P. 151 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15001-15100   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-assa 15001*	Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
Definition	df-asp 15002*	Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
Definition	df-ascl 15003*	Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unity element. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))))
Theorem	isassa 15004*	The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
Theorem	assalem 15005	The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
Theorem	assaass 15006	Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
Theorem	assaassr 15007	Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))
Theorem	assalmod 15008	An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
Theorem	assaring 15009	An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
Theorem	assasca 15010	The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by SN, 2-Mar-2025.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
Theorem	assa2ass 15011	Left- and right-associative property of an associative algebra. Notice that the scalars are commuted! (Contributed by AV, 14-Aug-2019.) (Proof shortened by Zhi Wang, 11-Sep-2025.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ ∗ = (.r‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × (𝐶 · 𝑌)) = ((𝐶 ∗ 𝐴) · (𝑋 × 𝑌)))
Theorem	assa2ass2 15012	Left- and right-associative property of an associative algebra. Notice that the scalars are not commuted! (Contributed by Zhi Wang, 11-Sep-2025.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ ∗ = (.r‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × (𝐶 · 𝑌)) = ((𝐴 ∗ 𝐶) · (𝑋 × 𝑌)))
Theorem	isassad 15013*	Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ (𝜑 → 𝑉 = (Base‘𝑊))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐹))    &   ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))    &   ⊢ (𝜑 → × = (.r‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))    &   ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))    ⇒   ⊢ (𝜑 → 𝑊 ∈ AssAlg)
Theorem	issubassa3 15014	A subring that is also a subspace is a subalgebra. The key theorem is islss3 14718. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝑆 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ AssAlg)
Theorem	issubassa 15015	The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝑆 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)))
Theorem	assapropd 15016*	If two structures have the same components (properties), one is an associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))    &   ⊢ 𝑃 = (Base‘𝐹)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))
Theorem	aspval 15017*	Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
Theorem	asplss 15018	The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) ∈ 𝐿)
Theorem	aspid 15019	The algebraic span of a subalgebra is itself. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝐴‘𝑆) = 𝑆)
Theorem	aspsubrg 15020	The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) ∈ (SubRing‘𝑊))
Theorem	aspss 15021	Span preserves subset ordering. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆) → (𝐴‘𝑇) ⊆ (𝐴‘𝑆))
Theorem	aspssid 15022	A set of vectors is a subset of its span. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝐴‘𝑆))
Theorem	asclfval 15023*	Function value of the algebra scalar lifting function. (Contributed by Mario Carneiro, 8-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    ⇒   ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))
Theorem	asclvald 15024	Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐴‘𝑋) = (𝑋 · 1 ))
Theorem	asclfnd 15025	Functionality of the algebra scalar lifting function. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝐴 Fn 𝐾)
Theorem	asclf 15026	The algebra scalar lifting function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝜑 → 𝐴:𝐾⟶𝐵)
Theorem	asclghm 15027	The algebra scalar lifting function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐹 GrpHom 𝑊))
Theorem	asclelbas 15028	Lifted scalars are in the base set of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) (Proof shortened by Thierry Arnoux, 22-Sep-2025.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑊))
Theorem	ascl0 15029	The scalar 0 embedded into a left module corresponds to the 0 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐴‘(0g‘𝐹)) = (0g‘𝑊))
Theorem	ascl1 15030	The scalar 1 embedded into a left module corresponds to the 1 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐴‘(1r‘𝐹)) = (1r‘𝑊))
Theorem	asclmul1 15031	Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ × = (.r‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = (𝑅 · 𝑋))
Theorem	asclmul2 15032	Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ × = (.r‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑋 × (𝐴‘𝑅)) = (𝑅 · 𝑋))
Theorem	ascldimul 15033	The algebra scalar lifting function distributes over multiplication. (Contributed by Mario Carneiro, 8-Mar-2015.) (Proof shortened by SN, 5-Nov-2023.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ × = (.r‘𝑊)    &   ⊢ · = (.r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝐴‘(𝑅 · 𝑆)) = ((𝐴‘𝑅) × (𝐴‘𝑆)))
Theorem	asclinvg 15034	The group inverse (negation) of a lifted scalar is the lifted negation of the scalar. (Contributed by AV, 2-Sep-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐼 = (invg‘𝑅)    &   ⊢ 𝐽 = (invg‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐶 ∈ 𝐵) → (𝐽‘(𝐴‘𝐶)) = (𝐴‘(𝐼‘𝐶)))
Theorem	asclrhm 15035	The algebra scalar lifting function is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 RingHom 𝑊))
Theorem	rnascl 15036	The set of lifted scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ AssAlg → ran 𝐴 = (𝑁‘{ 1 }))
Theorem	issubassa2 15037	A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆 ∈ 𝐿 ↔ ran 𝐴 ⊆ 𝑆))
Theorem	rnasclsubrg 15038	The scalar multiples of the unit vector form a subring of the vectors. (Contributed by SN, 5-Nov-2023.)
⊢ 𝐶 = (algSc‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ (𝜑 → ran 𝐶 ∈ (SubRing‘𝑊))
Theorem	rnasclmulcl 15039	(Vector) multiplication is closed for scalar multiples of the unit vector. (Contributed by SN, 5-Nov-2023.)
⊢ 𝐶 = (algSc‘𝑊)    &   ⊢ × = (.r‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ ran 𝐶 ∧ 𝑌 ∈ ran 𝐶)) → (𝑋 × 𝑌) ∈ ran 𝐶)
Theorem	rnasclassa 15040	The scalar multiples of the unit vector form a subalgebra of the vectors. (Contributed by SN, 16-Nov-2023.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝑈 = (𝑊 ↾s ran 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ (𝜑 → 𝑈 ∈ AssAlg)
Theorem	ressascl 15041	The lifting of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = (algSc‘𝑋))
Theorem	asclpropd 15042*	If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on 1r can be discharged either by letting 𝑊 = V (if strong equality is known on ·𝑠) or assuming 𝐾 is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
⊢ 𝐹 = (Scalar‘𝐾)    &   ⊢ 𝐺 = (Scalar‘𝐿)    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐹))    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐺))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑊)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))    &   ⊢ (𝜑 → (1r‘𝐾) ∈ 𝑊)    ⇒   ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))
Theorem	assamulgscmlem1 15043	Lemma 1 for assamulgscm 15045 (induction base). (Contributed by AV, 26-Aug-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐺 = (mulGrp‘𝐹)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐻 = (mulGrp‘𝑊)    &   ⊢ 𝐸 = (.g‘𝐻)    ⇒   ⊢ (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → (0𝐸(𝐴 · 𝑋)) = ((0 ↑ 𝐴) · (0𝐸𝑋)))
Theorem	assamulgscmlem2 15044	Lemma for assamulgscm 15045 (induction step). (Contributed by AV, 26-Aug-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐺 = (mulGrp‘𝐹)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐻 = (mulGrp‘𝑊)    &   ⊢ 𝐸 = (.g‘𝐻)    ⇒   ⊢ (𝑦 ∈ ℕ0 → (((𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ AssAlg) → ((𝑦𝐸(𝐴 · 𝑋)) = ((𝑦 ↑ 𝐴) · (𝑦𝐸𝑋)) → ((𝑦 + 1)𝐸(𝐴 · 𝑋)) = (((𝑦 + 1) ↑ 𝐴) · ((𝑦 + 1)𝐸𝑋)))))
Theorem	assamulgscm 15045	Exponentiation of a scalar multiplication in an associative algebra: (𝑎 · 𝑋)↑𝑁 = (𝑎↑𝑁) × (𝑋↑𝑁). (Contributed by AV, 26-Aug-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐺 = (mulGrp‘𝐹)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐻 = (mulGrp‘𝑊)    &   ⊢ 𝐸 = (.g‘𝐻)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉)) → (𝑁𝐸(𝐴 · 𝑋)) = ((𝑁 ↑ 𝐴) · (𝑁𝐸𝑋)))
Theorem	asclmulg 15046	Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ↑ = (.g‘𝑊)    &   ⊢ ∗ = (.g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘(𝑁 ∗ 𝑋)) = (𝑁 ↑ (𝐴‘𝑋)))
8.2  Abstract multivariate polynomials
8.2.1  Definition and basic properties
Syntax	cmps 15047	Multivariate power series.
class mPwSer
Syntax	cmpl 15048	Multivariate polynomials.
class mPoly
Definition	df-psr 15049*	Define the algebra of power series over the index set 𝑖 and with coefficients from the ring 𝑟. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋{ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
Definition	df-mplcoe 15050*	Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). The index set (which has an element for each variable) is 𝑖, the coefficients are in ring 𝑟, and for each variable there is a "degree" such that the coefficient is zero for a term where the powers are all greater than those degrees. (Degree is in quotes because there is no guarantee that coefficients below that degree are nonzero, as we do not assume decidable equality for 𝑟). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 7-Oct-2025.)
⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}))
Theorem	reldmpsr 15051	The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ Rel dom mPwSer
Theorem	psrval 15052*	Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑂 = (TopOpen‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 𝐷))    &   ⊢ ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵))    &   ⊢ × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))    &   ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))    &   ⊢ (𝜑 → 𝐽 = (∏t‘(𝐷 × {𝑂})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
Theorem	fnpsr 15053	The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
⊢ mPwSer Fn (V × V)
Theorem	psrvalstrd 15054	The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → + ∈ 𝑌)    &   ⊢ (𝜑 → × ∈ 𝑍)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑃)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑄)    ⇒   ⊢ (𝜑 → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) Struct ⟨1, 9⟩)
Theorem	psrbag 15055*	Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)))
Theorem	psrbagf 15056*	A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
Theorem	psrbagfsupp 15057*	Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0)
Theorem	fczpsrbag 15058*	The constant function equal to zero is a finite bag. (Contributed by AV, 8-Jul-2019.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ 0) ∈ 𝐷)
Theorem	psrbaglesuppg 15059*	The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))
Theorem	psrbaglesupp 15060*	The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))
Theorem	psrbagfi 15061*	A finite index set gives a simpler expression for finite bags. (Contributed by Jim Kingdon, 23-Nov-2025.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ (𝐼 ∈ Fin → 𝐷 = (ℕ0 ↑𝑚 𝐼))
Theorem	psrbaglecl 15062*	The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 ∈ 𝐷)
Theorem	psrbagaddclfi 15063*	The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.) Shorten proof and remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝐼 ∈ Fin) → (𝐹 ∘𝑓 + 𝐺) ∈ 𝐷)
Theorem	psrbagcon 15064*	The analogue of the statement "0 ≤ 𝐺 ≤ 𝐹 implies 0 ≤ 𝐹 − 𝐺 ≤ 𝐹 " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ((𝐹 ∘𝑓 − 𝐺) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝐺) ∘𝑟 ≤ 𝐹))
Theorem	psrbagconcl 15065*	The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹}    ⇒   ⊢ ((𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝐹 ∘𝑓 − 𝑋) ∈ 𝑆)
Theorem	psrbagconf1o 15066*	Bag complementation is a bijection on the set of bags dominated by a given bag 𝐹. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹}    ⇒   ⊢ (𝐹 ∈ 𝐷 → (𝑥 ∈ 𝑆 ↦ (𝐹 ∘𝑓 − 𝑥)):𝑆–1-1-onto→𝑆)
Theorem	psrbasg 15067*	The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 8-Jul-2019.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 𝐷))
Theorem	psrelbas 15068*	An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋:𝐷⟶𝐾)
Theorem	psrelbasfi 15069	Simpler form of psrelbas 15068 when the index set is finite. (Contributed by Jim Kingdon, 27-Nov-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋:(ℕ0 ↑𝑚 𝐼)⟶𝐾)
Theorem	psrelbasfun 15070	An element of the set of power series is a function. (Contributed by AV, 17-Jul-2019.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝑋 ∈ 𝐵 → Fun 𝑋)
Theorem	psrplusgg 15071	The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝑆)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))
Theorem	psradd 15072	The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ✚ 𝑌) = (𝑋 ∘𝑓 + 𝑌))
Theorem	psraddcl 15073	Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Mgm)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	psr0cl 15074*	The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐵)
Theorem	psr0lid 15075*	The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐷 × { 0 }) + 𝑋) = 𝑋)
Theorem	psrnegcl 15076*	The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
Theorem	psrlinv 15077*	The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ + = (+g‘𝑆)    ⇒   ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))
Theorem	psrgrp 15078	The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by SN, 7-Feb-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝑆 ∈ Grp)
Theorem	psr0 15079*	The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑂 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝜑 → 0 = (𝐷 × {𝑂}))
Theorem	psrneg 15080*	The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑀 = (invg‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))
Theorem	psr1clfi 15081*	The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝐵)
Theorem	reldmmpl 15082	The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ Rel dom mPoly
Theorem	mplvalcoe 15083*	Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑃 = (𝑆 ↾s 𝑈))
Theorem	mplbascoe 15084*	Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑈 = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )})
Theorem	mplelbascoe 15085*	Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑋‘𝑏) = 0 ))))
Theorem	fnmpl 15086	mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
⊢ mPoly Fn (V × V)
Theorem	mplrcl 15087	Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V)
Theorem	mplval2g 15088	Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑃 = (𝑆 ↾s 𝑈))
Theorem	mplbasss 15089	The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ 𝑈 ⊆ 𝐵
Theorem	mplelf 15090*	A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋:𝐷⟶𝐾)
Theorem	mplsubgfilemm 15091*	Lemma for mplsubgfi 15094. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    ⇒   ⊢ (𝜑 → ∃𝑗 𝑗 ∈ 𝑈)
Theorem	mplsubgfilemcl 15092	Lemma for mplsubgfi 15094. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ + = (+g‘𝑆)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)
Theorem	mplsubgfileminv 15093	Lemma for mplsubgfi 15094. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ 𝑁 = (invg‘𝑆)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑈)
Theorem	mplsubgfi 15094	The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))
Theorem	mpl0fi 15095*	The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑂 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    ⇒   ⊢ (𝜑 → 0 = (𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ↦ 𝑂))
Theorem	mplplusgg 15096	Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑌 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ + = (+g‘𝑌)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → + = (+g‘𝑆))
Theorem	mpladd 15097	The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ✚ 𝑌) = (𝑋 ∘𝑓 + 𝑌))
Theorem	mplnegfi 15098	The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝑀 = (invg‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))
Theorem	mplgrpfi 15099	The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
PART 9  BASIC TOPOLOGY
9.1  Topology
9.1.1  Topological spaces A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.
9.1.1.1  Topologies
Syntax	ctop 15100	Syntax for the class of topologies.
class Top