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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremablfac1a 18201* The factors of ablfac1b 18202 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 18202* 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 18203* The factors of ablfac1b 18202 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 18204* Lemma for ablfac1eu 18205. (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 18205* The factorization of ablfac1b 18202 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 18206* Lemma for pgpfac1 18212. (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 18207* Lemma for pgpfac1 18212. (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 18208* Lemma for pgpfac1 18212. (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 18209* Lemma for pgpfac1 18212. (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 18210* Lemma for pgpfac1 18212. (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 18211* Lemma for pgpfac1 18212. (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 18212* 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 18213* Lemma for pgpfac 18216. (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 18214* Lemma for pgpfac 18216. (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 18215* Lemma for pgpfac 18216. (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 18216* Full factorization of a finite abelian p-group, by iterating pgpfac1 18212. 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 18217* Lemma for ablfac 18220. (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 18218* Lemma for ablfac 18220. (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 18219* Lemma for ablfac 18220. (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 18220* 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 18221* Choose generators for each cyclic group in ablfac 18220. (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))

10.4  Rings

10.4.1  Multiplicative Group

Syntaxcmgp 18222 Multiplicative group.
class mulGrp

Definitiondf-mgp 18223 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 18400 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 18286) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 9642). (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩))

Theoremfnmgp 18224 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp Fn V

Theoremmgpval 18225 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)       𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)

Theoremmgpplusg 18226 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)        · = (+g𝑀)

Theoremmgplem 18227 Lemma for mgpbas 18228. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 2       (𝐸𝑅) = (𝐸𝑀)

Theoremmgpbas 18228 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 18229 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 18351. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝑆 = (Scalar‘𝑅)       𝑆 = (Scalar‘𝑀)

Theoremmgptset 18230 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)       (TopSet‘𝑅) = (TopSet‘𝑀)

Theoremmgptopn 18231 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐽 = (TopOpen‘𝑅)       𝐽 = (TopOpen‘𝑀)

Theoremmgpds 18232 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐵 = (dist‘𝑅)       𝐵 = (dist‘𝑀)

Theoremmgpress 18233 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))

10.4.2  Ring unit

Syntaxcur 18234 Extend class notation with ring unit.
class 1r

Definitiondf-ur 18235 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 18223). See also dfur2 18237, 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 18236 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 18237* 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)))

10.4.2.1  Semirings

Syntaxcsrg 18238 Extend class notation with the class of all semirings.
class SRing

Definitiondf-srg 18239* 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 18240* 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 18241 A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
(𝑅 ∈ SRing → 𝑅 ∈ CMnd)

Theoremsrgmnd 18242 A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
(𝑅 ∈ SRing → 𝑅 ∈ Mnd)

Theoremsrgmgp 18243 A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ SRing → 𝐺 ∈ Mnd)

Theoremsrgi 18244 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 18245 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 18246 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 18247* 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 18248 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 18249 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 18250 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 18251 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 18252 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)

Theoremsrgidmlem 18253 Lemma for srglidm 18254 and srgridm 18255. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋))

Theoremsrglidm 18254 The unit element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 1 · 𝑋) = 𝑋)

Theoremsrgridm 18255 The unit element of a semiring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · 1 ) = 𝑋)

Theoremissrgid 18256* Properties showing that an element 𝐼 is the unity element of a semiring. (Contributed by NM, 7-Aug-2013.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ SRing → ((𝐼𝐵 ∧ ∀𝑥𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼))

Theoremsrgacl 18257 Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Theoremsrgcom 18258 Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremsrgrz 18259 The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · 0 ) = 0 )

Theoremsrglz 18260 The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )

Theoremsrgisid 18261* In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝑍𝐵)    &   ((𝜑𝑥𝐵) → (𝑍 · 𝑥) = 𝑍)       (𝜑𝑍 = 0 )

Theoremsrg1zr 18262 The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Theoremsrgen1zr 18263 The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)    &   𝑍 = (0g𝑅)       ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1𝑜 ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Theoremsrgmulgass 18264 An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.g𝑅)    &    × = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Theoremsrgpcomp 18265 If two elements of a semiring commute, they also commute if one of the elements is raised to a higher power. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))       (𝜑 → ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵)))

Theoremsrgpcompp 18266 If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (((𝑁 𝐴) × (𝐾 𝐵)) × 𝐴) = (((𝑁 + 1) 𝐴) × (𝐾 𝐵)))

Theoremsrgpcomppsc 18267 If two elements of a semiring commute, they also commute if the elements are raised to a higher power and a scalar multiplication is involved. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)    &    · = (.g𝑅)    &   (𝜑𝐶 ∈ ℕ0)       (𝜑 → ((𝐶 · ((𝑁 𝐴) × (𝐾 𝐵))) × 𝐴) = (𝐶 · (((𝑁 + 1) 𝐴) × (𝐾 𝐵))))

Theoremsrglmhm 18268* Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringlghm 18337. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 MndHom 𝑅))

Theoremsrgrmhm 18269* Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringrghm 18338. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 MndHom 𝑅))

Theoremsrgsummulcr 18270* A finite semiring sum multiplied by a constant, analogous to gsummulc1 18339. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘𝐴𝑋)) · 𝑌))

Theoremsgsummulcl 18271* A finite semiring sum multiplied by a constant, analogous to gsummulc2 18340. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘𝐴𝑋))))

Theoremsrg1expzeq1 18272 The exponentiation (by a nonnegative integer) of the unity element of a (semi)ring, analogous to mulgnn0z 17285. (Contributed by AV, 25-Nov-2019.)
𝐺 = (mulGrp‘𝑅)    &    · = (.g𝐺)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → (𝑁 · 1 ) = 1 )

10.4.2.2  The binomial theorem for semirings

In this section, we prove the binomial theorem for semirings, srgbinom 18278, which is a generalization of the binomial theorem for complex numbers, binom 14273: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)).

Notice that the binomial theorem would also hold in the non-unital case (that is, in a "rg") and actually, the additive unit is not needed in its proof either. Therefore, it could be proven for even more general cases. An example would be the integrable nonnegative (resp. positive) bounded functions on .

Special cases of the binomial theorem are csrgbinom 18279 (binomial theorem for commutative semirings) and crngbinom 18354 (binomial theorem for commutative rings).

Theoremsrgbinomlem1 18273 Lemma 1 for srgbinomlem 18277. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑 ∧ (𝐷 ∈ ℕ0𝐸 ∈ ℕ0)) → ((𝐷 𝐴) × (𝐸 𝐵)) ∈ 𝑆)

Theoremsrgbinomlem2 18274 Lemma 2 for srgbinomlem 18277. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑 ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℕ0𝐸 ∈ ℕ0)) → (𝐶 · ((𝐷 𝐴) × (𝐸 𝐵))) ∈ 𝑆)

Theoremsrgbinomlem3 18275* Lemma 3 for srgbinomlem 18277. (Contributed by AV, 23-Aug-2019.) (Proof shortened by AV, 27-Oct-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))       ((𝜑𝜓) → ((𝑁 (𝐴 + 𝐵)) × 𝐴) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ ((𝑁C𝑘) · ((((𝑁 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))

Theoremsrgbinomlem4 18276* Lemma 4 for srgbinomlem 18277. (Contributed by AV, 24-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))       ((𝜑𝜓) → ((𝑁 (𝐴 + 𝐵)) × 𝐵) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ ((𝑁C(𝑘 − 1)) · ((((𝑁 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))

Theoremsrgbinomlem 18277* Lemma for srgbinom 18278. Inductive step, analogous to binomlem 14272. (Contributed by AV, 24-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))       ((𝜑𝜓) → ((𝑁 + 1) (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ (((𝑁 + 1)C𝑘) · ((((𝑁 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))

Theoremsrgbinom 18278* The binomial theorem for commuting elements of a semiring: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)) (generalization of binom 14273). (Contributed by AV, 24-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)       (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))

Theoremcsrgbinom 18279* The binomial theorem for commutative semirings. (Contributed by AV, 24-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)       (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴𝑆𝐵𝑆)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))

10.4.3  Definition and basic properties of unital rings

Syntaxcrg 18280 Extend class notation with class of all (unital) rings.
class Ring

Syntaxccrg 18281 Extend class notation with class of all (unital) commutative rings.
class CRing

Definitiondf-ring 18282* Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. Definition 1 in [BourbakiAlg1] p. 92 or definition of a ring with identity in part Preliminaries of [Roman] p. 19. So that the additive structure must be abelian (see ringcom 18312), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡]𝑥𝑟𝑦𝑟𝑧𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}

Definitiondf-cring 18283 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}

Theoremisring 18284* The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))

Theoremringgrp 18285 A ring is a group. (Contributed by NM, 15-Sep-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Grp)

Theoremringmgp 18286 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ Ring → 𝐺 ∈ Mnd)

Theoremiscrng 18287 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))

Theoremcrngmgp 18288 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ CRing → 𝐺 ∈ CMnd)

Theoremringmnd 18289 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ Mnd)

Theoremringmgm 18290 A ring is a magma. (Contributed by AV, 31-Jan-2020.)
(𝑅 ∈ Ring → 𝑅 ∈ Mgm)

Theoremcrngring 18291 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ CRing → 𝑅 ∈ Ring)

Theoremmgpf 18292 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
(mulGrp ↾ Ring):Ring⟶Mnd

Theoremringi 18293 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))

Theoremringcl 18294 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) ∈ 𝐵)

Theoremcrngcom 18295 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) = (𝑌 · 𝑋))

Theoremiscrng2 18296* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 · 𝑦) = (𝑦 · 𝑥)))

Theoremringass 18297 Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))

Theoremringideu 18298* The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → ∃!𝑢𝐵𝑥𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥))

Theoremringdi 18299 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Theoremringdir 18300 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

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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 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