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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdprdsplit 19101 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &    = (LSSum‘𝐺)    &   (𝜑𝐺dom DProd 𝑆)       (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆𝐶)) (𝐺 DProd (𝑆𝐷))))
 
Theoremdmdprdpr 19102 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))       (𝜑 → (𝐺dom DProd {⟨∅, 𝑆⟩, ⟨1o, 𝑇⟩} ↔ (𝑆 ⊆ (𝑍𝑇) ∧ (𝑆𝑇) = { 0 })))
 
Theoremdprdpr 19103 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 26-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &    = (LSSum‘𝐺)    &   (𝜑𝑆 ⊆ (𝑍𝑇))    &   (𝜑 → (𝑆𝑇) = { 0 })       (𝜑 → (𝐺 DProd {⟨∅, 𝑆⟩, ⟨1o, 𝑇⟩}) = (𝑆 𝑇))
 
Theoremdpjlem 19104 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)       (𝜑 → (𝐺 DProd (𝑆 ↾ {𝑋})) = (𝑆𝑋))
 
Theoremdpjcntz 19105 The two subgroups that appear in dpjval 19109 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
 
Theoremdpjdisj 19106 The two subgroups that appear in dpjval 19109 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &    0 = (0g𝐺)       (𝜑 → ((𝑆𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = { 0 })
 
Theoremdpjlsm 19107 The two subgroups that appear in dpjval 19109 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &    = (LSSum‘𝐺)       (𝜑 → (𝐺 DProd 𝑆) = ((𝑆𝑋) (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
 
Theoremdpjfval 19108* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   𝑄 = (proj1𝐺)       (𝜑𝑃 = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
 
Theoremdpjval 19109 Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   𝑄 = (proj1𝐺)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑃𝑋) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
 
Theoremdpjf 19110 The 𝑋-th index projection is a function from the direct product to the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑃𝑋):(𝐺 DProd 𝑆)⟶(𝑆𝑋))
 
Theoremdpjidcl 19111* The key property of projections: the sum of all the projections of 𝐴 is 𝐴. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝐴 ∈ (𝐺 DProd 𝑆))    &    0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }       (𝜑 → ((𝑥𝐼 ↦ ((𝑃𝑥)‘𝐴)) ∈ 𝑊𝐴 = (𝐺 Σg (𝑥𝐼 ↦ ((𝑃𝑥)‘𝐴)))))
 
Theoremdpjeq 19112* Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝐴 ∈ (𝐺 DProd 𝑆))    &    0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑 → (𝑥𝐼𝐶) ∈ 𝑊)       (𝜑 → (𝐴 = (𝐺 Σg (𝑥𝐼𝐶)) ↔ ∀𝑥𝐼 ((𝑃𝑥)‘𝐴) = 𝐶))
 
Theoremdpjid 19113* The key property of projections: the sum of all the projections of 𝐴 is 𝐴. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝐴 ∈ (𝐺 DProd 𝑆))       (𝜑𝐴 = (𝐺 Σg (𝑥𝐼 ↦ ((𝑃𝑥)‘𝐴))))
 
Theoremdpjlid 19114 The 𝑋-th index projection acts as the identity on elements of the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)    &   (𝜑𝐴 ∈ (𝑆𝑋))       (𝜑 → ((𝑃𝑋)‘𝐴) = 𝐴)
 
Theoremdpjrid 19115 The 𝑌-th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)    &   (𝜑𝐴 ∈ (𝑆𝑋))    &    0 = (0g𝐺)    &   (𝜑𝑌𝐼)    &   (𝜑𝑌𝑋)       (𝜑 → ((𝑃𝑌)‘𝐴) = 0 )
 
Theoremdpjghm 19116 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑃𝑋) ∈ ((𝐺s (𝐺 DProd 𝑆)) GrpHom 𝐺))
 
Theoremdpjghm2 19117 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑃𝑋) ∈ ((𝐺s (𝐺 DProd 𝑆)) GrpHom (𝐺s (𝑆𝑋))))
 
10.2.14.6  The Fundamental Theorem of Abelian Groups
 
Theoremablfacrplem 19118* Lemma for ablfacrp2 19120. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑀}    &   𝐿 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   (𝜑 → (♯‘𝐵) = (𝑀 · 𝑁))       (𝜑 → ((♯‘𝐾) gcd 𝑁) = 1)
 
Theoremablfacrp 19119* A finite abelian group whose order factors into relatively prime integers, itself "factors" into two subgroups 𝐾, 𝐿 that have trivial intersection and whose product is the whole group. Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑀}    &   𝐿 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   (𝜑 → (♯‘𝐵) = (𝑀 · 𝑁))    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)       (𝜑 → ((𝐾𝐿) = { 0 } ∧ (𝐾 𝐿) = 𝐵))
 
Theoremablfacrp2 19120* The factors 𝐾, 𝐿 of ablfacrp 19119 have the expected orders (which allows for repeated application to decompose 𝐺 into subgroups of prime-power order). Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑀}    &   𝐿 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   (𝜑 → (♯‘𝐵) = (𝑀 · 𝑁))       (𝜑 → ((♯‘𝐾) = 𝑀 ∧ (♯‘𝐿) = 𝑁))
 
Theoremablfac1lem 19121* Lemma for ablfac1b 19123. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)    &   𝑀 = (𝑃↑(𝑃 pCnt (♯‘𝐵)))    &   𝑁 = ((♯‘𝐵) / 𝑀)       ((𝜑𝑃𝐴) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀 gcd 𝑁) = 1 ∧ (♯‘𝐵) = (𝑀 · 𝑁)))
 
Theoremablfac1a 19122* The factors of ablfac1b 19123 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)       ((𝜑𝑃𝐴) → (♯‘(𝑆𝑃)) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
 
Theoremablfac1b 19123* Any abelian group is the direct product of factors of prime power order (with the exact order further matching the prime factorization of the group order). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)       (𝜑𝐺dom DProd 𝑆)
 
Theoremablfac1c 19124* The factors of ablfac1b 19123 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)    &   𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   (𝜑𝐷𝐴)       (𝜑 → (𝐺 DProd 𝑆) = 𝐵)
 
Theoremablfac1eulem 19125* Lemma for ablfac1eu 19126. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)    &   𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   (𝜑𝐷𝐴)    &   (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵))    &   (𝜑 → dom 𝑇 = 𝐴)    &   ((𝜑𝑞𝐴) → 𝐶 ∈ ℕ0)    &   ((𝜑𝑞𝐴) → (♯‘(𝑇𝑞)) = (𝑞𝐶))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ Fin)       (𝜑 → ¬ 𝑃 ∥ (♯‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃})))))
 
Theoremablfac1eu 19126* The factorization of ablfac1b 19123 is unique, in that any other factorization into prime power factors (even if the exponents are different) must be equal to 𝑆. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)    &   𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   (𝜑𝐷𝐴)    &   (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵))    &   (𝜑 → dom 𝑇 = 𝐴)    &   ((𝜑𝑞𝐴) → 𝐶 ∈ ℕ0)    &   ((𝜑𝑞𝐴) → (♯‘(𝑇𝑞)) = (𝑞𝐶))       (𝜑𝑇 = 𝑆)
 
Theorempgpfac1lem1 19127* Lemma for pgpfac1 19133. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))       ((𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊))) → ((𝑆 𝑊) (𝐾‘{𝐶})) = 𝑈)
 
Theorempgpfac1lem2 19128* Lemma for pgpfac1 19133. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))    &   (𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊)))    &    · = (.g𝐺)       (𝜑 → (𝑃 · 𝐶) ∈ (𝑆 𝑊))
 
Theorempgpfac1lem3a 19129* Lemma for pgpfac1 19133. (Contributed by Mario Carneiro, 4-Jun-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))    &   (𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊)))    &    · = (.g𝐺)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ((𝑃 · 𝐶)(+g𝐺)(𝑀 · 𝐴)) ∈ 𝑊)       (𝜑 → (𝑃𝐸𝑃𝑀))
 
Theorempgpfac1lem3 19130* Lemma for pgpfac1 19133. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))    &   (𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊)))    &    · = (.g𝐺)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ((𝑃 · 𝐶)(+g𝐺)(𝑀 · 𝐴)) ∈ 𝑊)    &   𝐷 = (𝐶(+g𝐺)((𝑀 / 𝑃) · 𝐴))       (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
 
Theorempgpfac1lem4 19131* Lemma for pgpfac1 19133. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))    &   (𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊)))    &    · = (.g𝐺)       (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
 
Theorempgpfac1lem5 19132* Lemma for pgpfac1 19133. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠𝑈𝐴𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠)))       (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
 
Theorempgpfac1 19133* Factorization of a finite abelian p-group. There is a direct product decomposition of any abelian group of prime-power order where one of the factors is cyclic and generated by an element of maximal order. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝐵))
 
Theorempgpfaclem1 19134* Lemma for pgpfac 19137. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))    &   𝐻 = (𝐺s 𝑈)    &   𝐾 = (mrCls‘(SubGrp‘𝐻))    &   𝑂 = (od‘𝐻)    &   𝐸 = (gEx‘𝐻)    &    0 = (0g𝐻)    &    = (LSSum‘𝐻)    &   (𝜑𝐸 ≠ 1)    &   (𝜑𝑋𝑈)    &   (𝜑 → (𝑂𝑋) = 𝐸)    &   (𝜑𝑊 ∈ (SubGrp‘𝐻))    &   (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })    &   (𝜑 → ((𝐾‘{𝑋}) 𝑊) = 𝑈)    &   (𝜑𝑆 ∈ Word 𝐶)    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → (𝐺 DProd 𝑆) = 𝑊)    &   𝑇 = (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
 
Theorempgpfaclem2 19135* Lemma for pgpfac 19137. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))    &   𝐻 = (𝐺s 𝑈)    &   𝐾 = (mrCls‘(SubGrp‘𝐻))    &   𝑂 = (od‘𝐻)    &   𝐸 = (gEx‘𝐻)    &    0 = (0g𝐻)    &    = (LSSum‘𝐻)    &   (𝜑𝐸 ≠ 1)    &   (𝜑𝑋𝑈)    &   (𝜑 → (𝑂𝑋) = 𝐸)    &   (𝜑𝑊 ∈ (SubGrp‘𝐻))    &   (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })    &   (𝜑 → ((𝐾‘{𝑋}) 𝑊) = 𝑈)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
 
Theorempgpfaclem3 19136* Lemma for pgpfac 19137. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
 
Theorempgpfac 19137* Full factorization of a finite abelian p-group, by iterating pgpfac1 19133. There is a direct product decomposition of any abelian group of prime-power order into cyclic subgroups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
 
Theoremablfaclem1 19138* Lemma for ablfac 19141. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   𝑂 = (od‘𝐺)    &   𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})       (𝑈 ∈ (SubGrp‘𝐺) → (𝑊𝑈) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)})
 
Theoremablfaclem2 19139* Lemma for ablfac 19141. (Contributed by Mario Carneiro, 27-Apr-2016.) (Proof shortened by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   𝑂 = (od‘𝐺)    &   𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})    &   (𝜑𝐹:𝐴⟶Word 𝐶)    &   (𝜑 → ∀𝑦𝐴 (𝐹𝑦) ∈ (𝑊‘(𝑆𝑦)))    &   𝐿 = 𝑦𝐴 ({𝑦} × dom (𝐹𝑦))    &   (𝜑𝐻:(0..^(♯‘𝐿))–1-1-onto𝐿)       (𝜑 → (𝑊𝐵) ≠ ∅)
 
Theoremablfaclem3 19140* Lemma for ablfac 19141. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   𝑂 = (od‘𝐺)    &   𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})       (𝜑 → (𝑊𝐵) ≠ ∅)
 
Theoremablfac 19141* The Fundamental Theorem of (finite) Abelian Groups. Any finite abelian group is a direct product of cyclic p-groups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
 
Theoremablfac2 19142* Choose generators for each cyclic group in ablfac 19141. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &    · = (.g𝐺)    &   𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))       (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
 
10.2.15  Simple groups
 
10.2.15.1  Definition and basic properties
 
Syntaxcsimpg 19143 Extend class notation with the class of simple groups.
class SimpGrp
 
Definitiondf-simpg 19144 Define class of all simple groups. A simple group is a group (df-grp 18046) with exactly two normal subgroups. These are always the subgroup of all elements and the subgroup containing only the identity (simpgnsgbid 19156). (Contributed by Rohan Ridenour, 3-Aug-2023.)
SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
 
Theoremissimpg 19145 The predicate "is a simple group". (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
 
Theoremissimpgd 19146 Deduce a simple group from its properties. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑 → (NrmSGrp‘𝐺) ≈ 2o)       (𝜑𝐺 ∈ SimpGrp)
 
Theoremsimpggrp 19147 A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
 
Theoremsimpggrpd 19148 A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐺 ∈ SimpGrp)       (𝜑𝐺 ∈ Grp)
 
Theoremsimpg2nsg 19149 A simple group has two normal subgroups. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)
 
Theoremtrivnsimpgd 19150 Trivial groups are not simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐵 = { 0 })       (𝜑 → ¬ 𝐺 ∈ SimpGrp)
 
Theoremsimpgntrivd 19151 Simple groups are nontrivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → ¬ 𝐵 = { 0 })
 
Theoremsimpgnideld 19152* A simple group contains a nonidentity element. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → ∃𝑥𝐵 ¬ 𝑥 = 0 )
 
Theoremsimpgnsgd 19153 The only normal subgroups of a simple group are the group itself and the trivial group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵})
 
Theoremsimpgnsgeqd 19154 A normal subgroup of a simple group is either the whole group or the trivial subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ SimpGrp)    &   (𝜑𝐴 ∈ (NrmSGrp‘𝐺))       (𝜑 → (𝐴 = { 0 } ∨ 𝐴 = 𝐵))
 
Theorem2nsgsimpgd 19155* If any normal subgroup of a nontrivial group is either the trivial subgroup or the whole group, the group is simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑 → ¬ { 0 } = 𝐵)    &   ((𝜑𝑥 ∈ (NrmSGrp‘𝐺)) → (𝑥 = { 0 } ∨ 𝑥 = 𝐵))       (𝜑𝐺 ∈ SimpGrp)
 
Theoremsimpgnsgbid 19156 A nontrivial group is simple if and only if its normal subgroups are exactly the group itself and the trivial subgroup. (Contributed by Rohan Ridenour, 4-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑 → ¬ { 0 } = 𝐵)       (𝜑 → (𝐺 ∈ SimpGrp ↔ (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}))
 
10.2.15.2  Classification of abelian simple groups
 
Theoremablsimpnosubgd 19157 A subgroup of an abelian simple group containing a nonidentity element is the whole group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑆)    &   (𝜑 → ¬ 𝐴 = 0 )       (𝜑𝑆 = 𝐵)
 
Theoremablsimpg1gend 19158* An abelian simple group is generated by any non-identity element. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)    &   (𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐴 = 0 )    &   (𝜑𝐶𝐵)       (𝜑 → ∃𝑛 ∈ ℤ 𝐶 = (𝑛 · 𝐴))
 
Theoremablsimpgcygd 19159 An abelian simple group is cyclic. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
(𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑𝐺 ∈ CycGrp)
 
Theoremablsimpgfindlem1 19160* Lemma for ablsimpgfind 19163. An element of an abelian finite simple group which doesn't square to the identity has finite order. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (((𝜑𝑥𝐵) ∧ (2 · 𝑥) ≠ 0 ) → (𝑂𝑥) ≠ 0)
 
Theoremablsimpgfindlem2 19161* Lemma for ablsimpgfind 19163. An element of an abelian finite simple group which squares to the identity has finite order. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (((𝜑𝑥𝐵) ∧ (2 · 𝑥) = 0 ) → (𝑂𝑥) ≠ 0)
 
Theoremcycsubggenodd 19162* Relationship between the order of a subgroup and the order of a generator of the subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶 = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴)))       (𝜑 → (𝑂𝐴) = if(𝐶 ∈ Fin, (♯‘𝐶), 0))
 
Theoremablsimpgfind 19163 An abelian simple group is finite. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑𝐵 ∈ Fin)
 
Theoremfincygsubgd 19164* The subgroup referenced in fincygsubgodd 19165 is a subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐻 = (𝑛 ∈ ℤ ↦ (𝑛 · (𝐶 · 𝐴)))    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶 ∈ ℕ)       (𝜑 → ran 𝐻 ∈ (SubGrp‘𝐺))
 
Theoremfincygsubgodd 19165* Calculate the order of a subgroup of a finite cyclic group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐷 = ((♯‘𝐵) / 𝐶)    &   𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴))    &   𝐻 = (𝑛 ∈ ℤ ↦ (𝑛 · (𝐶 · 𝐴)))    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐴𝐵)    &   (𝜑 → ran 𝐹 = 𝐵)    &   (𝜑𝐶 ∥ (♯‘𝐵))    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶 ∈ ℕ)       (𝜑 → (♯‘ran 𝐻) = 𝐷)
 
Theoremfincygsubgodexd 19166* A finite cyclic group has subgroups of every possible order. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CycGrp)    &   (𝜑𝐶 ∥ (♯‘𝐵))    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶 ∈ ℕ)       (𝜑 → ∃𝑥 ∈ (SubGrp‘𝐺)(♯‘𝑥) = 𝐶)
 
Theoremprmgrpsimpgd 19167 A group of prime order is simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑 → (♯‘𝐵) ∈ ℙ)       (𝜑𝐺 ∈ SimpGrp)
 
Theoremablsimpgprmd 19168 An abelian simple group has prime order. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → (♯‘𝐵) ∈ ℙ)
 
Theoremablsimpgd 19169 An abelian group is simple if and only if its order is prime. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)       (𝜑 → (𝐺 ∈ SimpGrp ↔ (♯‘𝐵) ∈ ℙ))
 
10.3  Rings
 
10.3.1  Multiplicative Group
 
Syntaxcmgp 19170 Multiplicative group.
class mulGrp
 
Definitiondf-mgp 19171 Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and unitgrp 19348 shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" (ringmgp 19234) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 10472). (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩))
 
Theoremfnmgp 19172 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp Fn V
 
Theoremmgpval 19173 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)       𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
 
Theoremmgpplusg 19174 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)        · = (+g𝑀)
 
Theoremmgplem 19175 Lemma for mgpbas 19176. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 2       (𝐸𝑅) = (𝐸𝑀)
 
Theoremmgpbas 19176 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑀)
 
Theoremmgpsca 19177 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 19299. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝑆 = (Scalar‘𝑅)       𝑆 = (Scalar‘𝑀)
 
Theoremmgptset 19178 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)       (TopSet‘𝑅) = (TopSet‘𝑀)
 
Theoremmgptopn 19179 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐽 = (TopOpen‘𝑅)       𝐽 = (TopOpen‘𝑀)
 
Theoremmgpds 19180 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐵 = (dist‘𝑅)       𝐵 = (dist‘𝑀)
 
Theoremmgpress 19181 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑆 = (𝑅s 𝐴)    &   𝑀 = (mulGrp‘𝑅)       ((𝑅𝑉𝐴𝑊) → (𝑀s 𝐴) = (mulGrp‘𝑆))
 
10.3.2  Ring unit
 
Syntaxcur 19182 Extend class notation with ring unit.
class 1r
 
Definitiondf-ur 19183 Define the multiplicative neutral element of a ring. This definition works by extracting the 0g element, i.e. the neutral element in a group or monoid, and transferring it to the multiplicative monoid via the mulGrp function (df-mgp 19171). See also dfur2 19185, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
1r = (0g ∘ mulGrp)
 
Theoremringidval 19184 The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐺 = (mulGrp‘𝑅)    &    1 = (1r𝑅)        1 = (0g𝐺)
 
Theoremdfur2 19185* The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)        1 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)))
 
10.3.2.1  Semirings
 
Syntaxcsrg 19186 Extend class notation with the class of all semirings.
class SRing
 
Definitiondf-srg 19187* Define class of all semirings. A semiring is a set equipped with two everywhere-defined internal operations, whose first one is an additive commutative monoid structure and the second one is a multiplicative monoid structure, and where multiplication is (left- and right-) distributive over addition. Compared to the definition of a ring, this definition also adds that the additive identity is an absorbing element of the multiplicative law, as this cannot be deduced from distributivity alone. Definition of [Golan] p. 1. Note that our semirings are unital. Such semirings are sometimes called "rigs", being "rings without negatives". (Contributed by Thierry Arnoux, 21-Mar-2018.)
SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡][(0g𝑓) / 𝑛]𝑥𝑟 (∀𝑦𝑟𝑧𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}
 
Theoremissrg 19188* The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
 
Theoremsrgcmn 19189 A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
(𝑅 ∈ SRing → 𝑅 ∈ CMnd)
 
Theoremsrgmnd 19190 A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
(𝑅 ∈ SRing → 𝑅 ∈ Mnd)
 
Theoremsrgmgp 19191 A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
 
Theoremsrgi 19192 Properties of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
 
Theoremsrgcl 19193 Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) ∈ 𝐵)
 
Theoremsrgass 19194 Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
 
Theoremsrgideu 19195* The unit element of a semiring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ SRing → ∃!𝑢𝐵𝑥𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥))
 
Theoremsrgfcl 19196 Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵)
 
Theoremsrgdi 19197 Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
 
Theoremsrgdir 19198 Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
 
Theoremsrgidcl 19199 The unit element of a semiring belongs to the base set of the semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ SRing → 1𝐵)
 
Theoremsrg0cl 19200 The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ SRing → 0𝐵)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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