P. 387 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38601-38700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rngohom0 38601	A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑊 = (GId‘𝐽)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊)
Theorem	rngohomsub 38602	Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐻 = ( /𝑔 ‘𝐺)    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝐾 = ( /𝑔 ‘𝐽)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
Theorem	rngohomco 38603	The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsHom 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇))
Theorem	rngokerinj 38604	A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑊 = (GId‘𝐺)    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑌 = ran 𝐽    &   ⊢ 𝑍 = (GId‘𝐽)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑍}) = {𝑊}))
Definition	df-rngoiso 38605*	Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)})
Theorem	rngoisoval 38606*	The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑌 = ran 𝐽    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})
Theorem	isrngoiso 38607	The predicate "is a ring isomorphism between 𝑅 and 𝑆". (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑌 = ran 𝐽    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌)))
Theorem	rngoiso1o 38608	A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑌 = ran 𝐽    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
Theorem	rngoisohom 38609	A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
Theorem	rngoisocnv 38610	The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → ◡𝐹 ∈ (𝑆 RingOpsIso 𝑅))
Theorem	rngoisoco 38611	The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsIso 𝑇))
Definition	df-risc 38612*	Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
Theorem	isriscg 38613*	The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))
Theorem	isrisc 38614*	The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝑅 ∈ V    &   ⊢ 𝑆 ∈ V    ⇒   ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
Theorem	risc 38615*	The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
Theorem	risci 38616	Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅 ≃𝑟 𝑆)
Theorem	riscer 38617	Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ≃𝑟 Er dom ≃𝑟
21.24.19  Commutative rings
Syntax	ccm2 38618	Extend class notation with a class that adds commutativity to various flavors of rings.
class Com2
Definition	df-com2 38619*	A device to add commutativity to various sorts of rings. I use ran 𝑔 because I suppose 𝑔 has a neutral element and therefore is onto. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
⊢ Com2 = {⟨𝑔, ℎ⟩ ∣ ∀𝑎 ∈ ran 𝑔∀𝑏 ∈ ran 𝑔(𝑎ℎ𝑏) = (𝑏ℎ𝑎)}
Syntax	cfld 38620	Extend class notation with the class of all fields.
class Fld
Definition	df-fld 38621	Definition of a field. A field is a commutative division ring. (Contributed by FL, 6-Sep-2009.) (Revised by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
⊢ Fld = (DivRingOps ∩ Com2)
Syntax	ccring 38622	Extend class notation with the class of commutative rings.
class CRingOps
Definition	df-crngo 38623	Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ CRingOps = (RingOps ∩ Com2)
Theorem	iscom2 38624*	A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
Theorem	iscrngo 38625	The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
Theorem	iscrngo2 38626*	The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
Theorem	iscringd 38627*	Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ (𝜑 → 𝐺 ∈ AbelOp)    &   ⊢ (𝜑 → 𝑋 = ran 𝐺)    &   ⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))    ⇒   ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ CRingOps)
Theorem	flddivrng 38628	A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
Theorem	crngorngo 38629	A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
Theorem	crngocom 38630	The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))
Theorem	crngm23 38631	Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
Theorem	crngm4 38632	Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
Theorem	fldcrngo 38633	A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
Theorem	isfld2 38634	The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
Theorem	crngohomfo 38635	The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐽 = (1st ‘𝑆)    &   ⊢ 𝑌 = ran 𝐽    ⇒   ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ CRingOps)
21.24.20  Ideals
Syntax	cidl 38636	Extend class notation with the class of ideals.
class Idl
Syntax	cpridl 38637	Extend class notation with the class of prime ideals.
class PrIdl
Syntax	cmaxidl 38638	Extend class notation with the class of maximal ideals.
class MaxIdl
Definition	df-idl 38639*	Obsolete defintion, use df-2idl 21370 (or df-lidl 21313) instead. Define the class of (two-sided) ideals of a ring 𝑅. A subset of 𝑅 is an ideal if it contains 0, is closed under addition, and is closed under multiplication on either side by any element of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
⊢ Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1st ‘𝑟) ∣ ((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)))})
Definition	df-pridl 38640*	Obsolete defintion, use df-prmidl 21442 instead. 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 ispridl2 38667 and ispridlc 38699. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
⊢ PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
Definition	df-maxidl 38641*	Obsolete defintion, use df-mxidl 33721 instead. 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.) (New usage is discouraged.)
⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))})
Theorem	idlval 38642*	Obsolete theorem, use 2idlval 21371 instead. The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
Theorem	isidl 38643*	Obsolete theorem, use df2idl2 21377 instead. The predicate "is an ideal of the ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
Theorem	isidlc 38644*	Obsolete theorem, use df2idl2crng 21402 instead. The predicate "is an ideal of the commutative ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ CRingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼))))
Theorem	idlss 38645	Obsolete theorem, use 2idlss 21382 instead. An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)
Theorem	idlcl 38646	Obsolete theorem, use 2idllidld 21374 and lidlbasel 21318 instead. An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝑋)
Theorem	idl0cl 38647	Obsolete theorem, use ringrng 20369 and rng2idl0 21387 instead. An ideal contains 0. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝐼)
Theorem	idladdcl 38648	Obsolete theorem, use 2idllidld 21374 and lidlacl 21327 instead. An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)
Theorem	idllmulcl 38649	Obsolete theorem, use 2idllidld 21374 and lidlmcl 21331 instead. An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)
Theorem	idlrmulcl 38650	Obsolete theorem, use 2idlridld 21375 and lidlmcl 21331 instead. An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)
Theorem	idlnegcl 38651	Obsolete theorem, use 2idllidld 21374 and lidlnegcl 21328 instead. An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑁 = (inv‘𝐺)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)
Theorem	idlsubcl 38652	Obsolete theorem, use 2idllidld 21374 and lidlsubcl 21330 instead. An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)
Theorem	rngoidl 38653	Obsolete theorem, use 2idl1 21381 instead. A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
Theorem	0idl 38654	Obsolete theorem, use 2idl0 21380 instead. The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
Theorem	1idl 38655	Obsolete theorem, use 2idllidld 21374 and lidl1el 21332 instead. Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))
Theorem	0rngo 38656	Obsolete theorem, use 0ring01eqbi2 20617 instead. In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))
Theorem	divrngidl 38657	Obsolete theorem, use drngnidl 21358 instead. The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})
Theorem	intidl 38658	Obsolete theorem, use intlidl 33706 instead. The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ∈ (Idl‘𝑅))
Theorem	inidl 38659	Obsolete theorem, use inlidl 33707 instead. The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅))
Theorem	unichnidl 38660*	Obsolete theorem, use unichnlidl 21343 instead. The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ∈ (Idl‘𝑅))
Theorem	keridl 38661	Obsolete theorem, use kerlidl 21398 instead. The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑆)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅))
Theorem	pridlval 38662*	Obsolete theorem, use prmidlval 21443 instead. The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
Theorem	ispridl 38663*	Obsolete theorem, use isprmidl 21444 instead. The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
Theorem	pridlidl 38664	Obsolete theorem, use prmidlidl 21450 instead. A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))
Theorem	pridlnr 38665	Obsolete theorem, use prmidlnr 21445 instead. A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ≠ 𝑋)
Theorem	pridl 38666*	Obsolete theorem, use isprmidlc 21453 instead. The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐻 = (2nd ‘𝑅)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃))
Theorem	ispridl2 38667*	Obsolete theorem, use prmidl2 21447 instead. A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38699 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))
Theorem	maxidlval 38668*	Obsolete theorem, use mxidlval 33722 instead. The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))})
Theorem	ismaxidl 38669*	Obsolete theorem, use ismxidl 33723 instead. The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))))
Theorem	maxidlidl 38670	Obsolete theorem, use mxidlidl 33724 instead. A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
Theorem	maxidlnr 38671	Obsolete theorem, use mxidlnr 33725 instead. A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋)
Theorem	maxidlmax 38672	Obsolete theorem, use mxidlmax 33726 instead. A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))
Theorem	maxidln1 38673	Obsolete theorem, use mxidln1 33727 instead. One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)
Theorem	maxidln0 38674	Obsolete theorem, use mxidlnzr 33728 instead. A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍)
21.24.21  Prime rings and integral domains
Syntax	cprrng 38675	Extend class notation with the class of prime rings.
class PrRing
Syntax	cdmn 38676	Extend class notation with the class of domains.
class Dmn
Definition	df-prrngo 38677	Obsolete definition, use df-prmring 49077 instead. Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)}
Definition	df-dmn 38678	Obsolete definition, use df-idom 20782 resp. dfidom2 49085 instead. Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
⊢ Dmn = (PrRing ∩ Com2)
Theorem	isprrngo 38679	Obsolete theorem, use isprmrng 49078 instead. The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
Theorem	prrngorngo 38680	Obsolete theorem, use prmrngring 49080 instead. A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
Theorem	smprngopr 38681	Obsolete theorem, use smprngprmrng 49081 instead. A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)
Theorem	divrngpr 38682	Obsolete theorem, use drngprmrng 49082 instead. A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)
Theorem	isdmn 38683	Obsolete theorem, use isidom2 49086 instead. The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
Theorem	isdmn2 38684	Obsolete theorem, use isidom2 49086 instead. The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
Theorem	dmncrng 38685	Obsolete theorem, use idomcringd 20812 instead. A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
Theorem	dmnrngo 38686	Obsolete theorem, use idomringd 20813 instead. A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)
Theorem	flddmn 38687	Obsolete theorem, use fldidom 20856 instead. A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn)
21.24.22  Ideal generators
Syntax	cigen 38688	Extend class notation with the ideal generation function.
class IdlGen
Definition	df-igen 38689*	Obsolete definition, use df-rsp 21314 instead. Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
Theorem	igenval 38690*	Obsolete theorem, use rspvalint 21350 instead. The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
Theorem	igenss 38691	Obsolete theorem, use rspssid 21346 instead. A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
Theorem	igenidl 38692	Obsolete theorem, use rspcl 21345 instead. The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
Theorem	igenmin 38693	Obsolete theorem, use rspssp 21349 instead. The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
Theorem	igenidl2 38694	Obsolete theorem, use rspidlid 33667 instead. The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)
Theorem	igenval2 38695*	Obsolete theorem, use rspprop 21351 instead. The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))
Theorem	prnc 38696*	Obsolete theorem, use rspsn0 21353 instead (holds for arbitrary rings). A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
Theorem	isfldidl 38697	Obsolete theorem, use isfieldidl 21367 instead. Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝐾)    &   ⊢ 𝐻 = (2nd ‘𝐾)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Theorem	isfldidl2 38698	Obsolete theorem, use isfieldidl2 21368 instead. Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝐾)    &   ⊢ 𝐻 = (2nd ‘𝐾)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Theorem	ispridlc 38699*	Obsolete theorem, use isprmidlc 21453 instead. The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))))
Theorem	pridlc 38700	Obsolete theorem, use prmidlc 21454 instead. Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))