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	mvrcl 21901	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 21902	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 21903	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 21904	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 21905	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 21906	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 21907*	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 21908*	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 21909*	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 21910*	Lemma for mplsubg 21911 and mpllss 21912. (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 21911	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 21912	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 21913*	Lemma for mplsubrg 21914. (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 21914	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 21915*	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 21916	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 21917	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 21918	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 21919	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 21920*	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 21921*	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 21922	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 21923	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 21924*	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 21925*	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 21926	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 21927	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 21928	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 21929	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 21930	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 21931	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 21932	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 21933	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 21934	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 21935	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 21936	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 21937	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 21938	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 21939	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 21940	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 21941	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 21942*	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 21943*	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 21944*	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 21945*	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 21946*	Lemma for mplcoe4 21978. (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 21947*	Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 21948), 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 21948*	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 21949	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 21950*	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 21951*	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 21952	Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
⊢ Rel dom ordPwSer
Theorem	opsrval 21953*	The value of the "ordered power series" function. This is the same as mPwSer psrval 21824, 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 21954*	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 21955	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 21956	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 21957	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 21958	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 21959	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 21960	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 21961	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 21962*	Lemma for opsrtos 21964. (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 21963*	Lemma for opsrtos 21964. (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 21964	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 21965	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 21966	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 21967	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 21968*	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 21969*	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 21970*	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 21971*	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 21972	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 21973	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 21974	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 21975*	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 21976*	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 21977*	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 21978*	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 21979	Evaluation of a multivariate polynomial in a subring.
class evalSub
Syntax	cevl 21980	Evaluation of a multivariate polynomial.
class eval
Definition	df-evls 21981*	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 21982*	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 21983*	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 21984*	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	psrbagev2 21985*	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	evlslem2 21986*	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 21987*	Lemma for evlseu 21990. 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 21988*	Lemma for evlseu 21990. 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 21989*	Lemma for evlseu 21990, 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 21990*	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 21991	Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
⊢ Rel dom evalSub
Theorem	mpfrcl 21992	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 21993*	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 21994*	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 21995	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 21996	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 21997*	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)))
Theorem	evlsgsumadd 21998*	Polynomial evaluation maps (additive) group sums to group sums. (Contributed by SN, 13-Feb-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))
Theorem	evlsgsummul 21999*	Polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by SN, 13-Feb-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   ⊢ 𝐻 = (mulGrp‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 )    ⇒   ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))
Theorem	evlspw 22000	Polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   ⊢ 𝐻 = (mulGrp‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘𝐻)(𝑄‘𝑋)))