P. 203 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ablfacrp 20201*	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 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑀}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)    &   ⊢ (𝜑 → (♯‘𝐵) = (𝑀 · 𝑁))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (𝜑 → ((𝐾 ∩ 𝐿) = { 0 } ∧ (𝐾 ⊕ 𝐿) = 𝐵))
Theorem	ablfacrp2 20202*	The factors 𝐾, 𝐿 of ablfacrp 20201 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 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑀}    &   ⊢ 𝐿 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)    &   ⊢ (𝜑 → (♯‘𝐵) = (𝑀 · 𝑁))    ⇒   ⊢ (𝜑 → ((♯‘𝐾) = 𝑀 ∧ (♯‘𝐿) = 𝑁))
Theorem	ablfac1lem 20203*	Lemma for ablfac1b 20205. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ ℙ)    &   ⊢ 𝑀 = (𝑃↑(𝑃 pCnt (♯‘𝐵)))    &   ⊢ 𝑁 = ((♯‘𝐵) / 𝑀)    ⇒   ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀 gcd 𝑁) = 1 ∧ (♯‘𝐵) = (𝑀 · 𝑁)))
Theorem	ablfac1a 20204*	The factors of ablfac1b 20205 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ ℙ)    ⇒   ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘(𝑆‘𝑃)) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
Theorem	ablfac1b 20205*	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 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ ℙ)    ⇒   ⊢ (𝜑 → 𝐺dom DProd 𝑆)
Theorem	ablfac1c 20206*	The factors of ablfac1b 20205 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ ℙ)    &   ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵)
Theorem	ablfac1eulem 20207*	Lemma for ablfac1eu 20208. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ ℙ)    &   ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵))    &   ⊢ (𝜑 → dom 𝑇 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ¬ 𝑃 ∥ (♯‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃})))))
Theorem	ablfac1eu 20208*	The factorization of ablfac1b 20205 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 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ ℙ)    &   ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵))    &   ⊢ (𝜑 → dom 𝑇 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))    ⇒   ⊢ (𝜑 → 𝑇 = 𝑆)
Theorem	pgpfac1lem1 20209*	Lemma for pgpfac1 20215. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    &   ⊢ 𝑆 = (𝐾‘{𝐴})    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })    &   ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)    &   ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈)
Theorem	pgpfac1lem2 20210*	Lemma for pgpfac1 20215. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    &   ⊢ 𝑆 = (𝐾‘{𝐴})    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })    &   ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)    &   ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)))    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (𝜑 → (𝑃 · 𝐶) ∈ (𝑆 ⊕ 𝑊))
Theorem	pgpfac1lem3a 20211*	Lemma for pgpfac1 20215. (Contributed by Mario Carneiro, 4-Jun-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    &   ⊢ 𝑆 = (𝐾‘{𝐴})    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })    &   ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)    &   ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)))    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ((𝑃 · 𝐶)(+g‘𝐺)(𝑀 · 𝐴)) ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑃 ∥ 𝐸 ∧ 𝑃 ∥ 𝑀))
Theorem	pgpfac1lem3 20212*	Lemma for pgpfac1 20215. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    &   ⊢ 𝑆 = (𝐾‘{𝐴})    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })    &   ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)    &   ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)))    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ((𝑃 · 𝐶)(+g‘𝐺)(𝑀 · 𝐴)) ∈ 𝑊)    &   ⊢ 𝐷 = (𝐶(+g‘𝐺)((𝑀 / 𝑃) · 𝐴))    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
Theorem	pgpfac1lem4 20213*	Lemma for pgpfac1 20215. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    &   ⊢ 𝑆 = (𝐾‘{𝐴})    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })    &   ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)    &   ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)))    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
Theorem	pgpfac1lem5 20214*	Lemma for pgpfac1 20215. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    &   ⊢ 𝑆 = (𝐾‘{𝐴})    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
Theorem	pgpfac1 20215*	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 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))
Theorem	pgpfaclem1 20216*	Lemma for pgpfac 20219. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))    &   ⊢ 𝐻 = (𝐺 ↾s 𝑈)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))    &   ⊢ 𝑂 = (od‘𝐻)    &   ⊢ 𝐸 = (gEx‘𝐻)    &   ⊢ 0 = (0g‘𝐻)    &   ⊢ ⊕ = (LSSum‘𝐻)    &   ⊢ (𝜑 → 𝐸 ≠ 1)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → (𝑂‘𝑋) = 𝐸)    &   ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))    &   ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })    &   ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)    &   ⊢ (𝜑 → 𝑆 ∈ Word 𝐶)    &   ⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝑊)    &   ⊢ 𝑇 = (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
Theorem	pgpfaclem2 20217*	Lemma for pgpfac 20219. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))    &   ⊢ 𝐻 = (𝐺 ↾s 𝑈)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))    &   ⊢ 𝑂 = (od‘𝐻)    &   ⊢ 𝐸 = (gEx‘𝐻)    &   ⊢ 0 = (0g‘𝐻)    &   ⊢ ⊕ = (LSSum‘𝐻)    &   ⊢ (𝜑 → 𝐸 ≠ 1)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → (𝑂‘𝑋) = 𝐸)    &   ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))    &   ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })    &   ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
Theorem	pgpfaclem3 20218*	Lemma for pgpfac 20219. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
Theorem	pgpfac 20219*	Full factorization of a finite abelian p-group, by iterating pgpfac1 20215. 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) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
Theorem	ablfaclem1 20220*	Lemma for ablfac 20223. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})    ⇒   ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑊‘𝑈) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)})
Theorem	ablfaclem2 20221*	Lemma for ablfac 20223. (Contributed by Mario Carneiro, 27-Apr-2016.) (Proof shortened by Mario Carneiro, 3-May-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})    &   ⊢ (𝜑 → 𝐹:𝐴⟶Word 𝐶)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) ∈ (𝑊‘(𝑆‘𝑦)))    &   ⊢ 𝐿 = ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝐹‘𝑦))    &   ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐿))–1-1-onto→𝐿)    ⇒   ⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅)
Theorem	ablfaclem3 20222*	Lemma for ablfac 20223. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})    ⇒   ⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅)
Theorem	ablfac 20223*	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) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
Theorem	ablfac2 20224*	Choose generators for each cyclic group in ablfac 20223. (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))    ⇒   ⊢ (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
10.2.15  Simple groups
10.2.15.1  Definition and basic properties
Syntax	csimpg 20225	Extend class notation with the class of simple groups.
class SimpGrp
Definition	df-simpg 20226	Define class of all simple groups. A simple group is a group (df-grp 19066) with exactly two normal subgroups. These are always the subgroup of all elements and the subgroup containing only the identity (simpgnsgbid 20238). (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
Theorem	issimpg 20227	The predicate "is a simple group". (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
Theorem	issimpgd 20228	Deduce a simple group from its properties. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o)    ⇒   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)
Theorem	simpggrp 20229	A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
Theorem	simpggrpd 20230	A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	simpg2nsg 20231	A simple group has two normal subgroups. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)
Theorem	trivnsimpgd 20232	Trivial groups are not simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = { 0 })    ⇒   ⊢ (𝜑 → ¬ 𝐺 ∈ SimpGrp)
Theorem	simpgntrivd 20233	Simple groups are nontrivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → ¬ 𝐵 = { 0 })
Theorem	simpgnideld 20234*	A simple group contains a nonidentity element. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ¬ 𝑥 = 0 )
Theorem	simpgnsgd 20235	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵})
Theorem	simpgnsgeqd 20236	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    &   ⊢ (𝜑 → 𝐴 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → (𝐴 = { 0 } ∨ 𝐴 = 𝐵))
Theorem	2nsgsimpgd 20237*	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → ¬ { 0 } = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (NrmSGrp‘𝐺)) → (𝑥 = { 0 } ∨ 𝑥 = 𝐵))    ⇒   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)
Theorem	simpgnsgbid 20238	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → ¬ { 0 } = 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 ∈ SimpGrp ↔ (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}))
10.2.15.2  Classification of abelian simple groups
Theorem	ablsimpnosubgd 20239	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → ¬ 𝐴 = 0 )    ⇒   ⊢ (𝜑 → 𝑆 = 𝐵)
Theorem	ablsimpg1gend 20240*	An abelian simple group is generated by any non-identity element. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 = 0 )    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℤ 𝐶 = (𝑛 · 𝐴))
Theorem	ablsimpgcygd 20241	An abelian simple group is cyclic. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ CycGrp)
Theorem	ablsimpgfindlem1 20242*	Lemma for ablsimpgfind 20245. 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 = (0g‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) ≠ 0 ) → (𝑂‘𝑥) ≠ 0)
Theorem	ablsimpgfindlem2 20243*	Lemma for ablsimpgfind 20245. 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 = (0g‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (𝑂‘𝑥) ≠ 0)
Theorem	cycsubggenodd 20244*	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)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴)))    ⇒   ⊢ (𝜑 → (𝑂‘𝐴) = if(𝐶 ∈ Fin, (♯‘𝐶), 0))
Theorem	ablsimpgfind 20245	An abelian simple group is finite. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → 𝐵 ∈ Fin)
Theorem	fincygsubgd 20246*	The subgroup referenced in fincygsubgodd 20247 is a subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐻 = (𝑛 ∈ ℤ ↦ (𝑛 · (𝐶 · 𝐴)))    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    ⇒   ⊢ (𝜑 → ran 𝐻 ∈ (SubGrp‘𝐺))
Theorem	fincygsubgodd 20247*	Calculate the order of a subgroup of a finite cyclic group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐷 = ((♯‘𝐵) / 𝐶)    &   ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴))    &   ⊢ 𝐻 = (𝑛 ∈ ℤ ↦ (𝑛 · (𝐶 · 𝐴)))    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ran 𝐹 = 𝐵)    &   ⊢ (𝜑 → 𝐶 ∥ (♯‘𝐵))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    ⇒   ⊢ (𝜑 → (♯‘ran 𝐻) = 𝐷)
Theorem	fincygsubgodexd 20248*	A finite cyclic group has subgroups of every possible order. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CycGrp)    &   ⊢ (𝜑 → 𝐶 ∥ (♯‘𝐵))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (SubGrp‘𝐺)(♯‘𝑥) = 𝐶)
Theorem	prmgrpsimpgd 20249	A group of prime order is simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → (♯‘𝐵) ∈ ℙ)    ⇒   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)
Theorem	ablsimpgprmd 20250	An abelian simple group has prime order. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → (♯‘𝐵) ∈ ℙ)
Theorem	ablsimpgd 20251	An abelian group is simple if and only if its order is prime. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    ⇒   ⊢ (𝜑 → (𝐺 ∈ SimpGrp ↔ (♯‘𝐵) ∈ ℙ))
10.2.16  Totally ordered monoids and groups
Syntax	comnd 20252	Extend class notation with the class of all right ordered monoids.
class oMnd
Syntax	cogrp 20253	Extend class notation with the class of all right ordered groups.
class oGrp
Definition	df-omnd 20254*	Define class of all right ordered monoids. An ordered monoid is a monoid with a total ordering compatible with its operation. It is possible to use this definition also for left ordered monoids, by applying it to (oppg‘𝑀). (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ oMnd = {𝑔 ∈ Mnd ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑝][(le‘𝑔) / 𝑙](𝑔 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))}
Definition	df-ogrp 20255	Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ oGrp = (Grp ∩ oMnd)
Theorem	isomnd 20256*	A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    ⇒   ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
Theorem	isogrp 20257	A (left-)ordered group is a group with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
Theorem	ogrpgrp 20258	A left-ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)
⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
Theorem	omndmnd 20259	A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
Theorem	omndtos 20260	A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
Theorem	omndadd 20261	In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))
Theorem	omndaddr 20262	In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑍 + 𝑋) ≤ (𝑍 + 𝑌))
Theorem	omndadd2d 20263	In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑍)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))
Theorem	omndadd2rd 20264	In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑍)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑊)    &   ⊢ (𝜑 → (oppg‘𝑀) ∈ oMnd)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))
Theorem	submomnd 20265	A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd)
Theorem	omndmul2 20266	In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ · = (.g‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))
Theorem	omndmul3 20267	In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ · = (.g‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑃 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ≤ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 0 ≤ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋))
Theorem	omndmul 20268	In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ · = (.g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    ⇒   ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑁 · 𝑌))
Theorem	ogrpinv0le 20269	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ≤ = (le‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))
Theorem	ogrpsub 20270	In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ≤ = (le‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) ≤ (𝑌 − 𝑍))
Theorem	ogrpaddlt 20271	In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
Theorem	ogrpaddltbi 20272	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))
Theorem	ogrpaddltrd 20273	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 < 𝑌)    ⇒   ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌))
Theorem	ogrpaddltrbid 20274	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 < 𝑌 ↔ (𝑍 + 𝑋) < (𝑍 + 𝑌)))
Theorem	ogrpsublt 20275	In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 − 𝑍) < (𝑌 − 𝑍))
Theorem	ogrpinv0lt 20276	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 ))
Theorem	ogrpinvlt 20277	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ oGrp ∧ (oppg‘𝐺) ∈ oGrp) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝐼‘𝑌) < (𝐼‘𝑋)))
Theorem	gsumle 20278	A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 ∘r ≤ 𝐺)    ⇒   ⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺))
10.3  Rings
10.3.1  Multiplicative Group
Syntax	cmgp 20279	Multiplicative group.
class mulGrp
Definition	df-mgp 20280	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 20530 shows that we get a group if we restrict to the elements that have inverses. This allows to formalize such notions as "the multiplication operation of a ring is a monoid" (ringmgp 20384) or "the multiplicative identity" in terms of the identity of a monoid (df-ur 20327). (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))
Theorem	fnmgp 20281	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
⊢ mulGrp Fn V
Theorem	mgpval 20282	Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
Theorem	mgpplusg 20283	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ · = (+g‘𝑀)
Theorem	mgpbas 20284	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)
Theorem	mgpsca 20285	The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 20473. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑆 = (Scalar‘𝑅)    ⇒   ⊢ 𝑆 = (Scalar‘𝑀)
Theorem	mgptset 20286	Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (TopSet‘𝑅) = (TopSet‘𝑀)
Theorem	mgptopn 20287	Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ 𝐽 = (TopOpen‘𝑀)
Theorem	mgpds 20288	Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (dist‘𝑅)    ⇒   ⊢ 𝐵 = (dist‘𝑀)
Theorem	mgpress 20289	Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))
Theorem	prdsmgp 20290	The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑌)    &   ⊢ 𝑍 = (𝑆Xs(mulGrp ∘ 𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 Fn 𝐼)    ⇒   ⊢ (𝜑 → ((Base‘𝑀) = (Base‘𝑍) ∧ (+g‘𝑀) = (+g‘𝑍)))
Theorem	elmgplsm 20291*	Membership in a product of two subsets of a multiplication group. (Contributed by Thierry Arnoux, 20-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ × = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝐸 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐹 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 𝑍 = (𝑥 · 𝑦)))
Theorem	elmgplsmd 20292	Membership in a product of two subsets of a multiplication group, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ × = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝐸 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐹 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ (𝐸 × 𝐹))
10.3.2  Non-unital rings ("rngs") According to Wikipedia, "... in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 28-Mar-2025).
Syntax	crng 20293	Extend class notation with class of all non-unital rings.
class Rng
Definition	df-rng 20294*	Define the class of all non-unital rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.)
⊢ Rng = {𝑓 ∈ Abel ∣ ((mulGrp‘𝑓) ∈ Smgrp ∧ [(Base‘𝑓) / 𝑏][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
Theorem	isrng 20295*	The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ Smgrp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
Theorem	rngabl 20296	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
Theorem	rngmgp 20297	A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)
Theorem	rngmgpf 20298	Restricted functionality of the multiplicative group on non-unital rings (mgpf 20393 analog). (Contributed by AV, 22-Feb-2025.)
⊢ (mulGrp ↾ Rng):Rng⟶Smgrp
Theorem	rnggrp 20299	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
Theorem	rngass 20300	Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))