P. 145 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14401-14500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rspvalg 14401	Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ (𝑊 ∈ 𝑉 → (RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊)))
Theorem	lidlex 14402	Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
⊢ (𝑊 ∈ 𝑉 → (LIdeal‘𝑊) ∈ V)
Theorem	rspex 14403	Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
⊢ (𝑊 ∈ 𝑉 → (RSpan‘𝑊) ∈ V)
Theorem	lidlmex 14404	Existence of the set a left ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
⊢ 𝐼 = (LIdeal‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝐼 → 𝑊 ∈ V)
Theorem	lidlss 14405	An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐼 = (LIdeal‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵)
Theorem	lidlssbas 14406	The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
Theorem	lidlbas 14407	A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
Theorem	islidlm 14408*	Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝐼 ∈ 𝑈 ↔ (𝐼 ⊆ 𝐵 ∧ ∃𝑗 𝑗 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐼 ((𝑥 · 𝑎) + 𝑏) ∈ 𝐼))
Theorem	rnglidlmcl 14409	A (left) ideal containing the zero element is closed under left-multiplication by elements of the full non-unital ring. If the ring is not a unital ring, and the ideal does not contain the zero element of the ring, then the closure cannot be proven. (Contributed by AV, 18-Feb-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ 0 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)
Theorem	dflidl2rng 14410*	Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼))
Theorem	isridlrng 14411*	A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
Theorem	lidl0cl 14412	An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 0 ∈ 𝐼)
Theorem	lidlacl 14413	An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 + 𝑌) ∈ 𝐼)
Theorem	lidlnegcl 14414	An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝑁‘𝑋) ∈ 𝐼)
Theorem	lidlsubg 14415	An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ (SubGrp‘𝑅))
Theorem	lidlsubcl 14416	An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 − 𝑌) ∈ 𝐼)
Theorem	dflidl2 14417*	Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼)))
Theorem	lidl0 14418	Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
Theorem	lidl1 14419	Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
Theorem	rspcl 14420	The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ∈ 𝑈)
Theorem	rspssid 14421	The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ (𝐾‘𝐺))
Theorem	rsp0 14422	The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 })
Theorem	rspssp 14423	The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ⊆ 𝐼) → (𝐾‘𝐺) ⊆ 𝐼)
Theorem	lidlrsppropdg 14424*	The left ideals and ring span of a ring depend only on the ring components. Here 𝑊 is expected to be either 𝐵 (when closure is available) or V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑌)    ⇒   ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿)))
Theorem	rnglidlmmgm 14425	The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm)
Theorem	rnglidlmsgrp 14426	The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
Theorem	rnglidlrng 14427	A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 𝑈 ∈ (SubGrp‘𝑅) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)
7.6.3  Two-sided ideals and quotient rings
Syntax	c2idl 14428	Ring two-sided ideal function.
class 2Ideal
Definition	df-2idl 14429	Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
Theorem	2idlmex 14430	Existence of the set a two-sided ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
⊢ 𝑇 = (2Ideal‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝑇 → 𝑊 ∈ V)
Theorem	2idlval 14431	Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝐽 = (LIdeal‘𝑂)    &   ⊢ 𝑇 = (2Ideal‘𝑅)    ⇒   ⊢ 𝑇 = (𝐼 ∩ 𝐽)
Theorem	2idlvalg 14432	Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝐽 = (LIdeal‘𝑂)    &   ⊢ 𝑇 = (2Ideal‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑇 = (𝐼 ∩ 𝐽))
Theorem	isridl 14433*	A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼)))
Theorem	2idlelb 14434	Membership in a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝐽 = (LIdeal‘𝑂)    &   ⊢ 𝑇 = (2Ideal‘𝑅)    ⇒   ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))
Theorem	2idllidld 14435	A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
Theorem	2idlridld 14436	A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑂))
Theorem	df2idl2rng 14437*	Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
⊢ 𝑈 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼)))
Theorem	df2idl2 14438*	Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
⊢ 𝑈 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼))))
Theorem	ridl0 14439	Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
Theorem	ridl1 14440	Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
Theorem	2idl0 14441	Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝐼)
Theorem	2idl1 14442	Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝐼)
Theorem	2idlss 14443	A two-sided ideal is a subset of the base set. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐼 = (2Ideal‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵)
Theorem	2idlbas 14444	The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 𝐵 = (Base‘𝐽)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐼)
Theorem	2idlelbas 14445	The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 𝐵 = (Base‘𝐽)    ⇒   ⊢ (𝜑 → (𝐵 ∈ (LIdeal‘𝑅) ∧ 𝐵 ∈ (LIdeal‘(oppr‘𝑅))))
Theorem	rng2idlsubrng 14446	A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
Theorem	rng2idlnsg 14447	A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
Theorem	rng2idl0 14448	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
Theorem	rng2idlsubgsubrng 14449	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
Theorem	rng2idlsubgnsg 14450	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
Theorem	rng2idlsubg0 14451	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
Theorem	2idlcpblrng 14452	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝑅 ~QG 𝑆)    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
Theorem	2idlcpbl 14453	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝑅 ~QG 𝑆)    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
Theorem	qus2idrng 14454	The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 14456 analog). (Contributed by AV, 23-Feb-2025.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng)
Theorem	qus1 14455	The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈)))
Theorem	qusring 14456	If 𝑆 is a two-sided ideal in 𝑅, then 𝑈 = 𝑅 / 𝑆 is a ring, called the quotient ring of 𝑅 by 𝑆. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
Theorem	qusrhm 14457*	If 𝑆 is a two-sided ideal in 𝑅, then the "natural map" from elements to their cosets is a ring homomorphism from 𝑅 to 𝑅 / 𝑆. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝑅 ~QG 𝑆))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))
Theorem	qusmul2 14458	Value of the ring operation in a quotient ring. (Contributed by Thierry Arnoux, 1-Sep-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
Theorem	crngridl 14459	In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂))
Theorem	crng2idl 14460	In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))
Theorem	qusmulrng 14461	Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14462. Similar to qusmul2 14458. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
⊢ ∼ = (𝑅 ~QG 𝑆)    &   ⊢ 𝐻 = (𝑅 /s ∼ )    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝐻)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	quscrng 14462	The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing)
7.6.4  Principal ideal rings. Divisibility in the integers
Theorem	rspsn 14463*	Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥})
7.7  The complex numbers as an algebraic extensible structure
7.7.1  Definition and basic properties
Syntax	cpsmet 14464	Extend class notation with the class of all pseudometric spaces.
class PsMet
Syntax	cxmet 14465	Extend class notation with the class of all extended metric spaces.
class ∞Met
Syntax	cmet 14466	Extend class notation with the class of all metrics.
class Met
Syntax	cbl 14467	Extend class notation with the metric space ball function.
class ball
Syntax	cfbas 14468	Extend class definition to include the class of filter bases.
class fBas
Syntax	cfg 14469	Extend class definition to include the filter generating function.
class filGen
Syntax	cmopn 14470	Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
Syntax	cmetu 14471	Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
Definition	df-psmet 14472*	Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
Definition	df-xmet 14473*	Define the set of all extended metrics on a given base set. The definition is similar to df-met 14474, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
Definition	df-met 14474*	Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. (Contributed by NM, 25-Aug-2006.)
⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
Definition	df-bl 14475*	Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
Definition	df-mopn 14476	Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.)
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Definition	df-fbas 14477*	Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
Definition	df-fg 14478*	Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
Definition	df-metu 14479*	Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
Theorem	blfn 14480	The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
⊢ ball Fn V
Theorem	mopnset 14481	Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
⊢ (𝐷 ∈ 𝑉 → (MetOpen‘𝐷) ∈ V)
Theorem	cndsex 14482	The standard distance function on the complex numbers is a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
⊢ (abs ∘ − ) ∈ V
Theorem	cntopex 14483	The standard topology on the complex numbers is a set. (Contributed by Jim Kingdon, 25-Sep-2025.)
⊢ (MetOpen‘(abs ∘ − )) ∈ V
Theorem	metuex 14484	Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
⊢ (𝐴 ∈ 𝑉 → (metUnif‘𝐴) ∈ V)
Syntax	ccnfld 14485	Extend class notation with the field of complex numbers.
class ℂfld
Definition	df-cnfld 14486*	The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator. The contract of this set is defined entirely by cnfldex 14488, cnfldadd 14491, cnfldmul 14493, cnfldcj 14494, cnfldtset 14495, cnfldle 14496, cnfldds 14497, and cnfldbas 14489. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) Use maps-to notation for addition and multiplication. (Revised by GG, 31-Mar-2025.) (New usage is discouraged.)
⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
Theorem	cnfldstr 14487	The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
⊢ ℂfld Struct ⟨1, ;13⟩
Theorem	cnfldex 14488	The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
⊢ ℂfld ∈ V
Theorem	cnfldbas 14489	The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
⊢ ℂ = (Base‘ℂfld)
Theorem	mpocnfldadd 14490*	The addition operation of the field of complex numbers. Version of cnfldadd 14491 using maps-to notation, which does not require ax-addf 8089. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
Theorem	cnfldadd 14491	The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
⊢ + = (+g‘ℂfld)
Theorem	mpocnfldmul 14492*	The multiplication operation of the field of complex numbers. Version of cnfldmul 14493 using maps-to notation, which does not require ax-mulf 8090. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
Theorem	cnfldmul 14493	The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
⊢ · = (.r‘ℂfld)
Theorem	cnfldcj 14494	The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.)
⊢ ∗ = (*𝑟‘ℂfld)
Theorem	cnfldtset 14495	The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 31-Mar-2025.)
⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
Theorem	cnfldle 14496	The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14486. (Revised by GG, 31-Mar-2025.)
⊢ ≤ = (le‘ℂfld)
Theorem	cnfldds 14497	The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14486. (Revised by GG, 31-Mar-2025.)
⊢ (abs ∘ − ) = (dist‘ℂfld)
Theorem	cncrng 14498	The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
⊢ ℂfld ∈ CRing
Theorem	cnring 14499	The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ ℂfld ∈ Ring
Theorem	cnfld0 14500	Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 0 = (0g‘ℂfld)