P. 149 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14801-14900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lsslmod 14801	A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LMod)
Theorem	lsslss 14802	The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑇 = (LSubSp‘𝑋)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑉 ∈ 𝑇 ↔ (𝑉 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑈)))
Theorem	islss4 14803*	A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝑆 ↔ (𝑈 ∈ (SubGrp‘𝑊) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝑈 (𝑎 · 𝑏) ∈ 𝑈)))
Theorem	lss1d 14804*	One-dimensional subspace (or zero-dimensional if 𝑋 is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∈ 𝑆)
Theorem	lssintclm 14805*	The intersection of an inhabited set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ⊆ 𝑆 ∧ ∃𝑤 𝑤 ∈ 𝐴) → ∩ 𝐴 ∈ 𝑆)
Theorem	lssincl 14806	The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆)
Syntax	clspn 14807	Extend class notation with span of a set of vectors.
class LSpan
Definition	df-lsp 14808*	Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
Theorem	lspfval 14809*	The span function for a left vector space (or a left module). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
Theorem	lspf 14810	The span function on a left module maps subsets to subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → 𝑁:𝒫 𝑉⟶𝑆)
Theorem	lspval 14811*	The span of a set of vectors (in a left module). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
Theorem	lspcl 14812	The span of a set of vectors is a subspace. (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ∈ 𝑆)
Theorem	lspsncl 14813	The span of a singleton is a subspace (frequently used special case of lspcl 14812). (Contributed by NM, 17-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆)
Theorem	lspprcl 14814	The span of a pair is a subspace (frequently used special case of lspcl 14812). (Contributed by NM, 11-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆)
Theorem	lsptpcl 14815	The span of an unordered triple is a subspace (frequently used special case of lspcl 14812). (Contributed by NM, 22-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌, 𝑍}) ∈ 𝑆)
Theorem	lspex 14816	Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
⊢ (𝑊 ∈ 𝑋 → (LSpan‘𝑊) ∈ V)
Theorem	lspsnsubg 14817	The span of a singleton is an additive subgroup (frequently used special case of lspcl 14812). (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
Theorem	lspid 14818	The span of a subspace is itself. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈)
Theorem	lspssv 14819	A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ⊆ 𝑉)
Theorem	lspss 14820	Span preserves subset ordering. (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈))
Theorem	lspssid 14821	A set of vectors is a subset of its span. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ (𝑁‘𝑈))
Theorem	lspidm 14822	The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈))
Theorem	lspun 14823	The span of union is the span of the union of spans. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ⊆ 𝑉 ∧ 𝑈 ⊆ 𝑉) → (𝑁‘(𝑇 ∪ 𝑈)) = (𝑁‘((𝑁‘𝑇) ∪ (𝑁‘𝑈))))
Theorem	lspssp 14824	If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ 𝑈)
Theorem	lspsnss 14825	The span of the singleton of a subspace member is included in the subspace. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
Theorem	ellspsn3 14826	A member of the span of the singleton of a vector is a member of a subspace containing the vector. (Contributed by NM, 4-Jul-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋}))    ⇒   ⊢ (𝜑 → 𝑌 ∈ 𝑈)
Theorem	lspprss 14827	The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈)
Theorem	lspsnid 14828	A vector belongs to the span of its singleton. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋}))
Theorem	ellspsn6 14829	Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)))
Theorem	ellspsn5b 14830	Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈))
Theorem	ellspsn5 14831	Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈)
Theorem	lspprid1 14832	A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑌}))
Theorem	lspprid2 14833	A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑌}))
Theorem	lspprvacl 14834	The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌}))
Theorem	lssats2 14835*	A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}))
Theorem	lspsneli 14836	A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 2-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋}))
Theorem	lspsn 14837*	Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})
Theorem	ellspsn 14838*	Member of span of the singleton of a vector. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑈 ∈ (𝑁‘{𝑋}) ↔ ∃𝑘 ∈ 𝐾 𝑈 = (𝑘 · 𝑋)))
Theorem	lspsnvsi 14839	Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅 · 𝑋)}) ⊆ (𝑁‘{𝑋}))
Theorem	lspsnss2 14840*	Comparable spans of singletons must have proportional vectors. (Contributed by NM, 7-Jun-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ ∃𝑘 ∈ 𝐾 𝑋 = (𝑘 · 𝑌)))
Theorem	lspsnneg 14841	Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑀 = (invg‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑀‘𝑋)}) = (𝑁‘{𝑋}))
Theorem	lspsnsub 14842	Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) = (𝑁‘{(𝑌 − 𝑋)}))
Theorem	lspsn0 14843	Span of the singleton of the zero vector. (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 })
Theorem	lsp0 14844	Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })
Theorem	lspuni0 14845	Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → ∪ (𝑁‘∅) = 0 )
Theorem	lspun0 14846	The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘𝑋))
Theorem	lspsneq0 14847	Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
Theorem	lspsneq0b 14848	Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑋 = 0 ↔ 𝑌 = 0 ))
Theorem	lmodindp1 14849	Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 )
Theorem	lsslsp 14850	Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑀 = (LSpan‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑋)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑁‘𝐺) = (𝑀‘𝐺))
Theorem	lss0v 14851	The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑍 = (0g‘𝑋)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑍 = 0 )
Theorem	lsspropdg 14852*	If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised 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 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))    &   ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))
Theorem	lsppropd 14853*	If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
⊢ (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 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))    &   ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
7.6  Subring algebras and ideals
7.6.1  Subring algebras
Syntax	csra 14854	Extend class notation with the subring algebra generator.
class subringAlg
Syntax	crglmod 14855	Extend class notation with the left module induced by a ring over itself.
class ringLMod
Definition	df-sra 14856*	Any ring can be regarded as a left algebra over any of its subrings. The function subringAlg associates with any ring and any of its subrings the left algebra consisting in the ring itself regarded as a left algebra over the subring. It has an inner product which is simply the ring product. (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), (𝑤 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑤)⟩)))
Definition	df-rgmod 14857	Any ring can be regarded as a left algebra over itself. The function ringLMod associates with any ring the left algebra consisting in the ring itself regarded as a left algebra over itself. It has an inner product which is simply the ring product. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
Theorem	sraval 14858	Lemma for srabaseg 14860 through sravscag 14864. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
Theorem	sralemg 14859	Lemma for srabaseg 14860 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   ⊢ (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   ⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))
Theorem	srabaseg 14860	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.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (Base‘𝑊) = (Base‘𝐴))
Theorem	sraaddgg 14861	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.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝐴))
Theorem	sramulrg 14862	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.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (.r‘𝑊) = (.r‘𝐴))
Theorem	srascag 14863	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.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
Theorem	sravscag 14864	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.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
Theorem	sraipg 14865	The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (.r‘𝑊) = (·𝑖‘𝐴))
Theorem	sratsetg 14866	Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))
Theorem	sraex 14867	Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	sratopng 14868	Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (TopOpen‘𝑊) = (TopOpen‘𝐴))
Theorem	sradsg 14869	Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (dist‘𝑊) = (dist‘𝐴))
Theorem	sraring 14870	Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring)
Theorem	sralmod 14871	The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    ⇒   ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
Theorem	sralmod0g 14872	The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 0 = (0g‘𝑊))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 0 = (0g‘𝐴))
Theorem	issubrgd 14873*	Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷))    &   ⊢ (𝜑 → 0 = (0g‘𝐼))    &   ⊢ (𝜑 → + = (+g‘𝐼))    &   ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼))    &   ⊢ (𝜑 → 0 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷)    &   ⊢ (𝜑 → 1 = (1r‘𝐼))    &   ⊢ (𝜑 → · = (.r‘𝐼))    &   ⊢ (𝜑 → 1 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 · 𝑦) ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼))
Theorem	rlmfn 14874	ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ ringLMod Fn V
Theorem	rlmvalg 14875	Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑊 ∈ 𝑉 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
Theorem	rlmbasg 14876	Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(ringLMod‘𝑅)))
Theorem	rlmplusgg 14877	Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(ringLMod‘𝑅)))
Theorem	rlm0g 14878	Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘(ringLMod‘𝑅)))
Theorem	rlmsubg 14879	Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ (𝑅 ∈ 𝑉 → (-g‘𝑅) = (-g‘(ringLMod‘𝑅)))
Theorem	rlmmulrg 14880	Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) = (.r‘(ringLMod‘𝑅)))
Theorem	rlmscabas 14881	Scalars in the ring module have the same base set. (Contributed by Jim Kingdon, 29-Apr-2025.)
⊢ (𝑅 ∈ 𝑋 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅))))
Theorem	rlmvscag 14882	Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)))
Theorem	rlmtopng 14883	Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ 𝑉 → (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅)))
Theorem	rlmdsg 14884	Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ 𝑉 → (dist‘𝑅) = (dist‘(ringLMod‘𝑅)))
Theorem	rlmlmod 14885	The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
Theorem	rlmvnegg 14886	Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
⊢ (𝑅 ∈ 𝑉 → (invg‘𝑅) = (invg‘(ringLMod‘𝑅)))
Theorem	ixpsnbasval 14887*	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.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ (Base‘𝑅))})
7.6.2  Ideals and spans
Syntax	clidl 14888	Ring left-ideal function.
class LIdeal
Syntax	crsp 14889	Ring span function.
class RSpan
Definition	df-lidl 14890	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)
Definition	df-rsp 14891	Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ RSpan = (LSpan ∘ ringLMod)
Theorem	lidlvalg 14892	Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑊 ∈ 𝑉 → (LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊)))
Theorem	rspvalg 14893	Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ (𝑊 ∈ 𝑉 → (RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊)))
Theorem	lidlex 14894	Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
⊢ (𝑊 ∈ 𝑉 → (LIdeal‘𝑊) ∈ V)
Theorem	rspex 14895	Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
⊢ (𝑊 ∈ 𝑉 → (RSpan‘𝑊) ∈ V)
Theorem	lidlmex 14896	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 14897	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 14898	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 14899	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 14900*	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))