P. 215 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mplelsfi 21401	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 → 𝑅 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 0 )
Theorem	mplval2 21402	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 21403	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 21404*	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 21405*	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 21406*	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 21407*	Lemma for mplsubg 21408 and mpllss 21409. (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 21408	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 21409	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 21410*	Lemma for mplsubrg 21411. (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 21411	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 21412*	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	mpladd 21413	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 21414	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 21415*	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 21416*	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 21417	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 21418	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 21419*	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 21420*	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	mvrcl 21421	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	mplgrp 21422	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 21423	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 21424	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 21425	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 21426	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 21427	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	ressmplbas2 21428	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 21429	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 21430	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 21431	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 21432	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 21433	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 21434	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 21435	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 21436*	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 21437*	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 21438*	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 21439*	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 21440*	Lemma for mplcoe4 21479. (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 21441*	Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 21442), 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 21442*	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 21443	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 21444*	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 21445*	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 21446	Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
⊢ Rel dom ordPwSer
Theorem	opsrval 21447*	The value of the "ordered power series" function. This is the same as mPwSer psrval 21317, 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 21448*	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 21449	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 21450	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	opsrbaslemOLD 21451	Obsolete version of opsrbaslem 21450 as of 1-Nov-2024. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝑁 < ;10    ⇒   ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂))
Theorem	opsrbas 21452	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	opsrbasOLD 21453	Obsolete version of opsrbaslem 21450 as of 1-Nov-2024. The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑂))
Theorem	opsrplusg 21454	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	opsrplusgOLD 21455	Obsolete version of opsrplusg 21454 as of 1-Nov-2024. The addition operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑂))
Theorem	opsrmulr 21456	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	opsrmulrOLD 21457	Obsolete version of opsrmulr 21456 as of 1-Nov-2024. The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑂))
Theorem	opsrvsca 21458	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	opsrvscaOLD 21459	Obsolete version of opsrvsca 21458 as of 1-Nov-2024. The scalar product of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑂))
Theorem	opsrsca 21460	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	opsrscaOLD 21461	Obsolete version of opsrsca 21460 as of 1-Nov-2024. The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 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 21462*	Lemma for opsrtos 21464. (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 21463*	Lemma for opsrtos 21464. (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 21464	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 21465	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 21466	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 21467	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	mvrf2 21468	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	mplmon2 21469*	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 21470*	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 21471*	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 21472*	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 21473	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 21474	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 21475	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 21476*	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 21477*	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 21478*	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 21479*	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 21480	Evaluation of a multivariate polynomial in a subring.
class evalSub
Syntax	cevl 21481	Evaluation of a multivariate polynomial.
class eval
Definition	df-evls 21482*	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 21483*	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 21484*	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 21485*	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 ))
Theorem	psrbagev1OLD 21486*	Obsolete version of psrbagev1 21485 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 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 → 𝐼 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 ))
Theorem	psrbagev2 21487*	Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened 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)    &   ⊢ (𝜑 → 𝑇 ∈ CMnd)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)    ⇒   ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶)
Theorem	psrbagev2OLD 21488*	Obsolete version of psrbagev2 21487 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = (.g‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ CMnd)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶)
Theorem	evlslem2 21489*	A linear function on the polynomial ring which is multiplicative on scaled monomials is generally multiplicative. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 11-Apr-2024.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐸 ∈ (𝑃 GrpHom 𝑆))    &   ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷))) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑗 ∘f + 𝑖), ((𝑥‘𝑗)(.r‘𝑅)(𝑦‘𝑖)), 0 ))) = ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(.r‘𝑃)𝑦)) = ((𝐸‘𝑥) · (𝐸‘𝑦)))
Theorem	evlslem3 21490*	Lemma for evlseu 21493. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.) (Revised by AV, 11-Apr-2024.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = (mulGrp‘𝑆)    &   ⊢ ↑ = (.g‘𝑇)    &   ⊢ · = (.r‘𝑆)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐸 = (𝑝 ∈ 𝐵 ↦ (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝐸‘(𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝐴, 𝐻, 0 ))) = ((𝐹‘𝐻) · (𝑇 Σg (𝐴 ∘f ↑ 𝐺))))
Theorem	evlslem6 21491*	Lemma for evlseu 21493. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.) (Revised by AV, 11-Apr-2024.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = (mulGrp‘𝑆)    &   ⊢ ↑ = (.g‘𝑇)    &   ⊢ · = (.r‘𝑆)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐸 = (𝑝 ∈ 𝐵 ↦ (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))):𝐷⟶𝐶 ∧ (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) finSupp (0g‘𝑆)))
Theorem	evlslem1 21492*	Lemma for evlseu 21493, give a formula for (the unique) polynomial evaluation homomorphism. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 26-Jul-2019.) (Revised by AV, 11-Apr-2024.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = (mulGrp‘𝑆)    &   ⊢ ↑ = (.g‘𝑇)    &   ⊢ · = (.r‘𝑆)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐸 = (𝑝 ∈ 𝐵 ↦ (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ (𝜑 → (𝐸 ∈ (𝑃 RingHom 𝑆) ∧ (𝐸 ∘ 𝐴) = 𝐹 ∧ (𝐸 ∘ 𝑉) = 𝐺))
Theorem	evlseu 21493*	For a given interpretation of the variables 𝐺 and of the scalars 𝐹, this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 11-Apr-2024.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)    ⇒   ⊢ (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
Theorem	reldmevls 21494	Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
⊢ Rel dom evalSub
Theorem	mpfrcl 21495	Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    ⇒   ⊢ (𝑋 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
Theorem	evlsval 21496*	Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.) (Revised by AV, 18-Sep-2021.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑉 = (𝐼 mVar 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼))    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))    &   ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)))    ⇒   ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌)))
Theorem	evlsval2 21497*	Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Revised by AV, 18-Sep-2021.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑉 = (𝐼 mVar 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼))    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))    &   ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)))    ⇒   ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))
Theorem	evlsrhm 21498	Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼))    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))
Theorem	evlssca 21499	Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    ⇒   ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋}))
Theorem	evlsvar 21500*	Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋)))