P. 220 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	resspsrmul 21901	A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPwSer 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌))
Theorem	resspsrvsca 21902	A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPwSer 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌))
Theorem	subrgpsr 21903	A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPwSer 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))
Theorem	psrascl 21904*	Value of the scalar injection into the power series algebra. (Contributed by SN, 18-May-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 )))
Theorem	psrasclcl 21905	A scalar is lifted into a member of the power series. (Contributed by SN, 25-Apr-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝐴‘𝐶) ∈ 𝐵)
Theorem	mvrfval 21906*	Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑌)    ⇒   ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
Theorem	mvrval 21907*	Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
Theorem	mvrval2 21908*	Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
Theorem	mvrid 21909*	The 𝑋𝑖-th coefficient of the term 𝑋𝑖 is 1. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((𝑉‘𝑋)‘(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1 )
Theorem	mvrf 21910	The power series variable function is a function from the index set to elements of the power series structure representing 𝑋𝑖 for each 𝑖. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑉:𝐼⟶𝐵)
Theorem	mvrf1 21911	The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 1 ≠ 0 )    ⇒   ⊢ (𝜑 → 𝑉:𝐼–1-1→𝐵)
Theorem	mvrcl2 21912	A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵)
Theorem	reldmmpl 21913	The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ Rel dom mPoly
Theorem	mplval 21914*	Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }    ⇒   ⊢ 𝑃 = (𝑆 ↾s 𝑈)
Theorem	mplbas 21915*	Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    ⇒   ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }
Theorem	mplelbas 21916	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.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    ⇒   ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 ))
Theorem	mvrcl 21917	A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵)
Theorem	mvrf2 21918	The power series/polynomial variable function maps indices to polynomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑉:𝐼⟶𝐵)
Theorem	mplrcl 21919	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	mplelsfi 21920	A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 25-Jun-2019.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 0 )
Theorem	mplval2 21921	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)    ⇒   ⊢ 𝑃 = (𝑆 ↾s 𝑈)
Theorem	mplbasss 21922	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 21923*	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	mplsubglem 21924*	If 𝐴 is an ideal of sets (a nonempty collection closed under subset and binary union) of the set 𝐷 of finite bags (the primary applications being 𝐴 = Fin and 𝐴 = 𝒫 𝐵 for some 𝐵), then the set of all power series whose coefficient functions are supported on an element of 𝐴 is a subgroup of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 16-Jul-2019.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∪ 𝑦) ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))
Theorem	mpllsslem 21925*	If 𝐴 is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set 𝐷 of finite bags (the primary applications being 𝐴 = Fin and 𝐴 = 𝒫 𝐵 for some 𝐵), then the set of all power series whose coefficient functions are supported on an element of 𝐴 is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 16-Jul-2019.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∪ 𝑦) ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆))
Theorem	mplsubglem2 21926*	Lemma for mplsubg 21927 and mpllss 21928. (Contributed by AV, 16-Jul-2019.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑈 = {𝑔 ∈ (Base‘𝑆) ∣ (𝑔 supp (0g‘𝑅)) ∈ Fin})
Theorem	mplsubg 21927	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 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))
Theorem	mpllss 21928	The set of polynomials is closed under scalar multiplication, i.e. it is a linear subspace of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆))
Theorem	mplsubrglem 21929*	Lemma for mplsubrg 21930. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by AV, 18-Jul-2019.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = ( ∘f + “ ((𝑋 supp 0 ) × (𝑌 supp 0 )))    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋(.r‘𝑆)𝑌) ∈ 𝑈)
Theorem	mplsubrg 21930	The set of polynomials is closed under multiplication, i.e. it is a subring of the set of power series. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑆))
Theorem	mpl0 21931*	The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑂 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    ⇒   ⊢ (𝜑 → 0 = (𝐷 × {𝑂}))
Theorem	mplplusg 21932	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)    ⇒   ⊢ + = (+g‘𝑆)
Theorem	mplmulr 21933	Value of multiplication 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)    &   ⊢ · = (.r‘𝑌)    ⇒   ⊢ · = (.r‘𝑆)
Theorem	mpladd 21934	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	mplneg 21935	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 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))
Theorem	mplmul 21936*	The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝐹‘𝑥) · (𝐺‘(𝑘 ∘f − 𝑥)))))))
Theorem	mpl1 21937*	The identity element of the ring of polynomials. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑈 = (1r‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 )))
Theorem	mplsca 21938	The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
Theorem	mplvsca2 21939	The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    ⇒   ⊢ · = ( ·𝑠 ‘𝑆)
Theorem	mplvsca 21940*	The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ ∙ = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹))
Theorem	mplvscaval 21941*	The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ ∙ = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌)))
Theorem	mplgrp 21942	The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
Theorem	mpllmod 21943	The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod)
Theorem	mplring 21944	The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
Theorem	mpllvec 21945	The polynomial ring is a vector space. (Contributed by SN, 29-Feb-2024.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ DivRing) → 𝑃 ∈ LVec)
Theorem	mplcrng 21946	The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing)
Theorem	mplassa 21947	The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ AssAlg)
Theorem	mplringd 21948	The polynomial ring is a ring. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑃 ∈ Ring)
Theorem	mpllmodd 21949	The polynomial ring is a left module. (Contributed by SN, 12-Mar-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑃 ∈ LMod)
Theorem	ressmplbas2 21950	The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑊 = (𝐼 mPwSer 𝐻)    &   ⊢ 𝐶 = (Base‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝜑 → 𝐵 = (𝐶 ∩ 𝐾))
Theorem	ressmplbas 21951	A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
Theorem	ressmpladd 21952	A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘𝑃)𝑌))
Theorem	ressmplmul 21953	A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌))
Theorem	ressmplvsca 21954	A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌))
Theorem	subrgmpl 21955	A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))
Theorem	subrgmvr 21956	The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    ⇒   ⊢ (𝜑 → 𝑉 = (𝐼 mVar 𝐻))
Theorem	subrgmvrf 21957	The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    ⇒   ⊢ (𝜑 → 𝑉:𝐼⟶𝐵)
Theorem	mplmon 21958*	A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
Theorem	mplmonmul 21959*	The product of two monomials adds the exponent vectors together. For example, the product of (𝑥↑2)(𝑦↑2) with (𝑦↑1)(𝑧↑3) is (𝑥↑2)(𝑦↑3)(𝑧↑3), where the exponent vectors ⟨2, 2, 0⟩ and ⟨0, 1, 3⟩ are added to give ⟨2, 3, 3⟩. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    &   ⊢ · = (.r‘𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )))
Theorem	mplcoe1 21960*	Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = (𝑃 Σg (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
Theorem	mplcoe3 21961*	Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 18-Jul-2019.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋)))
Theorem	mplcoe5lem 21962*	Lemma for mplcoe4 21994. (Contributed by AV, 7-Oct-2019.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐼)    ⇒   ⊢ (𝜑 → ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
Theorem	mplcoe5 21963*	Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 21964), it is sufficient that 𝑅 is a ring and all the variables of the multivariate polynomial commute. (Contributed by AV, 7-Oct-2019.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)))    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
Theorem	mplcoe2 21964*	Decompose a monomial into a finite product of powers of variables. (The assumption that 𝑅 is a commutative ring is not strictly necessary, because the submonoid of monomials is in the center of the multiplicative monoid of polynomials, but it simplifies the proof.) (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2019.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
Theorem	mplbas2 21965	An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐴 = (AlgSpan‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))
Theorem	ltbval 21966*	Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 𝐶 = (𝑇 <bag 𝐼)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧 ∈ 𝐼 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐼 (𝑧𝑇𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})
Theorem	ltbwe 21967*	The finite bag order is a well-order, given a well-order of the index set. (Contributed by Mario Carneiro, 2-Jun-2015.)
⊢ 𝐶 = (𝑇 <bag 𝐼)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 We 𝐼)    ⇒   ⊢ (𝜑 → 𝐶 We 𝐷)
Theorem	reldmopsr 21968	Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
⊢ Rel dom ordPwSer
Theorem	opsrval 21969*	The value of the "ordered power series" function. This is the same as mPwSer psrval 21840, but with the addition of a well-order on 𝐼 we can turn a strict order on 𝑅 into a strict order on the power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ < = (lt‘𝑅)    &   ⊢ 𝐶 = (𝑇 <bag 𝐼)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → 𝑂 = (𝑆 sSet ⟨(le‘ndx), ≤ ⟩))
Theorem	opsrle 21970*	An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ < = (lt‘𝑅)    &   ⊢ 𝐶 = (𝑇 <bag 𝐼)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ ≤ = (le‘𝑂)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))})
Theorem	opsrval2 21971	Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ ≤ = (le‘𝑂)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → 𝑂 = (𝑆 sSet ⟨(le‘ndx), ≤ ⟩))
Theorem	opsrbaslem 21972	Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (le‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂))
Theorem	opsrbas 21973	The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑂))
Theorem	opsrplusg 21974	The addition operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑂))
Theorem	opsrmulr 21975	The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑂))
Theorem	opsrvsca 21976	The scalar product operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑂))
Theorem	opsrsca 21977	The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂))
Theorem	opsrtoslem1 21978*	Lemma for opsrtos 21980. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Toset)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   ⊢ (𝜑 → 𝑇 We 𝐼)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ < = (lt‘𝑅)    &   ⊢ 𝐶 = (𝑇 <bag 𝐼)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜓 ↔ ∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))))    &   ⊢ ≤ = (le‘𝑂)    ⇒   ⊢ (𝜑 → ≤ = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))
Theorem	opsrtoslem2 21979*	Lemma for opsrtos 21980. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Toset)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   ⊢ (𝜑 → 𝑇 We 𝐼)    &   ⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ < = (lt‘𝑅)    &   ⊢ 𝐶 = (𝑇 <bag 𝐼)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜓 ↔ ∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))))    &   ⊢ ≤ = (le‘𝑂)    ⇒   ⊢ (𝜑 → 𝑂 ∈ Toset)
Theorem	opsrtos 21980	The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Toset)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   ⊢ (𝜑 → 𝑇 We 𝐼)    ⇒   ⊢ (𝜑 → 𝑂 ∈ Toset)
Theorem	opsrso 21981	The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Toset)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   ⊢ (𝜑 → 𝑇 We 𝐼)    &   ⊢ ≤ = (lt‘𝑂)    &   ⊢ 𝐵 = (Base‘𝑂)    ⇒   ⊢ (𝜑 → ≤ Or 𝐵)
Theorem	opsrcrng 21982	The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → 𝑂 ∈ CRing)
Theorem	opsrassa 21983	The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → 𝑂 ∈ AssAlg)
Theorem	mplmon2 21984*	Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))
Theorem	psrbag0 21985*	The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐷)
Theorem	psrbagsn 21986*	A singleton bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝐾, 1, 0)) ∈ 𝐷)
Theorem	mplascl 21987*	Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 )))
Theorem	mplasclf 21988	The scalar injection is a function into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝐴:𝐾⟶𝐵)
Theorem	subrgascl 21989	The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝐶 = (algSc‘𝑈)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇))
Theorem	subrgasclcl 21990	The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (𝐼 mPoly 𝐻)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝐴‘𝑋) ∈ 𝐵 ↔ 𝑋 ∈ 𝑇))
Theorem	mplmon2cl 21991*	A scaled monomial is a polynomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )) ∈ 𝐵)
Theorem	mplmon2mul 21992*	Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 𝐹, 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (𝐹 · 𝐺), 0 )))
Theorem	mplind 21993*	Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. The commutativity condition is stronger than strictly needed. (Contributed by Stefan O'Rear, 11-Mar-2015.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝑌 = (𝐼 mPoly 𝑅)    &   ⊢ + = (+g‘𝑌)    &   ⊢ · = (.r‘𝑌)    &   ⊢ 𝐶 = (algSc‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 + 𝑦) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 · 𝑦) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐶‘𝑥) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐻)
Theorem	mplcoe4 21994*	Decompose a polynomial into a finite sum of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = (𝑃 Σg (𝑘 ∈ 𝐷 ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, (𝑋‘𝑘), 0 )))))
11.3.2  Polynomial evaluation
Syntax	ces 21995	Evaluation of a multivariate polynomial in a subring.
class evalSub
Syntax	cevl 21996	Evaluation of a multivariate polynomial.
class eval
Definition	df-evls 21997*	Define the evaluation map for the polynomial algebra. The function ((𝐼 evalSub 𝑆)‘𝑅):𝑉⟶(𝑆 ↑m (𝑆 ↑m 𝐼)) makes sense when 𝐼 is an index set, 𝑆 is a ring, 𝑅 is a subring of 𝑆, and where 𝑉 is the set of polynomials in (𝐼 mPoly 𝑅). This function maps an element of the formal polynomial algebra (with coefficients in 𝑅) to a function from assignments 𝐼⟶𝑆 of the variables to elements of 𝑆 formed by evaluating the polynomial with the given assignments. (Contributed by Stefan O'Rear, 11-Mar-2015.)
⊢ evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))))
Definition	df-evl 21998*	A simplification of evalSub when the evaluation ring is the same as the coefficient ring. (Contributed by Stefan O'Rear, 19-Mar-2015.)
⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
Theorem	evlslem4 21999*	The support of a tensor product of ring element families is contained in the product of the supports. (Contributed by Stefan O'Rear, 8-Mar-2015.) (Revised by AV, 18-Jul-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑋 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))
Theorem	psrbagev1 22000*	A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = (.g‘𝑇)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ CMnd)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)    ⇒   ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 ))