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Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlspprat 19201* A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if 𝑧 is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ∃𝑧𝑉 𝑈 = (𝑁‘{𝑧}))

Theoremislbs2 19202* An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LVec → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))))

Theoremislbs3 19203* An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LVec → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑠(𝑠𝐵 → (𝑁𝑠) ⊊ 𝑉))))

Theoremlbsacsbs 19204 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 19202. (Contributed by David Moews, 1-May-2017.)
𝐴 = (LSubSp‘𝑊)    &   𝑁 = (mrCls‘𝐴)    &   𝑋 = (Base‘𝑊)    &   𝐼 = (mrInd‘𝐴)    &   𝐽 = (LBasis‘𝑊)       (𝑊 ∈ LVec → (𝑆𝐽 ↔ (𝑆𝐼 ∧ (𝑁𝑆) = 𝑋)))

Theoremlvecdim 19205 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 19192 and lbsacsbs 19204 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 17230. (Contributed by David Moews, 1-May-2017.)
𝐽 = (LBasis‘𝑊)       ((𝑊 ∈ LVec ∧ 𝑆𝐽𝑇𝐽) → 𝑆𝑇)

Theoremlbsextlem1 19206* Lemma for lbsext 19211. The set 𝑆 is the set of all linearly independent sets containing 𝐶; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐶𝑉)    &   (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}       (𝜑𝑆 ≠ ∅)

Theoremlbsextlem2 19207* Lemma for lbsext 19211. Since 𝐴 is a chain (actually, we only need it to be closed under binary union), the union 𝑇 of the spans of each individual element of 𝐴 is a subspace, and it contains all of 𝐴 (except for our target vector 𝑥- we are trying to make 𝑥 a linear combination of all the other vectors in some set from 𝐴). (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐶𝑉)    &   (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   𝑃 = (LSubSp‘𝑊)    &   (𝜑𝐴𝑆)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → [] Or 𝐴)    &   𝑇 = 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))       (𝜑 → (𝑇𝑃 ∧ ( 𝐴 ∖ {𝑥}) ⊆ 𝑇))

Theoremlbsextlem3 19208* Lemma for lbsext 19211. A chain in 𝑆 has an upper bound in 𝑆. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐶𝑉)    &   (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   𝑃 = (LSubSp‘𝑊)    &   (𝜑𝐴𝑆)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → [] Or 𝐴)    &   𝑇 = 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))       (𝜑 𝐴𝑆)

Theoremlbsextlem4 19209* Lemma for lbsext 19211. lbsextlem3 19208 satisfies the conditions for the application of Zorn's lemma zorn 9367 (thus invoking AC), and so there is a maximal linearly independent set extending 𝐶. Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐶𝑉)    &   (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   (𝜑 → 𝒫 𝑉 ∈ dom card)       (𝜑 → ∃𝑠𝐽 𝐶𝑠)

Theoremlbsextg 19210* For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 17-May-2015.)
𝐽 = (LBasis‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (((𝑊 ∈ LVec ∧ 𝒫 𝑉 ∈ dom card) ∧ 𝐶𝑉 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠𝐽 𝐶𝑠)

Theoremlbsext 19211* For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝐽 = (LBasis‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LVec ∧ 𝐶𝑉 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠𝐽 𝐶𝑠)

Theoremlbsexg 19212 Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.)
𝐽 = (LBasis‘𝑊)       ((CHOICE𝑊 ∈ LVec) → 𝐽 ≠ ∅)

Theoremlbsex 19213 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝐽 = (LBasis‘𝑊)       (𝑊 ∈ LVec → 𝐽 ≠ ∅)

Theoremlvecprop2d 19214* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 19215 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   𝐹 = (Scalar‘𝐾)    &   𝐺 = (Scalar‘𝐿)    &   (𝜑𝑃 = (Base‘𝐹))    &   (𝜑𝑃 = (Base‘𝐺))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(.r𝐹)𝑦) = (𝑥(.r𝐺)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))       (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))

Theoremlvecpropd 19215* If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑𝐹 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   𝑃 = (Base‘𝐹)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))       (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))

10.8  Ideals

10.8.1  The subring algebra; ideals

Syntaxcsra 19216 Extend class notation with the subring algebra generator.
class subringAlg

Syntaxcrglmod 19217 Extend class notation with the left module induced by a ring over itself.
class ringLMod

Syntaxclidl 19218 Ring left-ideal function.
class LIdeal

Syntaxcrsp 19219 Ring span function.
class RSpan

Definitiondf-sra 19220* Given any subring of a ring, we can construct a left-algebra by regarding the elements of the subring as scalars and the ring itself as a set of vectors. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)))

Definitiondf-rgmod 19221 Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))

Definitiondf-lidl 19222 Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. (Contributed by Stefan O'Rear, 31-Mar-2015.)
LIdeal = (LSubSp ∘ ringLMod)

Definitiondf-rsp 19223 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
RSpan = (LSpan ∘ ringLMod)

Theoremsraval 19224 Lemma for srabase 19226 through sravsca 19230. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))

Theoremsralem 19225 Lemma for srabase 19226 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝑁 < 5 ∨ 8 < 𝑁)       (𝜑 → (𝐸𝑊) = (𝐸𝐴))

Theoremsrabase 19226 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (Base‘𝑊) = (Base‘𝐴))

Theoremsraaddg 19227 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (+g𝑊) = (+g𝐴))

Theoremsramulr 19228 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = (.r𝐴))

Theoremsrasca 19229 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (𝑊s 𝑆) = (Scalar‘𝐴))

Theoremsravsca 19230 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = ( ·𝑠𝐴))

Theoremsraip 19231 The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = (·𝑖𝐴))

Theoremsratset 19232 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))

Theoremsratopn 19233 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (TopOpen‘𝑊) = (TopOpen‘𝐴))

Theoremsrads 19234 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (dist‘𝑊) = (dist‘𝐴))

Theoremsralmod 19235 The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)       (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Theoremsralmod0 19236 The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑0 = (0g𝑊))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑0 = (0g𝐴))

Theoremissubrngd2 19237* Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑆 = (𝐼s 𝐷))    &   (𝜑0 = (0g𝐼))    &   (𝜑+ = (+g𝐼))    &   (𝜑𝐷 ⊆ (Base‘𝐼))    &   (𝜑0𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ((𝜑𝑥𝐷) → ((invg𝐼)‘𝑥) ∈ 𝐷)    &   (𝜑1 = (1r𝐼))    &   (𝜑· = (.r𝐼))    &   (𝜑1𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 · 𝑦) ∈ 𝐷)    &   (𝜑𝐼 ∈ Ring)       (𝜑𝐷 ∈ (SubRing‘𝐼))

Theoremrlmfn 19238 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod Fn V

Theoremrlmval 19239 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))

Theoremlidlval 19240 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊))

Theoremrspval 19241 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊))

Theoremrlmval2 19242 Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝑊𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))

Theoremrlmbas 19243 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(Base‘𝑅) = (Base‘(ringLMod‘𝑅))

Theoremrlmplusg 19244 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(+g𝑅) = (+g‘(ringLMod‘𝑅))

Theoremrlm0 19245 Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
(0g𝑅) = (0g‘(ringLMod‘𝑅))

Theoremrlmsub 19246 Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
(-g𝑅) = (-g‘(ringLMod‘𝑅))

Theoremrlmmulr 19247 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(.r𝑅) = (.r‘(ringLMod‘𝑅))

Theoremrlmsca 19248 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
(𝑅𝑋𝑅 = (Scalar‘(ringLMod‘𝑅)))

Theoremrlmsca2 19249 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))

Theoremrlmvsca 19250 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(.r𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅))

Theoremrlmtopn 19251 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))

Theoremrlmds 19252 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(dist‘𝑅) = (dist‘(ringLMod‘𝑅))

Theoremrlmlmod 19253 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
(𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)

Theoremrlmlvec 19254 The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ DivRing → (ringLMod‘𝑅) ∈ LVec)

Theoremrlmvneg 19255 Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
(invg𝑅) = (invg‘(ringLMod‘𝑅))

Theoremrlmscaf 19256 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅))

Theoremixpsnbasval 19257* The value of an infinite Cartesian product of the base of a left module over a ring with a singleton. (Contributed by AV, 3-Dec-2018.)
((𝑅𝑉𝑋𝑊) → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓𝑋) ∈ (Base‘𝑅))})

Theoremlidlss 19258 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐵 = (Base‘𝑊)    &   𝐼 = (LIdeal‘𝑊)       (𝑈𝐼𝑈𝐵)

Theoremislidl 19259* Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       (𝐼𝑈 ↔ (𝐼𝐵𝐼 ≠ ∅ ∧ ∀𝑥𝐵𝑎𝐼𝑏𝐼 ((𝑥 · 𝑎) + 𝑏) ∈ 𝐼))

Theoremlidl0cl 19260 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → 0𝐼)

Theoremlidlacl 19261 An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    + = (+g𝑅)       (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ (𝑋𝐼𝑌𝐼)) → (𝑋 + 𝑌) ∈ 𝐼)

Theoremlidlnegcl 19262 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝑁 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋𝐼) → (𝑁𝑋) ∈ 𝐼)

Theoremlidlsubg 19263 An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → 𝐼 ∈ (SubGrp‘𝑅))

Theoremlidlsubcl 19264 An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝑈 = (LIdeal‘𝑅)    &    = (-g𝑅)       (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ (𝑋𝐼𝑌𝐼)) → (𝑋 𝑌) ∈ 𝐼)

Theoremlidlmcl 19265 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ (𝑋𝐵𝑌𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)

Theoremlidl1el 19266 An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ( 1𝐼𝐼 = 𝐵))

Theoremlidl0 19267 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } ∈ 𝑈)

Theoremlidl1 19268 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑈)

Theoremlidlacs 19269 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐵 = (Base‘𝑊)    &   𝐼 = (LIdeal‘𝑊)       (𝑊 ∈ Ring → 𝐼 ∈ (ACS‘𝐵))

Theoremrspcl 19270 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 ∧ 𝐺𝐵) → (𝐾𝐺) ∈ 𝑈)

Theoremrspssid 19271 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 ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))

Theoremrsp1 19272 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐾‘{ 1 }) = 𝐵)

Theoremrsp0 19273 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 })

Theoremrspssp 19274 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 ∧ 𝐼𝑈𝐺𝐼) → (𝐾𝐺) ⊆ 𝐼)

Theoremmrcrsp 19275 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐹 = (mrCls‘𝑈)       (𝑅 ∈ Ring → 𝐾 = 𝐹)

Theoremlidlnz 19276* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐼 ≠ { 0 }) → ∃𝑥𝐼 𝑥0 )

Theoremdrngnidl 19277 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ DivRing → 𝑈 = {{ 0 }, 𝐵})

Theoremlidlrsppropd 19278* 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿)))

10.8.2  Two-sided ideals and quotient rings

Syntaxc2idl 19279 Ring two-sided ideal function.
class 2Ideal

Definitiondf-2idl 19280 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‘(oppr𝑟))))

Theorem2idlval 19281 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝐽 = (LIdeal‘𝑂)    &   𝑇 = (2Ideal‘𝑅)       𝑇 = (𝐼𝐽)

Theorem2idlcpbl 19282 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → ((𝐴𝐸𝐶𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))

Theoremqus1 19283 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r𝑈)))

Theoremqusring 19284 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 ∧ 𝑆𝐼) → 𝑈 ∈ Ring)

Theoremqusrhm 19285* 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 = (𝑥𝑋 ↦ [𝑥](𝑅 ~QG 𝑆))       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))

Theoremcrngridl 19286 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr𝑅)       (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂))

Theoremcrng2idl 19287 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))

Theoremquscrng 19288 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (LIdeal‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑆𝐼) → 𝑈 ∈ CRing)

10.8.3  Principal ideal rings. Divisibility in the integers

Syntaxclpidl 19289 Ring left-principal-ideal function.
class LPIdeal

Syntaxclpir 19290 Class of left principal ideal rings.
class LPIR

Definitiondf-lpidl 19291* Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal = (𝑤 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})

Definitiondf-lpir 19292 Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}

Theoremlpival 19293* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})

Theoremislpidl 19294* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))

Theoremlpi0 19295 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } ∈ 𝑃)

Theoremlpi1 19296 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑃)

Theoremislpir 19297 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))

Theoremlpiss 19298 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ Ring → 𝑃𝑈)

Theoremislpir2 19299 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈𝑃))

Theoremlpirring 19300 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR → 𝑅 ∈ Ring)

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