P. 202 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dpjf 20101	The 𝑋-th index projection is a function from the direct product to the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → dom 𝑆 = 𝐼)    &   ⊢ 𝑃 = (𝐺dProj𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋))
Theorem	dpjidcl 20102*	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.)
⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → dom 𝑆 = 𝐼)    &   ⊢ 𝑃 = (𝐺dProj𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))))
Theorem	dpjeq 20103*	Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → dom 𝑆 = 𝐼)    &   ⊢ 𝑃 = (𝐺dProj𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }    &   ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐶) ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶))
Theorem	dpjid 20104*	The key property of projections: the sum of all the projections of 𝐴 is 𝐴. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → dom 𝑆 = 𝐼)    &   ⊢ 𝑃 = (𝐺dProj𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆))    ⇒   ⊢ (𝜑 → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))
Theorem	dpjlid 20105	The 𝑋-th index projection acts as the identity on elements of the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → dom 𝑆 = 𝐼)    &   ⊢ 𝑃 = (𝐺dProj𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋))    ⇒   ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴)
Theorem	dpjrid 20106	The 𝑌-th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → dom 𝑆 = 𝐼)    &   ⊢ 𝑃 = (𝐺dProj𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑋)    ⇒   ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 )
Theorem	dpjghm 20107	The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → dom 𝑆 = 𝐼)    &   ⊢ 𝑃 = (𝐺dProj𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝑃‘𝑋) ∈ ((𝐺 ↾s (𝐺 DProd 𝑆)) GrpHom 𝐺))
Theorem	dpjghm2 20108	The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (𝜑 → 𝐺dom DProd 𝑆)    &   ⊢ (𝜑 → dom 𝑆 = 𝐼)    &   ⊢ 𝑃 = (𝐺dProj𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝑃‘𝑋) ∈ ((𝐺 ↾s (𝐺 DProd 𝑆)) GrpHom (𝐺 ↾s (𝑆‘𝑋))))
10.2.14.6  The Fundamental Theorem of Abelian Groups
Theorem	ablfacrplem 20109*	Lemma for ablfacrp2 20111. (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))    ⇒   ⊢ (𝜑 → ((♯‘𝐾) gcd 𝑁) = 1)
Theorem	ablfacrp 20110*	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 20111*	The factors 𝐾, 𝐿 of ablfacrp 20110 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 20112*	Lemma for ablfac1b 20114. 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 20113*	The factors of ablfac1b 20114 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 20114*	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 20115*	The factors of ablfac1b 20114 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 20116*	Lemma for ablfac1eu 20117. (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 20117*	The factorization of ablfac1b 20114 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 20118*	Lemma for pgpfac1 20124. (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 20119*	Lemma for pgpfac1 20124. (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 20120*	Lemma for pgpfac1 20124. (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 20121*	Lemma for pgpfac1 20124. (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 20122*	Lemma for pgpfac1 20124. (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 20123*	Lemma for pgpfac1 20124. (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 20124*	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 20125*	Lemma for pgpfac 20128. (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 20126*	Lemma for pgpfac 20128. (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 20127*	Lemma for pgpfac 20128. (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 20128*	Full factorization of a finite abelian p-group, by iterating pgpfac1 20124. 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 20129*	Lemma for ablfac 20132. (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 20130*	Lemma for ablfac 20132. (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 20131*	Lemma for ablfac 20132. (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 20132*	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 20133*	Choose generators for each cyclic group in ablfac 20132. (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 20134	Extend class notation with the class of simple groups.
class SimpGrp
Definition	df-simpg 20135	Define class of all simple groups. A simple group is a group (df-grp 18976) with exactly two normal subgroups. These are always the subgroup of all elements and the subgroup containing only the identity (simpgnsgbid 20147). (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
Theorem	issimpg 20136	The predicate "is a simple group". (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
Theorem	issimpgd 20137	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 20138	A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
Theorem	simpggrpd 20139	A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	simpg2nsg 20140	A simple group has two normal subgroups. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)
Theorem	trivnsimpgd 20141	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 20142	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 20143*	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 20144	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 20145	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 20146*	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 20147	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 20148	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 20149*	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 20150	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 20151*	Lemma for ablsimpgfind 20154. 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 20152*	Lemma for ablsimpgfind 20154. 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 20153*	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 20154	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 20155*	The subgroup referenced in fincygsubgodd 20156 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 20156*	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 20157*	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 20158	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 20159	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 20160	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))
10.3  Rings
10.3.1  Multiplicative Group
Syntax	cmgp 20161	Multiplicative group.
class mulGrp
Definition	df-mgp 20162	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 20409 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 20266) or "the multiplicative identity" in terms of the identity of a monoid (df-ur 20209). (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))
Theorem	fnmgp 20163	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
⊢ mulGrp Fn V
Theorem	mgpval 20164	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 20165	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ · = (+g‘𝑀)
Theorem	mgplemOLD 20166	Obsolete version of setsplusg 19390 as of 18-Oct-2024. Lemma for mgpbas 20167. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝑁 ≠ 2    ⇒   ⊢ (𝐸‘𝑅) = (𝐸‘𝑀)
Theorem	mgpbas 20167	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	mgpbasOLD 20168	Obsolete version of mgpbas 20167 as of 18-Oct-2024. Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ 𝐵 = (Base‘𝑀)
Theorem	mgpsca 20169	The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 20350. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑆 = (Scalar‘𝑅)    ⇒   ⊢ 𝑆 = (Scalar‘𝑀)
Theorem	mgpscaOLD 20170	Obsolete version of mgpsca 20169 as of 18-Oct-2024. The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 20350. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑆 = (Scalar‘𝑅)    ⇒   ⊢ 𝑆 = (Scalar‘𝑀)
Theorem	mgptset 20171	Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (TopSet‘𝑅) = (TopSet‘𝑀)
Theorem	mgptsetOLD 20172	Obsolete version of mgptset 20171 as of 18-Oct-2024. Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (TopSet‘𝑅) = (TopSet‘𝑀)
Theorem	mgptopn 20173	Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ 𝐽 = (TopOpen‘𝑀)
Theorem	mgpds 20174	Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (dist‘𝑅)    ⇒   ⊢ 𝐵 = (dist‘𝑀)
Theorem	mgpdsOLD 20175	Obsolete version of mgpds 20174 as of 18-Oct-2024. Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (dist‘𝑅)    ⇒   ⊢ 𝐵 = (dist‘𝑀)
Theorem	mgpress 20176	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	mgpressOLD 20177	Obsolete version of mgpress 20176 as of 18-Oct-2024. Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))
Theorem	prdsmgp 20178	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)))
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 20179	Extend class notation with class of all non-unital rings.
class Rng
Definition	df-rng 20180*	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 20181*	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 20182	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
Theorem	rngmgp 20183	A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)
Theorem	rngmgpf 20184	Restricted functionality of the multiplicative group on non-unital rings (mgpf 20275 analog). (Contributed by AV, 22-Feb-2025.)
⊢ (mulGrp ↾ Rng):Rng⟶Smgrp
Theorem	rnggrp 20185	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
Theorem	rngass 20186	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)))
Theorem	rngdi 20187	Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
Theorem	rngdir 20188	Distributive law for the multiplication operation of a non-unital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
Theorem	rngacl 20189	Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	rng0cl 20190	The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → 0 ∈ 𝐵)
Theorem	rngcl 20191	Closure of the multiplication operation of a non-unital ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵)
Theorem	rnglz 20192	The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 20316. (Revised by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 )
Theorem	rngrz 20193	The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 20317. (Revised by AV, 16-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
Theorem	rngmneg1 20194	Negation of a product in a non-unital ring (mulneg1 11726 analog). In contrast to ringmneg1 20327, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌)))
Theorem	rngmneg2 20195	Negation of a product in a non-unital ring (mulneg2 11727 analog). In contrast to ringmneg2 20328, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌)))
Theorem	rngm2neg 20196	Double negation of a product in a non-unital ring (mul2neg 11729 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 20329. (Revised by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌))
Theorem	rngansg 20197	Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
⊢ (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
Theorem	rngsubdi 20198	Ring multiplication distributes over subtraction. (subdi 11723 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 20330. (Revised by AV, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍)))
Theorem	rngsubdir 20199	Ring multiplication distributes over subtraction. (subdir 11724 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 20331. (Revised by AV, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍)))
Theorem	isrngd 20200*	Properties that determine a non-unital ring. (Contributed by AV, 14-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → + = (+g‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Abel)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    ⇒   ⊢ (𝜑 → 𝑅 ∈ Rng)