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Type | Label | Description |
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Statement | ||
Theorem | psrridm 19801* | The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by AV, 8-Jul-2019.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 )) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ · = (.r‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · 𝑈) = 𝑋) | ||
Theorem | psrass1 19802* | Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ × = (.r‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍))) | ||
Theorem | psrdi 19803* | Distributive law for the ring of power series (left-distributivity). (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ × = (.r‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ + = (+g‘𝑆) ⇒ ⊢ (𝜑 → (𝑋 × (𝑌 + 𝑍)) = ((𝑋 × 𝑌) + (𝑋 × 𝑍))) | ||
Theorem | psrdir 19804* | Distributive law for the ring of power series (right-distributivity). (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ × = (.r‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ + = (+g‘𝑆) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) × 𝑍) = ((𝑋 × 𝑍) + (𝑌 × 𝑍))) | ||
Theorem | psrass23l 19805* | Associative identity for the ring of power series. Part of psrass23 19807 which does not require the scalar ring to be commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 14-Aug-2019.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ × = (.r‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌))) | ||
Theorem | psrcom 19806* | Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ × = (.r‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑌 × 𝑋)) | ||
Theorem | psrass23 19807* | Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 25-Nov-2019.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ × = (.r‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) ⇒ ⊢ (𝜑 → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))) | ||
Theorem | psrring 19808 | The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑆 ∈ Ring) | ||
Theorem | psr1 19809* | The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑈 = (1r‘𝑆) ⇒ ⊢ (𝜑 → 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))) | ||
Theorem | psrcrng 19810 | The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝑆 ∈ CRing) | ||
Theorem | psrassa 19811 | The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝑆 ∈ AssAlg) | ||
Theorem | resspsrbas 19812 | A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝑈 = (𝐼 mPwSer 𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ 𝑃 = (𝑆 ↾s 𝐵) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝑃)) | ||
Theorem | resspsradd 19813 | A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝑈 = (𝐼 mPwSer 𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ 𝑃 = (𝑆 ↾s 𝐵) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘𝑃)𝑌)) | ||
Theorem | resspsrmul 19814 | A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝑈 = (𝐼 mPwSer 𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ 𝑃 = (𝑆 ↾s 𝐵) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌)) | ||
Theorem | resspsrvsca 19815 | A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝑈 = (𝐼 mPwSer 𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ 𝑃 = (𝑆 ↾s 𝐵) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) | ||
Theorem | subrgpsr 19816 | A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝑈 = (𝐼 mPwSer 𝐻) & ⊢ 𝐵 = (Base‘𝑈) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆)) | ||
Theorem | mvrfval 19817* | 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 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))) | ||
Theorem | mvrval 19818* | 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 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))) | ||
Theorem | mvrval2 19819* | 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 19820* | 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 19821 | 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 19822 | 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 19823 | 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 19824 | The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ Rel dom mPoly | ||
Theorem | mplval 19825* | 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 19826* | 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 19827 | 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 | mplval2 19828 | 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 19829 | 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 19830* | 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 19831* | 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 19832* | 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 19833* | Lemma for mplsubg 19834 and mpllss 19835. (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 19834 | 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 19835 | 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 19836* | Lemma for mplsubrg 19837. (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 19837 | 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 19838* | 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 19839 | 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 | mplmul 19840* | 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 19841* | 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 19842 | 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 19843 | 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 19844* | 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 19845* | 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 19846 | 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 19847 | 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 19848 | 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 19849 | The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring) | ||
Theorem | mplcrng 19850 | 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 19851 | 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 19852 | 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 19853 | 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 19854 | 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 19855 | 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 19856 | 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 19857 | 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 19858 | 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 19859 | 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 19860* | 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 19861* | 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 19862* | 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 19863* | 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 19864* | Lemma for mplcoe4 19899. (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 19865* | Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 19866), 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 19866* | 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 19867 | 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 19868* | 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 19869* | 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 19870 | Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.) |
⊢ Rel dom ordPwSer | ||
Theorem | opsrval 19871* | The value of the "ordered power series" function. This is the same as mPwSer psrval 19759, 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 19872* | 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 19873 | 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 19874 | 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 19875 | 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 19876 | 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 19877 | 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 19878 | 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 19879 | 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 19880* | Lemma for opsrtos 19882. (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 19881* | Lemma for opsrtos 19882. (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 19882 | 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 19883 | 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 19884 | 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 19885 | 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 19886 | 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 19887 | 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 19888 | 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 19889* | 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 19890* | 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 19891* | 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 19892* | 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 19893 | 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 19894 | 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 19895 | 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 19896* | 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 19897* | 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 19898* | 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 ∈ V) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐻) | ||
Theorem | mplcoe4 19899* | 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 19900 | Evaluation of a multivariate polynomial in a subring. |
class evalSub |
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