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Theorem List for Metamath Proof Explorer - 17801-17900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremodinv 17801 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 17802* 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 17803* 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 17804* 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 17805 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 17806 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 17807 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 17808 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 17809 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 17810* 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 17811* 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 17812 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 17813 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 17814 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 17815* 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 17816* 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 17817* 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))

Theoremgexcl 17818 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 17819 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 17820* 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 17821 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 17822* 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 ))

Theoremgexdvds2 17823* 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 17824 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 17825* 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 17826 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 17827 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 17828 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 17829 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 17830* 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 17831 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝑃 pGrp 𝐺𝑃 ∈ ℙ)

Theorempgpgrp 17832 Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝑃 pGrp 𝐺𝐺 ∈ Grp)

Theorempgpfi1 17833 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 17834 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 17835 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 17836* Lemma for sylow1 17841. 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 17837* Lemma for sylow1 17841. 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 17838* Lemma for sylow1 17841. 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 17839* Lemma for sylow1 17841. 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 17840* Lemma for sylow1 17841. 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 17841* 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 17842* 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 17843* The converse to pgpfi1 17833. 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 17844 Alternate version of pgpfi 17843. (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 17845 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 17846* 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 17847 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 17848 A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))

Theoremslwispgp 17849 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 17850 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 17851 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 17852* 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 17853 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 17854 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 17855* Lemma for sylow2a 17857. 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 17856* Lemma for sylow2a 17857. 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 17857* 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 17858* Lemma for sylow2b 17861. 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 17859* Lemma for sylow2b 17861. 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 17860* Sylow's second theorem. Putting together the results of sylow2a 17857 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 17861* 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 17862 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 17863 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 17864* Sylow's second theorem. See also sylow2b 17861 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 17863). 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 17865* Lemma for sylow3 17871, 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 17866* Lemma for sylow3 17871, 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 17867* Lemma for sylow3 17871, 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 17868* Lemma for sylow3 17871, 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 17869* Lemma for sylow3 17871, second part. Reduce the group action of sylow3lem1 17865 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 17870* Lemma for sylow3 17871, second part. Using the lemma sylow2a 17857, 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 17871 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 17872 Extend class notation with subgroup sum.
class LSSum

Syntaxcpj1 17873 Extend class notation with left projection.
class proj1

Definitiondf-lsm 17874* 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 17875* 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 17876* 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 17877* Subspace sum value (for a group or vector space). Extended domain version of lsmval 17886. (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 17878* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 17887. (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 17879 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 17880 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 17881 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 17882 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 17883 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 17884 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 17885 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 17886* 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 17887* 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 17888 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 17889* Subgroup sum membership analogue of lsmelval 17887 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 17890 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 17891 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 17892 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 17893 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 17894 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))

Theoremlsmub2 17895 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)) → 𝑈 ⊆ (𝑇 𝑈))

Theoremlsmunss 17896 Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇𝑈) ⊆ (𝑇 𝑈))

Theoremlsmless1 17897 Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → (𝑆 𝑈) ⊆ (𝑇 𝑈))

Theoremlsmless2 17898 Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → (𝑆 𝑇) ⊆ (𝑆 𝑈))

Theoremlsmless12 17899 Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → (𝑅 𝑇) ⊆ (𝑆 𝑈))

Theoremlsmidm 17900 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 𝑈) = 𝑈)

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