P. 340 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nsgqusf1o 33901*	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 33902*	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 33903	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)
21.3.10.39  Ideals
Theorem	intlidl 33904	The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 8-Jun-2024.)
⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∩ 𝐶 ∈ (LIdeal‘𝑅))
Theorem	inlidl 33905	The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 28-Jun-2026.)
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (LIdeal‘𝑅))
Theorem	pidlnzb 33906	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 33907	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 33908	The ideal 𝐼 generated by an element 𝑋 of a commutative ring 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 ∈ CRing)    ⇒   ⊢ (𝜑 → (𝐼 = 𝐵 ↔ 𝑋 ∈ 𝑈))
Theorem	rhmquskerlem 33909*	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 33910*	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 33911	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 33912*	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 33913*	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 33914	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 33915	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 33916	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	drngidlhash 33917	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))
21.3.10.40  Maximal Ideals
Syntax	cmxidl 33918	Extend class notation with the class of maximal ideals.
class MaxIdeal
Definition	df-mxidl 33919*	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 33920*	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 33921*	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 33922	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 33923	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 33924	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 33925	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 33926	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 33927	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 33928	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 33929	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 33930	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 33931	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 33932*	The set 𝑃 used in the proof of ssmxidl 33933 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 33933*	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	drng0mxidl 33934	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 33935	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 33936	If a ring's only maximal ideal is the zero ideal, it is a division ring. See also drngmxidl 33935. (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 33937*	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 33938*	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 33939*	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 33940	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 33941	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 33942	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 33943	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 33944	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 33945	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 33946	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 33947	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 33948	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 33949	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 33950	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 33951	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))
Theorem	qsdrngilem 33952*	Lemma for qsdrngi 33953. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄))
Theorem	qsdrngi 33953	A quotient by a maximal left and maximal right ideal is a division ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))    ⇒   ⊢ (𝜑 → 𝑄 ∈ DivRing)
Theorem	qsdrnglem2 33954	Lemma for qsdrng 33955. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ DivRing)    &   ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑀 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐽 ∖ 𝑀))    ⇒   ⊢ (𝜑 → 𝐽 = 𝐵)
Theorem	qsdrng 33955	An ideal 𝑀 is both left and right maximal if and only if the factor ring 𝑄 is a division ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))    ⇒   ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))
Theorem	qsfld 33956	An ideal 𝑀 in the commutative ring 𝑅 is maximal if and only if the factor ring 𝑄 is a field. (Contributed by Thierry Arnoux, 13-Mar-2025.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝑄 ∈ Field ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
Theorem	mxidlprmALT 33957	Every maximal ideal is prime - alternative proof. (Contributed by Thierry Arnoux, 15-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    ⇒   ⊢ (𝜑 → 𝑀 ∈ (PrmIdeal‘𝑅))
21.3.10.41  Local rings
Theorem	drnglring 33958	A division ring is a local ring. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ (𝜑 → 𝐹 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝐹 ∈ LRing)
Theorem	dflring2 33959*	Alternate definition of a local ring. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝑈 ∨ ( 1 − 𝑥) ∈ 𝑈)))
Theorem	dflringlem 33960	Lemma for dflring3 33963. If a ring 𝑅 has a single maximal ideal 𝑀, then any element 𝑋 outside of 𝑀 is a unit. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    &   ⊢ (𝜑 → (MaxIdeal‘𝑅) = {𝑀})    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑀))    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑈)
Theorem	dflringlem2 33961	Lemma for dflring3 33963. In a commutative local ring 𝑅, the set (𝐵 ∖ 𝑈) of non-units is an ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ LRing)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
Theorem	dflringlem3 33962	Lemma for dflring3 33963. In a commutative local ring 𝑅, the set (𝐵 ∖ 𝑈) of non-units is a maximal ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ LRing)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑈) ∈ (MaxIdeal‘𝑅))
Theorem	dflring3 33963	Alternate definition of a local ring: local rings have a single maximal ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1o))
Theorem	dflring4 33964	Alternate definition of a local ring: the set (𝐵 ∖ 𝑈) of non-units is an ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)))
Theorem	fldlring 33965	A field is a local ring. (Contributed by Thierry Arnoux, 3-Jun-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐹 ∈ Field)    ⇒   ⊢ (𝜑 → 𝐹 ∈ LRing)
21.3.10.42  The semiring of ideals of a ring
Syntax	cidlsrg 33966	Extend class notation with the semiring of ideals of a ring.
class IDLsrg
Definition	df-idlsrg 33967*	Define a structure for the ideals of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ IDLsrg = (𝑟 ∈ V ↦ ⦋(LIdeal‘𝑟) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩}))
Theorem	idlsrgstr 33968	A constructed semiring of ideals is a structure. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩})    ⇒   ⊢ 𝑊 Struct ⟨1, ;10⟩
Theorem	idlsrgval 33969*	Lemma for idlsrgbas 33970 through idlsrgtset 33974. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ⊗ = (LSSum‘𝐺)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩}))
Theorem	idlsrgbas 33970	Base of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐼 = (Base‘𝑆))
Theorem	idlsrgplusg 33971	Additive operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆))
Theorem	idlsrg0g 33972	The zero ideal is the additive identity of the semiring of ideals. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } = (0g‘𝑆))
Theorem	idlsrgmulr 33973*	Multiplicative operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ⊗ = (LSSum‘𝐺)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑖 ∈ 𝐵, 𝑗 ∈ 𝐵 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗))) = (.r‘𝑆))
Theorem	idlsrgtset 33974*	Topology component of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐽 = (TopSet‘𝑆))
Theorem	idlsrgmulrval 33975	Value of the ring multiplication for the ideals of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ · = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼 · 𝐽)))
Theorem	idlsrgmulrcl 33976	Ideals of a ring 𝑅 are closed under multiplication. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ∈ 𝐵)
Theorem	idlsrgmulrss1 33977	In a commutative ring, the product of two ideals is a subset of the first one. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐼)
Theorem	idlsrgmulrss2 33978	The product of two ideals is a subset of the second one. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐽)
Theorem	idlsrgmulrssin 33979	In a commutative ring, the product of two ideals is a subset of their intersection. (Contributed by Thierry Arnoux, 17-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ (𝐼 ∩ 𝐽))
Theorem	idlsrgmnd 33980	The ideals of a ring form a monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Mnd)
Theorem	idlsrgcmnd 33981	The ideals of a ring form a commutative monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑆 ∈ CMnd)
21.3.10.43  Prime Elements
Theorem	rprmval 33982*	The prime elements of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
Theorem	isrprm 33983*	Property for 𝑃 to be a prime element in the ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))))
Theorem	rprmcl 33984	A ring prime is an element of the base set. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐵)
Theorem	rprmdvds 33985	If a ring prime 𝑄 divides a product 𝑋 · 𝑌, then it divides either 𝑋 or 𝑌. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∥ (𝑋 · 𝑌))    ⇒   ⊢ (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌))
Theorem	rprmnz 33986	A ring prime is nonzero. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝑄 ≠ 0 )
Theorem	rprmnunit 33987	A ring prime is not a unit. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ¬ 𝑄 ∈ 𝑈)
Theorem	rsprprmprmidl 33988	In a commutative ring, ideals generated by prime elements are prime ideals. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ (RPrime‘𝑅))    ⇒   ⊢ (𝜑 → (𝐾‘{𝑃}) ∈ (PrmIdeal‘𝑅))
Theorem	rsprprmprmidlb 33989	An ideal generated by a single element is a prime iff that element is prime. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑃 ↔ (𝐾‘{𝑋}) ∈ (PrmIdeal‘𝑅)))
Theorem	rprmndvdsr1 33990	A ring prime element does not divide the ring multiplicative identity. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 1 = (1r‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ¬ 𝑄 ∥ 1 )
Theorem	rprmasso 33991	In an integral domain, the associate of a prime is a prime. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∥ 𝑌)    &   ⊢ (𝜑 → 𝑌 ∥ 𝑋)    ⇒   ⊢ (𝜑 → 𝑌 ∈ 𝑃)
Theorem	rprmasso2 33992	In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∥ 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝑌 ∥ 𝑋)
Theorem	rprmasso3 33993*	In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∥ 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ 𝑈 (𝑡 · 𝑋) = 𝑌)
Theorem	unitmulrprm 33994	A ring unit multiplied by a ring prime is a ring prime. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐼 · 𝑄) ∈ 𝑃)
Theorem	rprmndvdsru 33995	A ring prime element does not divide any ring unit. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ¬ 𝑄 ∥ 𝑇)
Theorem	rprmirredlem 33996	Lemma for rprmirred 33997. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑄 ≠ 0 )    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑈))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 = (𝑋 · 𝑌))    &   ⊢ (𝜑 → 𝑄 ∥ 𝑋)    ⇒   ⊢ (𝜑 → 𝑌 ∈ 𝑈)
Theorem	rprmirred 33997	In an integral domain, ring primes are irreducible. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑄 ∈ 𝐼)
Theorem	rprmirredb 33998	In a principal ideal domain, the converse of rprmirred 33997 holds, i.e. irreducible elements are prime. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ PID)    ⇒   ⊢ (𝜑 → 𝐼 = 𝑃)
Theorem	rprmdvdspow 33999	If a prime element divides a ring "power", it divides the term. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑄 ∥ (𝑁 ↑ 𝑋))    ⇒   ⊢ (𝜑 → 𝑄 ∥ 𝑋)
Theorem	rprmdvdsprod 34000*	If a prime element 𝑄 divides a product, then it divides one term. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 finSupp 1 )    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝑄 ∥ (𝑀 Σg 𝐹))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐹 supp 1 )𝑄 ∥ (𝐹‘𝑥))