct-2015.)
proj1 = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑤)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑤)𝑦)))))
 
Theoremlsmfval 17782* The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
 
Theoremlsmvalx 17783* Subspace sum value (for a group or vector space). Extended domain version of lsmval 17792. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
 
Theoremlsmelvalx 17784* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 17793. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
 
Theoremlsmelvalix 17785 Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
 
Theoremoppglsm 17786 The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑂 = (oppg𝐺)    &    = (LSSum‘𝐺)       (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)
 
Theoremlsmssv 17787 Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) ⊆ 𝐵)
 
Theoremlsmless1x 17788 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑅 𝑈) ⊆ (𝑇 𝑈))
 
Theoremlsmless2x 17789 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑅 𝑇) ⊆ (𝑅 𝑈))
 
Theoremlsmub1x 17790 Subgroup sum is an upper bound of its arguments. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝑇𝐵𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))
 
Theoremlsmub2x 17791 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈𝐵) → 𝑈 ⊆ (𝑇 𝑈))
 
Theoremlsmval 17792* Subgroup sum value (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
 
Theoremlsmelval 17793* Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
 
Theoremlsmelvali 17794 Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)       (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
 
Theoremlsmelvalm 17795* Subgroup sum membership analogue of lsmelval 17793 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (-g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))       (𝜑 → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 𝑧)))
 
Theoremlsmelvalmi 17796 Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (-g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋𝑇)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋 𝑌) ∈ (𝑇 𝑈))
 
Theoremlsmsubm 17797 The sum of two commuting submonoids is a submonoid. (Contributed by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
 
Theoremlsmsubg 17798 The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
 
Theoremlsmcom2 17799 Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))
 
Theoremlsmub1 17800 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))
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