P. 381 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38001-38100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	crngorngo 38001	A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
Theorem	crngocom 38002	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 38003	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 38004	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 38005	A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
Theorem	isfld2 38006	The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
Theorem	crngohomfo 38007	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 38008	Extend class notation with the class of ideals.
class Idl
Syntax	cpridl 38009	Extend class notation with the class of prime ideals.
class PrIdl
Syntax	cmaxidl 38010	Extend class notation with the class of maximal ideals.
class MaxIdl
Definition	df-idl 38011*	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.)
⊢ 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 38012*	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 38039 and ispridlc 38071. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 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 38013*	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.)
⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))})
Theorem	idlval 38014*	The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 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 38015*	The predicate "is an ideal of the ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 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 38016*	The predicate "is an ideal of the commutative ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 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 38017	An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)
Theorem	idlcl 38018	An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝑋)
Theorem	idl0cl 38019	An ideal contains 0. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝐼)
Theorem	idladdcl 38020	An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)
Theorem	idllmulcl 38021	An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)
Theorem	idlrmulcl 38022	An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)
Theorem	idlnegcl 38023	An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑁 = (inv‘𝐺)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)
Theorem	idlsubcl 38024	An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)
Theorem	rngoidl 38025	A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
Theorem	0idl 38026	The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
Theorem	1idl 38027	Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))
Theorem	0rngo 38028	In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))
Theorem	divrngidl 38029	The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})
Theorem	intidl 38030	The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ∈ (Idl‘𝑅))
Theorem	inidl 38031	The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅))
Theorem	unichnidl 38032*	The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ∈ (Idl‘𝑅))
Theorem	keridl 38033	The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑆)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅))
Theorem	pridlval 38034*	The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 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 38035*	The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 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 38036	A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))
Theorem	pridlnr 38037	A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ≠ 𝑋)
Theorem	pridl 38038*	The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 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 38039*	A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38071 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 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 38040*	The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ 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 38041*	The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ 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 38042	A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
Theorem	maxidlnr 38043	A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋)
Theorem	maxidlmax 38044	A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))
Theorem	maxidln1 38045	One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀)
Theorem	maxidln0 38046	A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 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 38047	Extend class notation with the class of prime rings.
class PrRing
Syntax	cdmn 38048	Extend class notation with the class of domains.
class Dmn
Definition	df-prrngo 38049	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.)
⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)}
Definition	df-dmn 38050	Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ Dmn = (PrRing ∩ Com2)
Theorem	isprrngo 38051	The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
Theorem	prrngorngo 38052	A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
Theorem	smprngopr 38053	A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 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 38054	A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)
Theorem	isdmn 38055	The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
Theorem	isdmn2 38056	The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
Theorem	dmncrng 38057	A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
Theorem	dmnrngo 38058	A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)
Theorem	flddmn 38059	A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn)
21.24.22  Ideal generators
Syntax	cigen 38060	Extend class notation with the ideal generation function.
class IdlGen
Definition	df-igen 38061*	Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
Theorem	igenval 38062*	The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
Theorem	igenss 38063	A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
Theorem	igenidl 38064	The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
Theorem	igenmin 38065	The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
Theorem	igenidl2 38066	The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)
Theorem	igenval2 38067*	The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 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 38068*	A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)})
Theorem	isfldidl 38069	Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ 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 38070	Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝐺 = (1st ‘𝐾)    &   ⊢ 𝐻 = (2nd ‘𝐾)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Theorem	ispridlc 38071*	The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 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 38072	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))
Theorem	pridlc2 38073	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃)
Theorem	pridlc3 38074	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ 𝑃))
Theorem	isdmn3 38075*	The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
Theorem	dmnnzd 38076	A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))
Theorem	dmncan1 38077	Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))
Theorem	dmncan2 38078	Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))
21.25  Mathbox for Giovanni Mascellani
21.25.1  Tools for automatic proof building The results in this section are mostly meant for being used by automatic proof building programs. As a result, they might appear less useful or meaningful than others to human beings.
Theorem	efald2 38079	A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (¬ 𝜑 → ⊥)    ⇒   ⊢ 𝜑
Theorem	notbinot1 38080	Simplification rule of negation across a biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (¬ (¬ 𝜑 ↔ 𝜓) ↔ (𝜑 ↔ 𝜓))
Theorem	bicontr 38081	Biconditional of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥)
Theorem	impor 38082	An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ 𝜒))
Theorem	orfa 38083	The falsum ⊥ can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((𝜑 ∨ ⊥) ↔ 𝜑)
Theorem	notbinot2 38084	Commutation rule between negation and biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (¬ (𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ 𝜓))
Theorem	biimpor 38085	A rewriting rule for biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((¬ 𝜑 ↔ 𝜓) ∨ 𝜒))
Theorem	orfa1 38086	Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ ⊥) → 𝜓)
Theorem	orfa2 38087	Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ⊥)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → 𝜓)
Theorem	bifald 38088	Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ ⊥))
Theorem	orsild 38089	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	orsird 38090	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	cnf1dd 38091	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒))    &   ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	cnf2dd 38092	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ¬ 𝜃))    &   ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	cnfn1dd 38093	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → (¬ 𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	cnfn2dd 38094	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜓 → (𝜒 ∨ ¬ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	or32dd 38095	A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃) ∨ 𝜏)))    ⇒   ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) ∨ 𝜃)))
Theorem	notornotel1 38096	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	notornotel2 38097	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → ¬ (𝜓 ∨ ¬ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	contrd 38098	A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (¬ 𝜓 → 𝜒))    &   ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	an12i 38099	An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
⊢ (𝜑 ∧ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 ∧ (𝜑 ∧ 𝜒))
Theorem	exmid2 38100	An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ ((𝜓 ∧ 𝜑) → 𝜒)    &   ⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜂) → 𝜒)