P. 219 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)

Type	Label	Description
Statement
11.2  Associative algebras
11.2.1  Definition and basic properties
Syntax	casa 21801	Associative algebra.
class AssAlg
Syntax	casp 21802	Algebraic span function.
class AlgSpan
Syntax	cascl 21803	Class of algebra scalar injection function.
class algSc
Definition	df-assa 21804*	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 21805*	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 21806*	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 21807*	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 21808	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 21809	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 21810	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 21811	An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
Theorem	assaring 21812	An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
Theorem	assasca 21813	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 21814	Left- and right-associative property of an associative algebra. Notice that the scalars are commuted! (Contributed by AV, 14-Aug-2019.)
⊢ 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	isassad 21815*	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 21816	A subring that is also a subspace is a subalgebra. The key theorem is islss3 20855. (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 21817	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	sraassab 21818	A subring algebra is an associative algebra if and only if the subring is included in the ring's center. (Contributed by SN, 21-Mar-2025.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    &   ⊢ 𝑍 = (Cntr‘(mulGrp‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑊))    ⇒   ⊢ (𝜑 → (𝐴 ∈ AssAlg ↔ 𝑆 ⊆ 𝑍))
Theorem	sraassa 21819	The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.) (Proof shortened by SN, 21-Mar-2025.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    ⇒   ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)
Theorem	sraassaOLD 21820	Obsolete version of sraassa 21819 as of 21-Mar-2025. (Contributed by Mario Carneiro, 6-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    ⇒   ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)
Theorem	rlmassa 21821	The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ CRing → (ringLMod‘𝑅) ∈ AssAlg)
Theorem	assapropd 21822*	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 21823*	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 21824	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 21825	The algebraic span of a subalgebra is itself. (spanid 31229 analog.) (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 21826	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 21827	Span preserves subset ordering. (spanss 31230 analog.) (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 21828	A set of vectors is a subset of its span. (spanss2 31227 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝐴‘𝑆))
Theorem	asclfval 21829*	Function value of the algebraic scalars 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	asclval 21830	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)    ⇒   ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋 · 1 ))
Theorem	asclfn 21831	Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ 𝐴 Fn 𝐾
Theorem	asclf 21832	The algebra scalars 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 21833	The algebra scalars 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	ascl0 21834	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 21835	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 21836	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 21837	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 21838	The algebra scalars 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 21839	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 21840	The scalar injection 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 21841	The set of injected 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 21842	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 21843	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 21844	(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 21845	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 21846	The injection 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 21847*	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	aspval2 21848	The algebraic closure is the ring closure when the generating set is expanded to include all scalars. EDITORIAL : In light of this, is AlgSpan independently needed? (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝐶 = (algSc‘𝑊)    &   ⊢ 𝑅 = (mrCls‘(SubRing‘𝑊))    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = (𝑅‘(ran 𝐶 ∪ 𝑆)))
Theorem	assamulgscmlem1 21849	Lemma 1 for assamulgscm 21851 (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 21850	Lemma for assamulgscm 21851 (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 21851	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 21852	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)))
Theorem	zlmassa 21853	The ℤ-module operation turns a ring into an associative algebra over ℤ. Also see zlmlmod 21469. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    ⇒   ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg)
11.3  Abstract multivariate polynomials
11.3.1  Definition and basic properties
Syntax	cmps 21854	Multivariate power series.
class mPwSer
Syntax	cmvr 21855	Multivariate power series variables.
class mVar
Syntax	cmpl 21856	Multivariate polynomials.
class mPoly
Syntax	cltb 21857	Ordering on terms of a multivariate polynomial.
class <bag
Syntax	copws 21858	Ordered set of power series.
class ordPwSer
Definition	df-psr 21859*	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-mvr 21860*	Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟)))))
Definition	df-mpl 21861*	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). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.)
⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g‘𝑟)}))
Definition	df-ltbag 21862*	Define a well-order on the set of all finite bags from the index set 𝑖 given a wellordering 𝑟 of 𝑖. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ <bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ ∃𝑧 ∈ 𝑖 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})
Definition	df-opsr 21863*	Define a total order on the set of all power series in 𝑠 from the index set 𝑖 given a wellordering 𝑟 of 𝑖 and a totally ordered base ring 𝑠. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ ⦋(𝑖 mPwSer 𝑠) / 𝑝⦌(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
Theorem	reldmpsr 21864	The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ Rel dom mPwSer
Theorem	psrval 21865*	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	psrvalstr 21866	The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) Struct ⟨1, 9⟩
Theorem	psrbag 21867*	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 21868*	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	psrbagfOLD 21869*	Obsolete version of psrbag 21867 as of 6-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0)
Theorem	psrbagfsupp 21870*	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	psrbagfsuppOLD 21871*	Obsolete version of psrbagfsupp 21870 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0)
Theorem	snifpsrbag 21872*	A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 8-Jul-2019.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) ∈ 𝐷)
Theorem	fczpsrbag 21873*	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	psrbaglesupp 21874*	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	psrbaglesuppOLD 21875*	Obsolete version of psrbaglesupp 21874 as of 5-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))
Theorem	psrbaglecl 21876*	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	psrbagleclOLD 21877*	Obsolete version of psrbaglecl 21876 as of 5-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → 𝐺 ∈ 𝐷)
Theorem	psrbagaddcl 21878*	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) → (𝐹 ∘f + 𝐺) ∈ 𝐷)
Theorem	psrbagaddclOLD 21879*	Obsolete version of psrbagaddcl 21878 as of 7-Aug-2024. (Contributed by Mario Carneiro, 9-Jan-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘f + 𝐺) ∈ 𝐷)
Theorem	psrbagcon 21880*	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	psrbagconOLD 21881*	Obsolete version of psrbagcon 21880 as of 5-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → ((𝐹 ∘f − 𝐺) ∈ 𝐷 ∧ (𝐹 ∘f − 𝐺) ∘r ≤ 𝐹))
Theorem	psrbaglefi 21882*	There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)
Theorem	psrbaglefiOLD 21883*	Obsolete version of psrbaglefi 21882 as of 6-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)
Theorem	psrbagconcl 21884*	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	psrbagconclOLD 21885*	Obsolete version of psrbagconcl 21884 as of 6-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝐹 ∘f − 𝑋) ∈ 𝑆)
Theorem	psrbagleadd1 21886*	The analogue of "𝑋 ≤ 𝐹 implies 𝑋 + 𝐺 ≤ 𝐹 + 𝐺 " (compare leadd1d 11840) for bags. (Contributed by SN, 2-May-2025.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   ⊢ 𝑇 = {𝑧 ∈ 𝐷 ∣ 𝑧 ∘r ≤ (𝐹 ∘f + 𝐺)}    ⇒   ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∘f + 𝐺) ∈ 𝑇)
Theorem	psrbagconf1o 21887*	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	psrbagconf1oOLD 21888*	Obsolete version of psrbagconf1o 21887 as of 6-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑥 ∈ 𝑆 ↦ (𝐹 ∘f − 𝑥)):𝑆–1-1-onto→𝑆)
Theorem	gsumbagdiaglemOLD 21889*	Obsolete version of gsumbagdiaglem 21892 as of 6-Aug-2024. (Contributed by Mario Carneiro, 5-Jan-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝑌 ∈ 𝑆 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑌)}))
Theorem	gsumbagdiagOLD 21890*	Obsolete version of gsumbagdiag 21893 as of 6-Aug-2024. (Contributed by Mario Carneiro, 5-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝑆, 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝑆, 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑘)} ↦ 𝑋)))
Theorem	psrass1lemOLD 21891*	Obsolete version of psrass1lem 21894 as of 7-Aug-2024. (Contributed by Mario Carneiro, 5-Jan-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑋 ∈ 𝐵)    &   ⊢ (𝑘 = (𝑛 ∘f − 𝑗) → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑗 ∈ 𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ↦ 𝑋)))))
Theorem	gsumbagdiaglem 21892*	Lemma for gsumbagdiag 21893. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 6-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝑌 ∈ 𝑆 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑌)}))
Theorem	gsumbagdiag 21893*	Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 15759 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 6-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝑆, 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝑆, 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑘)} ↦ 𝑋)))
Theorem	psrass1lem 21894*	A group sum commutation used by psrass1 21926. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑋 ∈ 𝐵)    &   ⊢ (𝑘 = (𝑛 ∘f − 𝑗) → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑗 ∈ 𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ↦ 𝑋)))))
Theorem	psrbas 21895*	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 → 𝐵 = (𝐾 ↑m 𝐷))
Theorem	psrelbas 21896*	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	psrelbasfun 21897	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	psrplusg 21898	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)    ⇒   ⊢ ✚ = ( ∘f + ↾ (𝐵 × 𝐵))
Theorem	psradd 21899	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 21900	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)