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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | odhash 18701 | 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 ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (♯‘(𝐾‘{𝐴})) = +∞) | ||
Theorem | odhash2 18702 | 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 ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(𝐾‘{𝐴})) = (𝑂‘𝐴)) | ||
Theorem | odhash3 18703 | 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 ∧ 𝐴 ∈ 𝑋 ∧ (𝐾‘{𝐴}) ∈ Fin) → (𝑂‘𝐴) = (♯‘(𝐾‘{𝐴}))) | ||
Theorem | odngen 18704* | A cyclic subgroup of size (𝑂‘𝐴) has (ϕ‘(𝑂‘𝐴)) generators. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘{𝑥 ∈ (𝐾‘{𝐴}) ∣ (𝑂‘𝑥) = (𝑂‘𝐴)}) = (ϕ‘(𝑂‘𝐴))) | ||
Theorem | gexval 18705* | Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) (Revised by AV, 26-Sep-2020.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) | ||
Theorem | gexlem1 18706* | 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) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ⇒ ⊢ (𝐺 ∈ 𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) | ||
Theorem | gexcl 18707 | The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐸 ∈ ℕ0) | ||
Theorem | gexid 18708 | Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝐸 · 𝐴) = 0 ) | ||
Theorem | gexlem2 18709* | 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) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ) → 𝐸 ∈ (1...𝑁)) | ||
Theorem | gexdvdsi 18710 | 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) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∥ 𝑁) → (𝑁 · 𝐴) = 0 ) | ||
Theorem | gexdvds 18711* | 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) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 )) | ||
Theorem | gexdvds2 18712* | 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) → (𝐸 ∥ 𝑁 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ 𝑁)) | ||
Theorem | gexod 18713 | 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 ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ 𝐸) | ||
Theorem | gexcl3 18714* | 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 ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐸 ∈ ℕ) | ||
Theorem | gexnnod 18715 | 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 ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ) | ||
Theorem | gexcl2 18716 | 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) | ||
Theorem | gexdvds3 18717 | 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)) | ||
Theorem | gex1 18718 | 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)) | ||
Theorem | ispgp 18719* | 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 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛))) | ||
Theorem | pgpprm 18720 | Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ) | ||
Theorem | pgpgrp 18721 | Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp) | ||
Theorem | pgpfi1 18722 | 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) = (𝑃↑𝑁) → 𝑃 pGrp 𝐺)) | ||
Theorem | pgp0 18723 | The identity subgroup is a 𝑃-group for every prime 𝑃. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 pGrp (𝐺 ↾s { 0 })) | ||
Theorem | subgpgp 18724 | A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑆)) | ||
Theorem | sylow1lem1 18725* | Lemma for sylow1 18730. The p-adic valuation of the size of 𝑆 is equal to the number of excess powers of 𝑃 in (♯‘𝑋) / (𝑃↑𝑁). (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ⇒ ⊢ (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) | ||
Theorem | sylow1lem2 18726* | Lemma for sylow1 18730. The function ⊕ is a group action on 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ⇒ ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆)) | ||
Theorem | sylow1lem3 18727* | Lemma for sylow1 18730. 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ (𝜑 → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) | ||
Theorem | sylow1lem4 18728* | Lemma for sylow1 18730. The stabilizer subgroup of any element of 𝑆 is at most 𝑃↑𝑁 in size. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵} ⇒ ⊢ (𝜑 → (♯‘𝐻) ≤ (𝑃↑𝑁)) | ||
Theorem | sylow1lem5 18729* | Lemma for sylow1 18730. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly 𝑃↑𝑁. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵} & ⊢ (𝜑 → (𝑃 pCnt (♯‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ⇒ ⊢ (𝜑 → ∃ℎ ∈ (SubGrp‘𝐺)(♯‘ℎ) = (𝑃↑𝑁)) | ||
Theorem | sylow1 18730* | Sylow's first theorem. If 𝑃↑𝑁 is a prime power that divides the cardinality of 𝐺, then 𝐺 has a supgroup with size 𝑃↑𝑁. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁)) | ||
Theorem | odcau 18731* | 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)) → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃) | ||
Theorem | pgpfi 18732* | The converse to pgpfi1 18722. 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) = (𝑃↑𝑛)))) | ||
Theorem | pgpfi2 18733 | Alternate version of pgpfi 18732. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))) | ||
Theorem | pgphash 18734 | The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) | ||
Theorem | isslw 18735* | 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)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))) | ||
Theorem | slwprm 18736 | 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) | ||
Theorem | slwsubg 18737 | A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) | ||
Theorem | slwispgp 18738 | Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑆 = (𝐺 ↾s 𝐾) ⇒ ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)) | ||
Theorem | slwpss 18739 | A proper superset of a Sylow subgroup is not a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑆 = (𝐺 ↾s 𝐾) ⇒ ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ¬ 𝑃 pGrp 𝑆) | ||
Theorem | slwpgp 18740 | A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑆 = (𝐺 ↾s 𝐻) ⇒ ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆) | ||
Theorem | pgpssslw 18741* | Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑆 = (𝐺 ↾s 𝐻) & ⊢ 𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ↦ (♯‘𝑥)) ⇒ ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻 ⊆ 𝑘) | ||
Theorem | slwn0 18742 | 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) ≠ ∅) | ||
Theorem | subgslw 18743 | 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 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻)) | ||
Theorem | sylow2alem1 18744* | Lemma for sylow2a 18746. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌)) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ = {𝐴}) | ||
Theorem | sylow2alem2 18745* | Lemma for sylow2a 18746. 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)) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) | ||
Theorem | sylow2a 18746* | 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)) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍))) | ||
Theorem | sylow2blem1 18747* | Lemma for sylow2b 18750. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝐾) & ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = [(𝐵 + 𝐶)] ∼ ) | ||
Theorem | sylow2blem2 18748* | Lemma for sylow2b 18750. 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) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝐾) & ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ⇒ ⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ ))) | ||
Theorem | sylow2blem3 18749* | Sylow's second theorem. Putting together the results of sylow2a 18746 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 𝑔 ∈ 𝑋 with ℎ𝑔𝐾 = 𝑔𝐾 for all ℎ ∈ 𝐻. This implies that invg(𝑔)ℎ𝑔 ∈ 𝐾, so ℎ is in the conjugated subgroup 𝑔𝐾invg(𝑔). (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝐾) & ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) & ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻)) & ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | ||
Theorem | sylow2b 18750* | Sylow's second theorem. Any 𝑃-group 𝐻 is a subgroup of a conjugated 𝑃-group 𝐾 of order 𝑃↑𝑛 ∥ (♯‘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) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻)) & ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | ||
Theorem | slwhash 18751 | 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) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺)) ⇒ ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) | ||
Theorem | fislw 18752 | 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)))))) | ||
Theorem | sylow2 18753* | Sylow's second theorem. See also sylow2b 18750 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 18752). This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | ||
Theorem | sylow3lem1 18754* | Lemma for sylow3 18760, first part. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ⇒ ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺))) | ||
Theorem | sylow3lem2 18755* | Lemma for sylow3 18760, 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾} & ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)} ⇒ ⊢ (𝜑 → 𝐻 = 𝑁) | ||
Theorem | sylow3lem3 18756* | Lemma for sylow3 18760, 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾} & ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)} ⇒ ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~QG 𝑁)))) | ||
Theorem | sylow3lem4 18757* | Lemma for sylow3 18760, 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾} & ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)} ⇒ ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋))))) | ||
Theorem | sylow3lem5 18758* | Lemma for sylow3 18760, second part. Reduce the group action of sylow3lem1 18754 to a given Sylow subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ⇒ ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾s 𝐾) GrpAct (𝑃 pSyl 𝐺))) | ||
Theorem | sylow3lem6 18759* | Lemma for sylow3 18760, second part. Using the lemma sylow2a 18746, 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) & ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ⇒ ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1) | ||
Theorem | sylow3 18760 | 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ 𝑁 = (♯‘(𝑃 pSyl 𝐺)) ⇒ ⊢ (𝜑 → (𝑁 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) ∧ (𝑁 mod 𝑃) = 1)) | ||
Syntax | clsm 18761 | Extend class notation with subgroup sum. |
class LSSum | ||
Syntax | cpj1 18762 | Extend class notation with left projection. |
class proj1 | ||
Definition | df-lsm 18763* | 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 (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)))) | ||
Definition | df-pj1 18764* | 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) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑤)𝑦))))) | ||
Theorem | lsmfval 18765* | 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) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))) | ||
Theorem | lsmvalx 18766* | Subspace sum value (for a group or vector space). Extended domain version of lsmval 18775. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) | ||
Theorem | lsmelvalx 18767* | Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 18776. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))) | ||
Theorem | lsmelvalix 18768 | 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) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈)) | ||
Theorem | oppglsm 18769 | The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.) (Proof shortened by AV, 2-Mar-2024.) |
⊢ 𝑂 = (oppg‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇) | ||
Theorem | lsmssv 18770 | Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) ⊆ 𝐵) | ||
Theorem | lsmless1x 18771 | 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) ⇒ ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑅 ⊆ 𝑇) → (𝑅 ⊕ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmless2x 18772 | 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) ⇒ ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈)) | ||
Theorem | lsmub1x 18773 | Subgroup sum is an upper bound of its arguments. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ⊆ 𝐵 ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmub2x 18774 | 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) ∧ 𝑈 ⊆ 𝐵) → 𝑈 ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmval 18775* | 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) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) | ||
Theorem | lsmelval 18776* | Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))) | ||
Theorem | lsmelvali 18777 | Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmelvalm 18778* | Subgroup sum membership analogue of lsmelval 18776 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ − = (-g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) | ||
Theorem | lsmelvalmi 18779 | Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ − = (-g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ 𝑇) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmsubm 18780 | 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 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺)) | ||
Theorem | lsmsubg 18781 | 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 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺)) | ||
Theorem | lsmcom2 18782 | Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) | ||
Theorem | smndlsmidm 18783 | The direct product is idempotent for submonoids. (Contributed by AV, 27-Dec-2023.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑈 ∈ (SubMnd‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈) | ||
Theorem | lsmub1 18784 | 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)) | ||
Theorem | lsmub2 18785 | 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)) | ||
Theorem | lsmunss 18786 | 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)) → (𝑇 ∪ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmless1 18787 | 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) ∧ 𝑆 ⊆ 𝑇) → (𝑆 ⊕ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmless2 18788 | 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) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) | ||
Theorem | lsmless12 18789 | 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)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → (𝑅 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) | ||
Theorem | lsmidm 18790 | Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (Proof shortened by AV, 27-Dec-2023.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈) | ||
Theorem | lsmidmOLD 18791 | Obsolete proof of lsmidm 18790 as of 13-Jan-2024. Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈) | ||
Theorem | lsmlub 18792 | The least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → ((𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈) ↔ (𝑆 ⊕ 𝑇) ⊆ 𝑈)) | ||
Theorem | lsmss1 18793 | Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → (𝑇 ⊕ 𝑈) = 𝑈) | ||
Theorem | lsmss1b 18794 | Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ 𝑈 ↔ (𝑇 ⊕ 𝑈) = 𝑈)) | ||
Theorem | lsmss2 18795 | Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑈 ⊆ 𝑇) → (𝑇 ⊕ 𝑈) = 𝑇) | ||
Theorem | lsmss2b 18796 | Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) | ||
Theorem | lsmass 18797 | Subgroup sum is associative. (Contributed by NM, 2-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑅 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → ((𝑅 ⊕ 𝑇) ⊕ 𝑈) = (𝑅 ⊕ (𝑇 ⊕ 𝑈))) | ||
Theorem | mndlsmidm 18798 | Subgroup sum is idempotent for monoids. This corresponds to the observation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → (𝐵 ⊕ 𝐵) = 𝐵) | ||
Theorem | lsm01 18799 | Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑋 ∈ (SubGrp‘𝐺) → (𝑋 ⊕ { 0 }) = 𝑋) | ||
Theorem | lsm02 18800 | Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑋 ∈ (SubGrp‘𝐺) → ({ 0 } ⊕ 𝑋) = 𝑋) |
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