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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mvrval2 20801* | Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) | ||
Theorem | mvrid 20802* | The 𝑋𝑖-th coefficient of the term 𝑋𝑖 is 1. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑉‘𝑋)‘(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1 ) | ||
Theorem | mvrf 20803 | The power series variable function is a function from the index set to elements of the power series structure representing 𝑋𝑖 for each 𝑖. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) | ||
Theorem | mvrf1 20804 | The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 1 ≠ 0 ) ⇒ ⊢ (𝜑 → 𝑉:𝐼–1-1→𝐵) | ||
Theorem | mvrcl2 20805 | A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) | ||
Theorem | reldmmpl 20806 | The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ Rel dom mPoly | ||
Theorem | mplval 20807* | Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } ⇒ ⊢ 𝑃 = (𝑆 ↾s 𝑈) | ||
Theorem | mplbas 20808* | Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) ⇒ ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } | ||
Theorem | mplelbas 20809 | Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) ⇒ ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 )) | ||
Theorem | mplrcl 20810 | 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 20811 | A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 finSupp 0 ) | ||
Theorem | mplval2 20812 | 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 20813 | 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 20814* | 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 20815* | 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 20816* | 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 20817* | Lemma for mplsubg 20818 and mpllss 20819. (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 20818 | 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 20819 | 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 20820* | Lemma for mplsubrg 20821. (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 20821 | 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 20822* | The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑂 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Grp) ⇒ ⊢ (𝜑 → 0 = (𝐷 × {𝑂})) | ||
Theorem | mpladd 20823 | 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 20824 | 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 20825* | 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 20826* | 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 20827 | 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 20828 | 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 20829* | 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 20830* | The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ ∙ = ( ·𝑠 ‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) | ||
Theorem | mvrcl 20831 | A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) | ||
Theorem | mplgrp 20832 | 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 20833 | 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 20834 | 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 20835 | 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 20836 | 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 20837 | The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ AssAlg) | ||
Theorem | ressmplbas2 20838 | 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 20839 | 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 20840 | 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 20841 | 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 20842 | 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 20843 | 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 20844 | 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 20845 | 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 20846* | 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 20847* | 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 20848* | 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 20849* | 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 20850* | Lemma for mplcoe4 20883. (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 20851* | Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 20852), 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 20852* | 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 20853 | 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 20854* | 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 20855* | 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 20856 | Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.) |
⊢ Rel dom ordPwSer | ||
Theorem | opsrval 20857* | The value of the "ordered power series" function. This is the same as mPwSer psrval 20728, 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 20858* | 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 20859 | 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 20860 | 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 20861 | 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 20862 | 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 20863 | 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 20864 | 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 20865 | 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 20866* | Lemma for opsrtos 20868. (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 20867* | Lemma for opsrtos 20868. (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 20868 | 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 20869 | 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 20870 | 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 20871 | The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → 𝑂 ∈ AssAlg) | ||
Theorem | mvrf2 20872 | 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 20873* | 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 20874* | 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 20875* | 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 20876* | 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 20877 | 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 20878 | 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 20879 | 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 20880* | 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 20881* | 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 20882* | 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 20883* | 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 20884 | Evaluation of a multivariate polynomial in a subring. |
class evalSub | ||
Syntax | cevl 20885 | Evaluation of a multivariate polynomial. |
class eval | ||
Definition | df-evls 20886* | 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 20887* | 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 20888* | 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 20889* | A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.) |
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = (.g‘𝑇) & ⊢ 0 = (0g‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ CMnd) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) | ||
Theorem | psrbagev1OLD 20890* | Obsolete version of psrbagev1 20889 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = (.g‘𝑇) & ⊢ 0 = (0g‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ CMnd) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) | ||
Theorem | psrbagev2 20891* | Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.) |
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = (.g‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ CMnd) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶) | ||
Theorem | psrbagev2OLD 20892* | Obsolete version of psrbagev2 20891 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = (.g‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ CMnd) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶) | ||
Theorem | evlslem2 20893* | 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 20894* | Lemma for evlseu 20897. 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 20895* | Lemma for evlseu 20897. 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 20896* | Lemma for evlseu 20897, 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 20897* | 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 20898 | Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
⊢ Rel dom evalSub | ||
Theorem | mpfrcl 20899 | 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 20900* | 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))) |
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