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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | subrgmvr 20701 | 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 20702 | 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 20703* | 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 20704* | 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 20705* | 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 20706* | 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 20707* | Lemma for mplcoe4 20742. (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 20708* | Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 20709), 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 20709* | 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 20710 | 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 20711* | 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 20712* | 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 20713 | Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.) |
⊢ Rel dom ordPwSer | ||
Theorem | opsrval 20714* | The value of the "ordered power series" function. This is the same as mPwSer psrval 20600, 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 20715* | 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 20716 | 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 20717 | 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) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 < ;10 ⇒ ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) | ||
Theorem | opsrbas 20718 | 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) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑂)) | ||
Theorem | opsrplusg 20719 | 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) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑂)) | ||
Theorem | opsrmulr 20720 | 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) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑂)) | ||
Theorem | opsrvsca 20721 | 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) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑂)) | ||
Theorem | opsrsca 20722 | 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) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂)) | ||
Theorem | opsrtoslem1 20723* | Lemma for opsrtos 20725. (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 20724* | Lemma for opsrtos 20725. (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 20725 | 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 20726 | 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 20727 | 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 20728 | 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 | mplrcl 20729 | 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 20730 | 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 | mvrf2 20731 | 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 20732* | 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 20733* | 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 20734* | 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 20735* | 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 20736 | 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 20737 | 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 20738 | 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 20739* | 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 20740* | 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 20741* | 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 20742* | 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 ))))) | ||
Syntax | ces 20743 | Evaluation of a multivariate polynomial in a subring. |
class evalSub | ||
Syntax | cevl 20744 | Evaluation of a multivariate polynomial. |
class eval | ||
Definition | df-evls 20745* | 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 20746* | 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 20747* | 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 | psrbagfsupp 20748* | Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) |
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ⇒ ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0) | ||
Theorem | psrbagev1 20749* | 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.) |
⊢ 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 20750* | 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.) |
⊢ 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 20751* | 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 20752* | Lemma for evlseu 20755. 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 20753* | Lemma for evlseu 20755. 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 20754* | Lemma for evlseu 20755, 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 20755* | 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 20756 | Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
⊢ Rel dom evalSub | ||
Theorem | mpfrcl 20757 | 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 20758* | 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 20759* | 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 20760 | 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 20761 | 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 20762* | 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 20763* | 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 20764* | 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 20765 | 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))) | ||
Theorem | evlsvarpw 20766 | Polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by SN, 21-Feb-2024.) |
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝑊 = (𝐼 mPoly 𝑈) & ⊢ 𝐺 = (mulGrp‘𝑊) & ⊢ ↑ = (.g‘𝐺) & ⊢ 𝑋 = ((𝐼 mVar 𝑈)‘𝑌) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝑃 = (𝑆 ↑s (𝐵 ↑m 𝐼)) & ⊢ 𝐻 = (mulGrp‘𝑃) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘𝐻)(𝑄‘𝑋))) | ||
Theorem | evlval 20767 | 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) | ||
Theorem | evlrhm 20768 | 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 = (𝑅 ↑s (𝐵 ↑m 𝐼)) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑄 ∈ (𝑊 RingHom 𝑇)) | ||
Theorem | evlsscasrng 20769 | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐶 = (algSc‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝑅) ⇒ ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋))) | ||
Theorem | evlsca 20770 | 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) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) | ||
Theorem | evlsvarsrng 20771 | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝐴) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑂‘(𝑉‘𝑋))) | ||
Theorem | evlvar 20772* | Simple polynomial evaluation maps variables to projections. (Contributed by AV, 12-Sep-2019.) |
⊢ 𝑄 = (𝐼 eval 𝑆) & ⊢ 𝑉 = (𝐼 mVar 𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) | ||
Theorem | mpfconst 20773 | Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝑅) ⇒ ⊢ (𝜑 → ((𝐵 ↑m 𝐼) × {𝑋}) ∈ 𝑄) | ||
Theorem | mpfproj 20774* | Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑓‘𝐽)) ∈ 𝑄) | ||
Theorem | mpfsubrg 20775 | 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) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)))) | ||
Theorem | mpff 20776 | Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ 𝑄 → 𝐹:(𝐵 ↑m 𝐼)⟶𝐵) | ||
Theorem | mpfaddcl 20777 | The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ + = (+g‘𝑆) ⇒ ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹 ∘f + 𝐺) ∈ 𝑄) | ||
Theorem | mpfmulcl 20778 | The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ · = (.r‘𝑆) ⇒ ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹 ∘f · 𝐺) ∈ 𝑄) | ||
Theorem | mpfind 20779* | 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) & ⊢ + = (+g‘𝑆) & ⊢ · = (.r‘𝑆) & ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁) & ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎) & ⊢ (𝑥 = ((𝐵 ↑m 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂)) & ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁)) & ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → 𝜒) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → 𝜃) & ⊢ (𝜑 → 𝐴 ∈ 𝑄) ⇒ ⊢ (𝜑 → 𝜌) | ||
Syntax | cslv 20780 | Select a subset of variables in a multivariate polynomial. |
class selectVars | ||
Syntax | cmhp 20781 | Multivariate polynomials. |
class mHomP | ||
Syntax | cpsd 20782 | Power series partial derivative function. |
class mPSDer | ||
Syntax | cai 20783 | Algebraically independent. |
class AlgInd | ||
Definition | df-selv 20784* | Define the "variable selection" function. The function ((𝐼 selectVars 𝑅)‘𝐽) maps elements of (𝐼 mPoly 𝑅) bijectively onto (𝐽 mPoly ((𝐼 ∖ 𝐽) 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 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ ⦋((𝑖 ∖ 𝑗) mPoly 𝑟) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝑖 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝑖 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥)))))))) | ||
Definition | df-mhp 20785* | 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 (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})) | ||
Definition | df-psd 20786* | Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0))))))))) | ||
Definition | df-algind 20787* | 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 (𝑤 ↾s 𝑘))) ↦ ((((𝑣 evalSub 𝑤)‘𝑘)‘𝑓)‘( I ↾ 𝑣)))}) | ||
Theorem | selvffval 20788* | Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝑗) mPoly 𝑅) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)))))))) | ||
Theorem | selvfval 20789* | Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) ⇒ ⊢ (𝜑 → ((𝐼 selectVars 𝑅)‘𝐽) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))) | ||
Theorem | selvval 20790* | Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))) | ||
Theorem | mhpfval 20791* | Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})) | ||
Theorem | mhpval 20792* | Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}}) | ||
Theorem | ismhp 20793* | Property of being a homogeneous polynomial. (Contributed by Steven Nguyen, 25-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) | ||
Theorem | ismhp2 20794* | Deduce a homogeneous polynomial from its properties. (Contributed by SN, 25-May-2024.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | ||
Theorem | mhpmpl 20795 | A homogeneous polynomial is a polynomial. (Contributed by Steven Nguyen, 25-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐵) | ||
Theorem | mhpdeg 20796* | All nonzero terms of a homogeneous polynomial have degree 𝑁. (Contributed by Steven Nguyen, 25-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) | ||
Theorem | mhp0cl 20797* | The zero polynomial is homogeneous. (Contributed by SN, 12-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐷 × { 0 }) ∈ (𝐻‘𝑁)) | ||
Theorem | mhpvarcl 20798 | A power series variable is a polynomial of degree 1. (Contributed by SN, 25-May-2024.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑉‘𝑋) ∈ (𝐻‘1)) | ||
Theorem | mhpaddcl 20799 | Homogeneous polynomials are closed under addition. (Contributed by SN, 26-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ + = (+g‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) & ⊢ (𝜑 → 𝑌 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐻‘𝑁)) | ||
Theorem | mhpinvcl 20800 | Homogeneous polynomials are closed under taking the opposite. (Contributed by SN, 12-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑀 = (invg‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁)) |
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