P. 335 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elringlsmd 33401	Membership in a product of two subsets of a ring, 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))
Theorem	ringlsmss 33402	Closure of the product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ × = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐹 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐸 × 𝐹) ⊆ 𝐵)
Theorem	ringlsmss1 33403	The product of an ideal 𝐼 of a commutative ring 𝑅 with some set E is a subset of the ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ × = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐸 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼)
Theorem	ringlsmss2 33404	The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ × = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝐸 × 𝐼) ⊆ 𝐼)
Theorem	lsmsnpridl 33405	The product of the ring with a single element is equal to the principal ideal generated by that element. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ × = (LSSum‘𝐺)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐵 × {𝑋}) = (𝐾‘{𝑋}))
Theorem	lsmsnidl 33406	The product of the ring with a single element is a principal ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ × = (LSSum‘𝐺)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐵 × {𝑋}) ∈ (LPIdeal‘𝑅))
Theorem	lsmidllsp 33407	The sum of two ideals is the ideal generated by their union. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝐼 ⊕ 𝐽) = (𝐾‘(𝐼 ∪ 𝐽)))
Theorem	lsmidl 33408	The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝐼 ⊕ 𝐽) ∈ (LIdeal‘𝑅))
Theorem	lsmssass 33409	Group sum is associative, subset version (see lsmass 19701). (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑅 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑅 ⊕ 𝑇) ⊕ 𝑈) = (𝑅 ⊕ (𝑇 ⊕ 𝑈)))
Theorem	grplsm0l 33410	Sumset with the identity singleton is the original set. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ({ 0 } ⊕ 𝐴) = 𝐴)
Theorem	grplsmid 33411	The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝐴) → ({𝑋} ⊕ 𝐴) = 𝐴)
21.3.9.37  The quotient map
Theorem	quslsm 33412	Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} ⊕ 𝑆))
Theorem	qusbas2 33413*	Alternate definition of the group quotient set, as the set of all cosets of the form ({𝑥} ⊕ 𝑁). (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ (SubGrp‘𝐺))    ⇒   ⊢ (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = ran (𝑥 ∈ 𝐵 ↦ ({𝑥} ⊕ 𝑁)))
Theorem	qus0g 33414	The identity element of a quotient group. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    ⇒   ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → (0g‘𝑄) = 𝑁)
Theorem	qusima 33415*	The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   ⊢ (𝜑 → 𝐻 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝐺))    ⇒   ⊢ (𝜑 → (𝐸‘𝐻) = (𝐹 “ 𝐻))
Theorem	qusrn 33416*	The natural map from elements to their cosets is surjective. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑈 = (𝐵 / (𝐺 ~QG 𝑁))    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → ran 𝐹 = 𝑈)
Theorem	nsgqus0 33417	A normal subgroup 𝑁 is a member of all subgroups 𝐹 of the quotient group by 𝑁. In fact, it is the identity element of the quotient group. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    ⇒   ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝐹)
Theorem	nsgmgclem 33418*	Lemma for nsgmgc 33419. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝑄))    ⇒   ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝐹} ∈ (SubGrp‘𝐺))
Theorem	nsgmgc 33419*	There is a monotone Galois connection between the lattice of subgroups of a group 𝐺 containing a normal subgroup 𝑁 and the lattice of subgroups of the quotient group 𝐺 / 𝑁. This is sometimes called the lattice theorem. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ 𝐽 = (𝑉MGalConn𝑊)    &   ⊢ 𝑉 = (toInc‘𝑆)    &   ⊢ 𝑊 = (toInc‘𝑇)    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → 𝐸𝐽𝐹)
Theorem	nsgqusf1olem1 33420*	Lemma for nsgqusf1o 33423. (Contributed by Thierry Arnoux, 4-Aug-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ ≤ = (le‘(toInc‘𝑆))    &   ⊢ ≲ = (le‘(toInc‘𝑇))    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
Theorem	nsgqusf1olem2 33421*	Lemma for nsgqusf1o 33423. (Contributed by Thierry Arnoux, 4-Aug-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ ≤ = (le‘(toInc‘𝑆))    &   ⊢ ≲ = (le‘(toInc‘𝑇))    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → ran 𝐸 = 𝑇)
Theorem	nsgqusf1olem3 33422*	Lemma for nsgqusf1o 33423. (Contributed by Thierry Arnoux, 4-Aug-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ ≤ = (le‘(toInc‘𝑆))    &   ⊢ ≲ = (le‘(toInc‘𝑇))    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → ran 𝐹 = 𝑆)
Theorem	nsgqusf1o 33423*	The canonical projection homomorphism 𝐸 defines a bijective correspondence between the set 𝑆 of subgroups of 𝐺 containing a normal subgroup 𝑁 and the subgroups of the quotient group 𝐺 / 𝑁. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ ≤ = (le‘(toInc‘𝑆))    &   ⊢ ≲ = (le‘(toInc‘𝑇))    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → 𝐸:𝑆–1-1-onto→𝑇)
Theorem	lmhmqusker 33424*	A surjective module homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMIso 𝐻))
Theorem	lmicqusker 33425	The image 𝐻 of a module homomorphism 𝐹 is isomorphic with the quotient module 𝑄 over 𝐹's kernel 𝐾. This is part of what is sometimes called the first isomorphism theorem for modules. (Contributed by Thierry Arnoux, 10-Mar-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))    ⇒   ⊢ (𝜑 → 𝑄 ≃𝑚 𝐻)
21.3.9.38  Ideals
Theorem	lidlmcld 33426	An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐼)
Theorem	intlidl 33427	The intersection of a nonempty collection of ideals is an ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∩ 𝐶 ∈ (LIdeal‘𝑅))
Theorem	0ringidl 33428	The zero ideal is the only ideal of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (LIdeal‘𝑅) = {{ 0 }})
Theorem	pidlnzb 33429	A principal ideal is nonzero iff it is generated by a nonzero elements (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 ↔ (𝐾‘{𝑋}) ≠ { 0 }))
Theorem	lidlunitel 33430	If an ideal 𝐼 contains a unit 𝐽, then it is the whole ring. (Contributed by Thierry Arnoux, 19-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 = 𝐵)
Theorem	unitpidl1 33431	The ideal 𝐼 generated by an element 𝑋 of an integral domain 𝑅 is the unit ideal 𝐵 iff 𝑋 is a ring unit. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐼 = (𝐾‘{𝑋})    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → (𝐼 = 𝐵 ↔ 𝑋 ∈ 𝑈))
Theorem	rhmquskerlem 33432*	The mapping 𝐽 induced by a ring homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝐺 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))
Theorem	rhmqusker 33433*	A surjective ring homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ CRing)    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingIso 𝐻))
Theorem	ricqusker 33434	The image 𝐻 of a ring homomorphism 𝐹 is isomorphic with the quotient ring 𝑄 over 𝐹's kernel 𝐾. This a part of what is sometimes called the first isomorphism theorem for rings. (Contributed by Thierry Arnoux, 10-Mar-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝑄 ≃𝑟 𝐻)
Theorem	elrspunidl 33435*	Elementhood in the span of a union of ideals. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 𝑁 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ⊆ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘∪ 𝑆) ↔ ∃𝑎 ∈ (𝐵 ↑m 𝑆)(𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg 𝑎) ∧ ∀𝑘 ∈ 𝑆 (𝑎‘𝑘) ∈ 𝑘)))
Theorem	elrspunsn 33436*	Membership to the span of an ideal 𝑅 and a single element 𝑋. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑁 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝐼))    ⇒   ⊢ (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))
Theorem	lidlincl 33437	Ideals are closed under intersection. (Contributed by Thierry Arnoux, 5-Jul-2024.)
⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑈) → (𝐼 ∩ 𝐽) ∈ 𝑈)
Theorem	idlinsubrg 33438	The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝑉 = (LIdeal‘𝑆)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ∈ 𝑉)
Theorem	rhmimaidl 33439	The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑇 = (LIdeal‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)
Theorem	drngidl 33440	A nonzero ring is a division ring if and only if its only left ideals are the zero ideal and the unit ideal. (Proposed by Gerard Lang, 13-Mar-2025.) (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → (𝑅 ∈ DivRing ↔ 𝑈 = {{ 0 }, 𝐵}))
Theorem	drngidlhash 33441	A ring is a division ring if and only if it admits exactly two ideals. (Proposed by Gerard Lang, 13-Mar-2025.) (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑅 ∈ DivRing ↔ (♯‘𝑈) = 2))
21.3.9.39  Prime Ideals
Syntax	cprmidl 33442	Extend class notation with the class of prime ideals.
class PrmIdeal
Definition	df-prmidl 33443*	Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵 ⊆ 𝐼 for ideals 𝐴 and 𝐵, either 𝐴 ⊆ 𝐼 or 𝐵 ⊆ 𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 33448 and isprmidlc 33454. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.)
⊢ PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
Theorem	prmidlval 33444*	The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
Theorem	isprmidl 33445*	The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
Theorem	prmidlnr 33446	A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ≠ 𝐵)
Theorem	prmidl 33447*	The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))
Theorem	prmidl2 33448*	A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38056 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
Theorem	idlmulssprm 33449	Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 ⊆ 𝑃 or 𝐽 ⊆ 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.)
⊢ × = (LSSum‘(mulGrp‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃)    ⇒   ⊢ (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))
Theorem	pridln1 33450	A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ¬ 1 ∈ 𝐼)
Theorem	prmidlidl 33451	A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))
Theorem	prmidlssidl 33452	Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
Theorem	cringm4 33453	Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊)))
Theorem	isprmidlc 33454*	The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))))
Theorem	prmidlc 33455	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ 𝐵 ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼 ∈ 𝑃 ∨ 𝐽 ∈ 𝑃))
Theorem	0ringprmidl 33456	The trivial ring does not have any prime ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (PrmIdeal‘𝑅) = ∅)
Theorem	prmidl0 33457	The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn)
Theorem	rhmpreimaprmidl 33458	The preimage of a prime ideal by a ring homomorphism is a prime ideal. (Contributed by Thierry Arnoux, 29-Jun-2024.)
⊢ 𝑃 = (PrmIdeal‘𝑅)    ⇒   ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃)
Theorem	qsidomlem1 33459	If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))
Theorem	qsidomlem2 33460	A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)
Theorem	qsidom 33461	An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅)))
Theorem	qsnzr 33462	A quotient of a non-zero ring by a proper ideal is a non-zero ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝑄 ∈ NzRing)
Theorem	ssdifidllem 33463*	Lemma for ssdifidl 33464: The set 𝑃 used in the proof of ssdifidl 33464 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    &   ⊢ (𝜑 → 𝑍 ⊆ 𝑃)    &   ⊢ (𝜑 → 𝑍 ≠ ∅)    &   ⊢ (𝜑 → [⊊] Or 𝑍)    ⇒   ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)
Theorem	ssdifidl 33464*	Let 𝑅 be a ring, and let 𝐼 be an ideal of 𝑅 disjoint with a set 𝑆. Then there exists an ideal 𝑖, maximal among the set 𝑃 of ideals containing 𝐼 and disjoint with 𝑆. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    ⇒   ⊢ (𝜑 → ∃𝑖 ∈ 𝑃 ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗)
Theorem	ssdifidlprm 33465*	If the set 𝑆 of ssdifidl 33464 is multiplicatively closed, then the ideal 𝑖 is prime. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    ⇒   ⊢ (𝜑 → ∃𝑖 ∈ 𝑃 (𝑖 ∈ (PrmIdeal‘𝑅) ∧ ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗))
21.3.9.40  Maximal Ideals
Syntax	cmxidl 33466	Extend class notation with the class of maximal ideals.
class MaxIdeal
Definition	df-mxidl 33467*	Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
Theorem	mxidlval 33468*	The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))})
Theorem	ismxidl 33469*	The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))
Theorem	mxidlidl 33470	A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
Theorem	mxidlnr 33471	A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
Theorem	mxidlmax 33472	A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))
Theorem	mxidln1 33473	One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1 ∈ 𝑀)
Theorem	mxidlnzr 33474	A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing)
Theorem	mxidlmaxv 33475	An ideal 𝐼 strictly containing a maximal ideal 𝑀 is the whole ring 𝐵. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑀 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀))    ⇒   ⊢ (𝜑 → 𝐼 = 𝐵)
Theorem	crngmxidl 33476	In a commutative ring, maximal ideals of the opposite ring coincide with maximal ideals. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝑀 = (MaxIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑀 = (MaxIdeal‘𝑂))
Theorem	mxidlprm 33477	Every maximal ideal is prime. Statement in [Lang] p. 92. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ × = (LSSum‘(mulGrp‘𝑅))    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
Theorem	mxidlirredi 33478	In an integral domain, the generator of a maximal ideal is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑀 = (𝐾‘{𝑋})    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    ⇒   ⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅))
Theorem	mxidlirred 33479	In a principal ideal domain, maximal ideals are exactly the ideals generated by irreducible elements. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑀 = (𝐾‘{𝑋})    &   ⊢ (𝜑 → 𝑅 ∈ PID)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))
Theorem	ssmxidllem 33480*	The set 𝑃 used in the proof of ssmxidl 33481 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)}    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝑃)    &   ⊢ (𝜑 → 𝑍 ≠ ∅)    &   ⊢ (𝜑 → [⊊] Or 𝑍)    ⇒   ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)
Theorem	ssmxidl 33481*	Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼 ⊆ 𝑚)
Theorem	drnglidl1ne0 33482	In a nonzero ring, the zero ideal is different of the unit ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝐵 ≠ { 0 })
Theorem	drng0mxidl 33483	In a division ring, the zero ideal is a maximal ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.)
⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅))
Theorem	drngmxidl 33484	The zero ideal is the only ideal of a division ring. (Contributed by Thierry Arnoux, 16-Mar-2025.)
⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }})
Theorem	drngmxidlr 33485	If a ring's only maximal ideal is the zero ideal, it is a division ring. See also drngmxidl 33484. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑀 = (MaxIdeal‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑀 = {{ 0 }})    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	krull 33486*	Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.)
⊢ (𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))
Theorem	mxidlnzrb 33487*	A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.)
⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)))
Theorem	krullndrng 33488*	Krull's theorem for non-division-rings: Existence of a nonzero maximal ideal. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑚 ≠ { 0 })
Theorem	opprabs 33489	The opposite ring of the opposite ring is the original ring. Note the conditions on this theorem, which makes it unpractical in case we only have e.g. 𝑅 ∈ Ring as a premise. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑅)    &   ⊢ (𝜑 → (.r‘ndx) ∈ dom 𝑅)    &   ⊢ (𝜑 → · Fn (𝐵 × 𝐵))    ⇒   ⊢ (𝜑 → 𝑅 = (oppr‘𝑂))
Theorem	oppreqg 33490	Group coset equivalence relation for the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
Theorem	opprnsg 33491	Normal subgroups of the opposite ring are the same as the original normal subgroups. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂)
Theorem	opprlidlabs 33492	The ideals of the opposite ring's opposite ring are the ideals of the original ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
Theorem	oppr2idl 33493	Two sided ideal of the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
Theorem	opprmxidlabs 33494	The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    ⇒   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))
Theorem	opprqusbas 33495	The base of the quotient of the opposite ring is the same as the base of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
Theorem	opprqusplusg 33496	The group operation of the quotient of the opposite ring is the same as the group operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   ⊢ 𝐸 = (Base‘𝑄)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    ⇒   ⊢ (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
Theorem	opprqus0g 33497	The group identity element of the quotient of the opposite ring is the same as the group identity element of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    ⇒   ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
Theorem	opprqusmulr 33498	The multiplication operation of the quotient of the opposite ring is the same as the multiplication operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐸 = (Base‘𝑄)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    ⇒   ⊢ (𝜑 → (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
Theorem	opprqus1r 33499	The ring unity of the quotient of the opposite ring is the same as the ring unity of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
Theorem	opprqusdrng 33500	The quotient of the opposite ring is a division ring iff the opposite of the quotient ring is. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))