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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmplcrng 19501 The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ CRing) → 𝑃 ∈ CRing)

Theoremmplassa 19502 The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ CRing) → 𝑃 ∈ AssAlg)

Theoremressmplbas2 19503 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 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑊 = (𝐼 mPwSer 𝐻)    &   𝐶 = (Base‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝜑𝐵 = (𝐶𝐾))

Theoremressmplbas 19504 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       (𝜑𝐵 = (Base‘𝑃))

Theoremressmpladd 19505 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(+g𝑈)𝑌) = (𝑋(+g𝑃)𝑌))

Theoremressmplmul 19506 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))

Theoremressmplvsca 19507 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝑇𝑌𝐵)) → (𝑋( ·𝑠𝑈)𝑌) = (𝑋( ·𝑠𝑃)𝑌))

Theoremsubrgmpl 19508 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)       ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

Theoremsubrgmvr 19509 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)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐻 = (𝑅s 𝑇)       (𝜑𝑉 = (𝐼 mVar 𝐻))

Theoremsubrgmvrf 19510 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)       (𝜑𝑉:𝐼𝐵)

Theoremmplmon 19511* A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐷)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)

Theoremmplmonmul 19512* 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 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐷)    &    · = (.r𝑃)    &   (𝜑𝑌𝐷)       (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = (𝑋𝑓 + 𝑌), 1 , 0 )))

Theoremmplcoe1 19513* Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐵 = (Base‘𝑃)    &    · = ( ·𝑠𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑𝑋 = (𝑃 Σg (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))

Theoremmplcoe3 19514* 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 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐼)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 (𝑉𝑋)))

Theoremmplcoe5lem 19515* Lemma for mplcoe4 19551. (Contributed by AV, 7-Oct-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑌𝐷)    &   (𝜑 → ∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)))    &   (𝜑𝑆𝐼)       (𝜑 → ran (𝑘𝑆 ↦ ((𝑌𝑘) (𝑉𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝑆 ↦ ((𝑌𝑘) (𝑉𝑘)))))

Theoremmplcoe5 19516* Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 19517), 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 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑌𝐷)    &   (𝜑 → ∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)))       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))

Theoremmplcoe2 19517* 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 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐷)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))

Theoremmplbas2 19518 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)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))

Theoremltbval 19519* Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑇𝑊)       (𝜑𝐶 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})

Theoremltbwe 19520* 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 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑇𝑊)    &   (𝜑𝑇 We 𝐼)       (𝜑𝐶 We 𝐷)

Theoremreldmopsr 19521 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
Rel dom ordPwSer

Theoremopsrval 19522* The value of the "ordered power series" function. This is the same as mPwSer psrval 19410, 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 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 = (𝑆 sSet ⟨(le‘ndx), ⟩))

Theoremopsrle 19523* 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 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &    = (le‘𝑂)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))})

Theoremopsrval2 19524 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 = (𝑆 sSet ⟨(le‘ndx), ⟩))

Theoremopsrbaslem 19525 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.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 10       (𝜑 → (𝐸𝑆) = (𝐸𝑂))

TheoremopsrbaslemOLD 19526 Obsolete version of opsrbaslem 19525 as of 9-Sep-2021. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 10       (𝜑 → (𝐸𝑆) = (𝐸𝑂))

Theoremopsrbas 19527 The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (Base‘𝑆) = (Base‘𝑂))

Theoremopsrplusg 19528 The addition operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (+g𝑆) = (+g𝑂))

Theoremopsrmulr 19529 The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (.r𝑆) = (.r𝑂))

Theoremopsrvsca 19530 The scalar product operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → ( ·𝑠𝑆) = ( ·𝑠𝑂))

Theoremopsrsca 19531 The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝑅 = (Scalar‘𝑂))

Theoremopsrtoslem1 19532* Lemma for opsrtos 19534. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Toset)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝑇 We 𝐼)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    < = (lt‘𝑅)    &   𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜓 ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))    &    = (le‘𝑂)       (𝜑 = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))

Theoremopsrtoslem2 19533* Lemma for opsrtos 19534. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Toset)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝑇 We 𝐼)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    < = (lt‘𝑅)    &   𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜓 ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))    &    = (le‘𝑂)       (𝜑𝑂 ∈ Toset)

Theoremopsrtos 19534 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Toset)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝑇 We 𝐼)       (𝜑𝑂 ∈ Toset)

Theoremopsrso 19535 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Toset)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝑇 We 𝐼)    &    = (lt‘𝑂)    &   𝐵 = (Base‘𝑂)       (𝜑 Or 𝐵)

Theoremopsrcrng 19536 The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 ∈ CRing)

Theoremopsrassa 19537 The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 ∈ AssAlg)

Theoremmplrcl 19538 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)       (𝑋𝐵𝐼 ∈ V)

Theoremmplelsfi 19539 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 = (0g𝑅)    &   (𝜑𝐹𝐵)    &   (𝜑𝑅𝑉)       (𝜑𝐹 finSupp 0 )

Theoremmvrf2 19540 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)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑉:𝐼𝐵)

Theoremmplmon2 19541* Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &    · = ( ·𝑠𝑃)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐾𝐷)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 · (𝑦𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))

Theorempsrbag0 19542* The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐼𝑉 → (𝐼 × {0}) ∈ 𝐷)

Theorempsrbagsn 19543* A singleton bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐼𝑉 → (𝑥𝐼 ↦ if(𝑥 = 𝐾, 1, 0)) ∈ 𝐷)

Theoremmplascl 19544* Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐴𝑋) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 )))

Theoremmplasclf 19545 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)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝐴:𝐾𝐵)

Theoremsubrgascl 19546 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 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐶 = (algSc‘𝑈)       (𝜑𝐶 = (𝐴𝑇))

Theoremsubrgasclcl 19547 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 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐵 = (Base‘𝑈)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝑋𝐾)       (𝜑 → ((𝐴𝑋) ∈ 𝐵𝑋𝑇))

Theoremmplmon2cl 19548* A scaled monomial is a polynomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐶 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝑋𝐶)    &   (𝜑𝐾𝐷)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )) ∈ 𝐵)

Theoremmplmon2mul 19549* Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐶 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)    &    = (.r𝑃)    &    · = (.r𝑅)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)    &   (𝜑𝐹𝐶)    &   (𝜑𝐺𝐶)       (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 𝐹, 0 )) (𝑦𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 ))) = (𝑦𝐷 ↦ if(𝑦 = (𝑋𝑓 + 𝑌), (𝐹 · 𝐺), 0 )))

Theoremmplind 19550* 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)    &    + = (+g𝑌)    &    · = (.r𝑌)    &   𝐶 = (algSc‘𝑌)    &   𝐵 = (Base‘𝑌)    &   ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥 + 𝑦) ∈ 𝐻)    &   ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥 · 𝑦) ∈ 𝐻)    &   ((𝜑𝑥𝐾) → (𝐶𝑥) ∈ 𝐻)    &   ((𝜑𝑥𝐼) → (𝑉𝑥) ∈ 𝐻)    &   (𝜑𝑋𝐵)    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑋𝐻)

Theoremmplcoe4 19551* Decompose a polynomial into a finite sum of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑𝑋 = (𝑃 Σg (𝑘𝐷 ↦ (𝑦𝐷 ↦ if(𝑦 = 𝑘, (𝑋𝑘), 0 )))))

10.10.2  Polynomial evaluation

Syntaxces 19552 Evaluation of a multivariate polynomial in a subring.
class evalSub

Syntaxcevl 19553 Evaluation of a multivariate polynomial.
class eval

Definitiondf-evls 19554* Define the evaluation map for the polynomial algebra. The function ((𝐼 evalSub 𝑆)‘𝑅):𝑉⟶(𝑆𝑚 (𝑆𝑚 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 𝐼𝑆 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 (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥)))))))

Definitiondf-evl 19555* 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)))

Theoremevlslem4 19556* 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 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑥𝐼) → 𝑋𝐵)    &   ((𝜑𝑦𝐽) → 𝑌𝐵)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)       (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))

Theorempsrbagfsupp 19557* Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.)
𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}       ((𝑋𝐷𝐼𝑉) → 𝑋 finSupp 0)

Theorempsrbagev1 19558* 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.)
𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐶 = (Base‘𝑇)    &    · = (.g𝑇)    &    0 = (0g𝑇)    &   (𝜑𝑇 ∈ CMnd)    &   (𝜑𝐵𝐷)    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝐼 ∈ V)       (𝜑 → ((𝐵𝑓 · 𝐺):𝐼𝐶 ∧ (𝐵𝑓 · 𝐺) finSupp 0 ))

Theorempsrbagev2 19559* Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 18-Jul-2019.)
𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐶 = (Base‘𝑇)    &    · = (.g𝑇)    &    0 = (0g𝑇)    &   (𝜑𝑇 ∈ CMnd)    &   (𝜑𝐵𝐷)    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝐼 ∈ V)       (𝜑 → (𝑇 Σg (𝐵𝑓 · 𝐺)) ∈ 𝐶)

Theoremevlslem2 19560* A linear function on the polynomial ring which is multiplicative on scaled monomials is generally multiplicative. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑆)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐸 ∈ (𝑃 GrpHom 𝑆))    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑗𝐷𝑖𝐷))) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = (𝑗𝑓 + 𝑖), ((𝑥𝑗)(.r𝑅)(𝑦𝑖)), 0 ))) = ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))       ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘(𝑥(.r𝑃)𝑦)) = ((𝐸𝑥) · (𝐸𝑦)))

Theoremevlslem6 19561* Lemma for evlseu 19564. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝑇 = (mulGrp‘𝑆)    &    = (.g𝑇)    &    · = (.r𝑆)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐸 = (𝑝𝐵 ↦ (𝑆 Σg (𝑏𝐷 ↦ ((𝐹‘(𝑝𝑏)) · (𝑇 Σg (𝑏𝑓 𝐺))))))    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑏𝐷 ↦ ((𝐹‘(𝑌𝑏)) · (𝑇 Σg (𝑏𝑓 𝐺)))):𝐷𝐶 ∧ (𝑏𝐷 ↦ ((𝐹‘(𝑌𝑏)) · (𝑇 Σg (𝑏𝑓 𝐺)))) finSupp (0g𝑆)))

Theoremevlslem3 19562* Lemma for evlseu 19564. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝑇 = (mulGrp‘𝑆)    &    = (.g𝑇)    &    · = (.r𝑆)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐸 = (𝑝𝐵 ↦ (𝑆 Σg (𝑏𝐷 ↦ ((𝐹‘(𝑝𝑏)) · (𝑇 Σg (𝑏𝑓 𝐺))))))    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)    &    0 = (0g𝑅)    &   (𝜑𝐴𝐷)    &   (𝜑𝐻𝐾)       (𝜑 → (𝐸‘(𝑥𝐷 ↦ if(𝑥 = 𝐴, 𝐻, 0 ))) = ((𝐹𝐻) · (𝑇 Σg (𝐴𝑓 𝐺))))

Theoremevlslem1 19563* Lemma for evlseu 19564, give a formula for (the unique) polynomial evaluation homomorphism. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 26-Jul-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝑇 = (mulGrp‘𝑆)    &    = (.g𝑇)    &    · = (.r𝑆)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐸 = (𝑝𝐵 ↦ (𝑆 Σg (𝑏𝐷 ↦ ((𝐹‘(𝑝𝑏)) · (𝑇 Σg (𝑏𝑓 𝐺))))))    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)    &   𝐴 = (algSc‘𝑃)       (𝜑 → (𝐸 ∈ (𝑃 RingHom 𝑆) ∧ (𝐸𝐴) = 𝐹 ∧ (𝐸𝑉) = 𝐺))

Theoremevlseu 19564* 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.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐶 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑃)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)       (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))

Theoremreldmevls 19565 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
Rel dom evalSub

Theoremmpfrcl 19566 Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)       (𝑋𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))

Theoremevlsval 19567* 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 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐵𝑚 𝐼))    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝑋 = (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥}))    &   𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)))       ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌)))

Theoremevlsval2 19568* 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 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐵𝑚 𝐼))    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝑋 = (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥}))    &   𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)))       ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))

Theoremevlsrhm 19569 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 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐵𝑚 𝐼))    &   𝐵 = (Base‘𝑆)       ((𝐼𝑉𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Theoremevlssca 19570 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 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = ((𝐵𝑚 𝐼) × {𝑋}))

Theoremevlsvar 19571* 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 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑋)))

Theoremevlval 19572 Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
𝑄 = (𝐼 eval 𝑅)    &   𝐵 = (Base‘𝑅)       𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)

Theoremevlrhm 19573 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = (𝐼 eval 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝑅s (𝐵𝑚 𝐼))       ((𝐼𝑉𝑅 ∈ CRing) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Theoremevlsscasrng 19574 The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 12-Sep-2019.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑂 = (𝐼 eval 𝑆)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝐼 mPoly 𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝐶 = (algSc‘𝑃)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = (𝑂‘(𝐶𝑋)))

Theoremevlsca 19575 Simple polynomial evaluation maps scalars to constant functions. (Contributed by AV, 12-Sep-2019.)
𝑄 = (𝐼 eval 𝑆)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑄‘(𝐴𝑋)) = ((𝐵𝑚 𝐼) × {𝑋}))

Theoremevlsvarsrng 19576 The evaluation of the variable of polynomials over subring yields the same result as evaluated as variable of the polynomials over the ring itself. (Contributed by AV, 12-Sep-2019.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑂 = (𝐼 eval 𝑆)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝐴)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑂‘(𝑉𝑋)))

Theoremevlvar 19577* Simple polynomial evaluation maps variables to projections. (Contributed by AV, 12-Sep-2019.)
𝑄 = (𝐼 eval 𝑆)    &   𝑉 = (𝐼 mVar 𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑋)))

Theoremmpfconst 19578 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → ((𝐵𝑚 𝐼) × {𝑋}) ∈ 𝑄)

Theoremmpfproj 19579* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐽𝐼)       (𝜑 → (𝑓 ∈ (𝐵𝑚 𝐼) ↦ (𝑓𝐽)) ∈ 𝑄)

Theoremmpfsubrg 19580 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) (Revised by AV, 19-Sep-2021.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)       ((𝐼𝑉𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆s ((Base‘𝑆) ↑𝑚 𝐼))))

Theoremmpff 19581 Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐵 = (Base‘𝑆)       (𝐹𝑄𝐹:(𝐵𝑚 𝐼)⟶𝐵)

Theoremmpfaddcl 19582 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &    + = (+g𝑆)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 + 𝐺) ∈ 𝑄)

Theoremmpfmulcl 19583 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &    · = (.r𝑆)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 · 𝐺) ∈ 𝑄)

Theoremmpfind 19584* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    · = (.r𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)    &   (𝑥 = ((𝐵𝑚 𝐼) × {𝑓}) → (𝜓𝜒))    &   (𝑥 = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) → (𝜓𝜃))    &   (𝑥 = 𝑓 → (𝜓𝜏))    &   (𝑥 = 𝑔 → (𝜓𝜂))    &   (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))    &   (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))    &   (𝑥 = 𝐴 → (𝜓𝜌))    &   ((𝜑𝑓𝑅) → 𝜒)    &   ((𝜑𝑓𝐼) → 𝜃)    &   (𝜑𝐴𝑄)       (𝜑𝜌)

10.10.3  Additional definitions for (multivariate) polynomials

Remark: There are no theorems using these definitions yet!

Syntaxcmhp 19585 Multivariate polynomials.
class mHomP

Syntaxcpsd 19586 Power series partial derivative function.
class mPSDer

Syntaxcslv 19587 Select a subset of variables in a multivariate polynomial.
class selectVars

Syntaxcai 19588 Algebraically independent.
class AlgInd

Definitiondf-mhp 19589* Define the subspaces of order- 𝑛 homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g𝑟)) ⊆ {𝑔 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔𝑗) = 𝑛}}))

Definitiondf-psd 19590* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑥) + 1)(.g𝑟)(𝑓‘(𝑘𝑓 + (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))))))

Definitiondf-selv 19591* Define the "variable selection" function. The function ((𝐼 selectVars 𝑅)‘𝐽) maps elements of (𝐼 mPoly 𝑅) bijectively onto (𝐽 mPoly ((𝐼𝐽) mPoly 𝑅)) in the natural way, for example if 𝐼 = {𝑥, 𝑦} and 𝐽 = {𝑦} it would map 1 + 𝑥 + 𝑦 + 𝑥𝑦 ∈ ({𝑥, 𝑦} mPoly ℤ) to (1 + 𝑥) + (1 + 𝑥)𝑦 ∈ ({𝑦} mPoly ({𝑥} mPoly ℤ)). This, for example, allows one to treat a multivariate polynomial as a univariate polynomial with coefficients in a polynomial ring with one less variable. (Contributed by Mario Carneiro, 21-Mar-2015.)
selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (𝑖 mPoly 𝑟) ↦ ((𝑖𝑗) mPoly 𝑟) / 𝑠(𝑥 ∈ (Scalar‘𝑠) ↦ (𝑥( ·𝑠𝑠)(1r𝑠))) / 𝑐((((𝑖 evalSub 𝑠)‘(𝑐s 𝑟))‘(𝑐𝑓))‘(𝑥𝑖 ↦ if(𝑥𝑗, ((𝑗 mVar ((𝑖𝑗) mPoly 𝑟))‘𝑥), (𝑐 ∘ (((𝑖𝑗) mVar 𝑟)‘𝑥))))))))

Definitiondf-algind 19592* Define the predicate "the set 𝑣 is algebraically independent in the algebra 𝑤". A collection of vectors is algebraically independent if no nontrivial polynomial with elements from the subset evaluates to zero. (Contributed by Mario Carneiro, 21-Mar-2015.)
AlgInd = (𝑤 ∈ V, 𝑘 ∈ 𝒫 (Base‘𝑤) ↦ {𝑣 ∈ 𝒫 (Base‘𝑤) ∣ Fun (𝑓 ∈ (Base‘(𝑣 mPoly (𝑤s 𝑘))) ↦ ((((𝑣 evalSub 𝑤)‘𝑘)‘𝑓)‘( I ↾ 𝑣)))})

10.10.4  Univariate polynomials

According to Wikipedia ("Polynomial", 23-Dec-2019, https://en.wikipedia.org/wiki/Polynomial) "A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial." In this sense univariate polynomials are defined as multivariate polynomials restricted to one indeterminate/polynomial variable in the following, see ply1bascl2 19622.

According to the definition in Wikipedia "a polynomial can either be zero or can be written as the sum of a finite number of nonzero terms. Each term consists of the product of a number - called the coefficient of the term - and a finite number of indeterminates, raised to nonnegative integer powers.". By this, a term of a univariate polynomial (often also called "polynomial term") is the product of a coefficient (usually a member of the underlying ring) and the variable, raised to a nonnegative integer power.

A (univariate) polynomial which has only one term is called (univariate) monomial - therefore, the notions "term" and "monomial" are often used synonymously, see also the definition in [Lang] p. 102. Sometimes, however, a monomial is defined as power product, "a product of powers of variables with nonnegative integer exponents", see Wikipedia ("Monomial", 23-Dec-2019, https://en.wikipedia.org/wiki/Mononomial). In [Lang] p. 101, such terms are called "primitive monomials". To avoid any ambiguity, the notion "primitive monomial" is used for such power products ("x^i") in the following, whereas the synonym for "term" ("ai x^i") will be "scaled monomial".

Syntaxcps1 19593 Univariate power series.
class PwSer1

Syntaxcv1 19594 The base variable of a univariate power series.
class var1

Syntaxcpl1 19595 Univariate polynomials.
class Poly1

Syntaxcco1 19596 Coefficient function for a univariate polynomial.
class coe1

Syntaxctp1 19597 Convert a univariate polynomial representation to multivariate.
class toPoly1

Definitiondf-psr1 19598 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))

Definitiondf-vr1 19599 Define the base element of a univariate power series (the 𝑋 element of the set 𝑅[𝑋] of polynomials and also the 𝑋 in the set 𝑅[[𝑋]] of power series). (Contributed by Mario Carneiro, 8-Feb-2015.)
var1 = (𝑟 ∈ V ↦ ((1𝑜 mVar 𝑟)‘∅))

Definitiondf-ply1 19600 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))))

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