P. 137 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13601-13700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lspsnel 13601*	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 13602	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 13603*	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 13604	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 13605	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 13606	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 13607	Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })
Theorem	lspuni0 13608	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 13609	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 13610	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 13611	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 13612	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 13613	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 13614	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 13615*	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 13616*	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 13617	Extend class notation with the subring algebra generator.
class subringAlg
Syntax	crglmod 13618	Extend class notation with the left module induced by a ring over itself.
class ringLMod
Definition	df-sra 13619*	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 13620	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 13621	Lemma for srabaseg 13623 through sravscag 13627. (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 13622	Lemma for srabaseg 13623 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 13623	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 13624	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 13625	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 13626	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 13627	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 13628	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 13629	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 13630	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 13631	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 13632	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 13633	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 13634	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 13635	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 13636*	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 13637	ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ ringLMod Fn V
Theorem	rlmvalg 13638	Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑊 ∈ 𝑉 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
Theorem	rlmbasg 13639	Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(ringLMod‘𝑅)))
Theorem	rlmplusgg 13640	Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(ringLMod‘𝑅)))
Theorem	rlm0g 13641	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 13642	Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ (𝑅 ∈ 𝑉 → (-g‘𝑅) = (-g‘(ringLMod‘𝑅)))
Theorem	rlmmulrg 13643	Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) = (.r‘(ringLMod‘𝑅)))
Theorem	rlmscabas 13644	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 13645	Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)))
Theorem	rlmtopng 13646	Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ 𝑉 → (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅)))
Theorem	rlmdsg 13647	Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ 𝑉 → (dist‘𝑅) = (dist‘(ringLMod‘𝑅)))
Theorem	rlmlmod 13648	The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
Theorem	rlmvnegg 13649	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 13650*	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 13651	Ring left-ideal function.
class LIdeal
Syntax	crsp 13652	Ring span function.
class RSpan
Definition	df-lidl 13653	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 13654	Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ RSpan = (LSpan ∘ ringLMod)
Theorem	lidlvalg 13655	Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (𝑊 ∈ 𝑉 → (LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊)))
Theorem	rspvalg 13656	Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ (𝑊 ∈ 𝑉 → (RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊)))
Theorem	lidlex 13657	Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
⊢ (𝑊 ∈ 𝑉 → (LIdeal‘𝑊) ∈ V)
Theorem	rspex 13658	Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
⊢ (𝑊 ∈ 𝑉 → (RSpan‘𝑊) ∈ V)
Theorem	lidlmex 13659	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 13660	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 13661	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 13662	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 13663*	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 13664	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 13665*	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 13666*	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 13667	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 13668	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 13669	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 13670	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 13671	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 13672*	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 13673	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 13674	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 13675	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 13676	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 13677	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 13678	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)
7.6.3  Two-sided ideals and quotient rings
Syntax	c2idl 13679	Ring two-sided ideal function.
class 2Ideal
Definition	df-2idl 13680	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	2idlvalg 13681	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	2idlmex 13682	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	isridl 13683*	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 13684	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 13685	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 13686	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 13687*	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 13688*	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 13689	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 13690	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 13691	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 13692	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 13693	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 13694	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 13695	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))))
7.7  The complex numbers as an algebraic extensible structure
7.7.1  Definition and basic properties
Syntax	cpsmet 13696	Extend class notation with the class of all pseudometric spaces.
class PsMet
Syntax	cxmet 13697	Extend class notation with the class of all extended metric spaces.
class ∞Met
Syntax	cmet 13698	Extend class notation with the class of all metrics.
class Met
Syntax	cbl 13699	Extend class notation with the metric space ball function.
class ball
Syntax	cfbas 13700	Extend class definition to include the class of filter bases.
class fBas