HomeTheorem List (p. 340 of 507)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 340 of 507) | < Previous Next > |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-31281) |
Hilbert Space Explorer
(31282-32804) |
Users' Mathboxes
(32805-50680) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ig1pmindeg 33901 | The polynomial ideal generator is of minimum degree. (Contributed by Thierry Arnoux, 19-Mar-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐺 = (idlGen1p‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑃)) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐷‘(𝐺‘𝐼)) ≤ (𝐷‘𝐹)) | ||
| Theorem | q1pdir 33902 | Distribution of univariate polynomial quotient over addition. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ / = (quot1p‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐶 ∈ 𝑁) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ + = (+g‘𝑃) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))) | ||
| Theorem | q1pvsca 33903 | Scalar multiplication property of the polynomial division. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ / = (quot1p‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐶 ∈ 𝑁) & ⊢ × = ( ·𝑠 ‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝐵 × 𝐴) / 𝐶) = (𝐵 × (𝐴 / 𝐶))) | ||
| Theorem | r1pvsca 33904 | Scalar multiplication property of the polynomial remainder operation. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ 𝐸 = (rem1p‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑁) & ⊢ × = ( ·𝑠 ‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝐵 × 𝐴)𝐸𝐷) = (𝐵 × (𝐴𝐸𝐷))) | ||
| Theorem | r1p0 33905 | Polynomial remainder operation of a division of the zero polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ 𝐸 = (rem1p‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐷 ∈ 𝑁) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ (𝜑 → ( 0 𝐸𝐷) = 0 ) | ||
| Theorem | r1pcyc 33906 | The polynomial remainder operation is periodic. See modcyc 13946. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ 𝐸 = (rem1p‘𝑅) & ⊢ + = (+g‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑁) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝐴 + (𝐶 · 𝐵))𝐸𝐵) = (𝐴𝐸𝐵)) | ||
| Theorem | r1padd1 33907 | Addition property of the polynomial remainder operation, similar to modadd1 13948. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ 𝐸 = (rem1p‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑁) & ⊢ (𝜑 → (𝐴𝐸𝐷) = (𝐵𝐸𝐷)) & ⊢ + = (+g‘𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶)𝐸𝐷) = ((𝐵 + 𝐶)𝐸𝐷)) | ||
| Theorem | r1plmhm 33908* | The univariate polynomial remainder function 𝐹 is a module homomorphism. Its image (𝐹 “s 𝑃) is sometimes called the "ring of remainders". (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝐸 = (rem1p‘𝑅) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑃 LMHom (𝐹 “s 𝑃))) | ||
| Theorem | r1pquslmic 33909* | The univariate polynomial remainder ring (𝐹 “s 𝑃) is module isomorphic with the quotient ring. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝐸 = (rem1p‘𝑅) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝑁) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝐾)) ⇒ ⊢ (𝜑 → 𝑄 ≃𝑚 (𝐹 “s 𝑃)) | ||
| Theorem | psrbasfsupp 33910 | Rewrite a finite support for nonnegative integers : For functions mapping a set 𝐼 to the nonnegative integers, having finite support can also be written as having a finite preimage of the positive integers. The latter expression is used for example in psrbas 22095, but with the former expression, theorems about finite support can be used more directly. (Contributed by Thierry Arnoux, 10-Jan-2026.) |
| ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ 𝑓 finSupp 0} ⇒ ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | ||
| Theorem | psrnzr 33911 | The ring of power series over a nonzero ring form a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ NzRing) ⇒ ⊢ (𝜑 → 𝑆 ∈ NzRing) | ||
| Theorem | mplnzr 33912 | The multivariate polynomials over a nonzero ring form a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ NzRing) ⇒ ⊢ (𝜑 → 𝑃 ∈ NzRing) | ||
| Theorem | 0mplrim 33913* | Build a ring isomorphism between multivariate polynomials with no variables and the underlying ring. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (∅ mPoly 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝑝‘∅)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingIso 𝑅)) | ||
| Theorem | 0mplric 33914 | Multivariate polynomials with no variables are isomorphic with the underlying ring. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (∅ mPoly 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑃 ≃𝑟 𝑅) | ||
| Theorem | mplasclco 33915* | Case where composing an algebra scalar lifting functions with a scalar leads to a scalar. This is useful when working with selectVars. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝑆 = (Base‘𝑅) & ⊢ 𝑂 = (𝐽 mPoly 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑄 = (𝐼 mPoly 𝑂) & ⊢ 𝐴 = (algSc‘𝑂) & ⊢ 𝐵 = (algSc‘𝑃) & ⊢ 𝐶 = (algSc‘𝑄) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ 𝐸 = {𝑗 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑗 “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) = (𝐶‘(𝐴‘𝑋))) | ||
| Theorem | selvascl 33916 | The "variable selection" function evaluated at a scalar. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) ⇒ ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘(𝐴‘𝑋)) = (𝐷‘𝑋)) | ||
| Theorem | selvply1rhmlema 33917* | Lemma for selvply1rhm 33924. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = ({𝑋} mPoly 𝑅) & ⊢ · = (.r‘𝑃) & ⊢ × = (.r‘𝑄) & ⊢ 𝑄 = (Poly1‘𝑅) & ⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘𝐹) ∈ (Base‘𝑄)) | ||
| Theorem | selvply1rhmlemb 33918* | Lemma for selvply1rhm 33924. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = ({𝑋} mPoly 𝑅) & ⊢ · = (.r‘𝑃) & ⊢ × = (.r‘𝑄) & ⊢ 𝑄 = (Poly1‘𝑅) & ⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘(𝐹 · 𝐺)) = ((𝑀‘𝐹) × (𝑀‘𝐺))) | ||
| Theorem | selvply1rhmlem1 33919* | Lemma for selvply1rhm 33924. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly1‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝐻:𝐵⟶(Base‘𝑄)) | ||
| Theorem | selvply1rhmlem2 33920* | Lemma for selvply1rhm 33924: Image of the ring unit by the mapping 𝐻 (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly1‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → (𝐻‘(1r‘𝑃)) = (1r‘𝑄)) | ||
| Theorem | selvply1rhmlem3 33921* | Lemma for selvply1rhm 33924. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly1‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ (ℕ0 ↑m 1o)) ⇒ ⊢ (𝜑 → ((𝐻‘𝐹)‘𝑁) = ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝐹)‘{⟨𝑋, (𝑁‘∅)⟩})) | ||
| Theorem | selvply1rhmlem4 33922* | Lemma for selvply1rhm 33924: The mapping 𝐻 is linear. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly1‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐻‘(𝐹(+g‘𝑃)𝐺)) = ((𝐻‘𝐹)(+g‘𝑄)(𝐻‘𝐺))) | ||
| Theorem | selvply1rhmlem5 33923* | Lemma for selvply1rhm 33924. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly1‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 𝑀 = (𝑞 ∈ (Base‘({𝑋} mPoly 𝑈)) ↦ (𝑠 ∈ (ℕ0 ↑m 1o) ↦ (𝑞‘{⟨𝑋, (𝑠‘∅)⟩}))) ⇒ ⊢ (𝜑 → (𝐻‘𝐹) = (𝑀‘(((𝐼 selectVars 𝑅)‘{𝑋})‘𝐹))) | ||
| Theorem | selvply1rhm 33924* | Build a ring homomorphism 𝐻 between the multivariate polynomials 𝑃 with variables in 𝐼 and the univariate polynomials 𝑄 in a single variable 𝑋 element of 𝐼. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly1‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝐻 ∈ (𝑃 RingHom 𝑄)) | ||
| Theorem | selvply1rhm0 33925* | The ring homomorphism 𝐻 built in selvply1rhm 33924 is injective. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly1‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ 0 = (0g‘𝑄) & ⊢ 𝑍 = (0g‘𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → (𝐻‘𝐹) = 0 ) ⇒ ⊢ (𝜑 → 𝐹 = 𝑍) | ||
| Theorem | mplidomlem 33926* | Lemma for mplidom 33927. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ 𝐻 = (𝑓 ∈ 𝐶 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩}))) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅) & ⊢ 𝑈 = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅) & ⊢ 𝑄 = (Poly1‘𝑈) ⇒ ⊢ (𝜑 → 𝑃 ∈ IDomn) | ||
| Theorem | mplidom 33927 | The multivariate polynomials over an integral domain form an integral domain. See ply1idom 26293. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → 𝑃 ∈ IDomn) | ||
| Syntax | cextv 33928 | Extend class notation with the "variable extension" function. |
| class extendVars | ||
| Definition | df-extv 33929* | Define the "variable extension" function. The function ((𝐼extendVars𝑅)‘𝐴) converts polynomials with variables indexed by (𝐼 ∖ {𝐴}) into polynomials indexed by 𝐼, and therefore maps elements of ((𝐼 ∖ {𝐴}) mPoly 𝑅) onto (𝐼 mPoly 𝑅). (Contributed by Thierry Arnoux, 20-Jan-2026.) |
| ⊢ extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟)))))) | ||
| Theorem | extvval 33930* | Value of the "variable extension" function. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐽 = (𝐼 ∖ {𝑎}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) ⇒ ⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 ))))) | ||
| Theorem | extvfval 33931* | The "variable extension" function evaluated for adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 )))) | ||
| Theorem | extvfv 33932* | The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) ⇒ ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 ))) | ||
| Theorem | extvfvv 33933* | The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 )) | ||
| Theorem | extvfvvcl 33934* | Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) ∈ 𝐵) | ||
| Theorem | extvfvcl 33935* | Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁) | ||
| Theorem | extvfvalf 33936* | The "variable extension" function maps polynomials with variables indexed in 𝐽 to polynomials with variables indexed in 𝐼. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴):𝑀⟶𝑁) | ||
| Theorem | mvrvalind 33937* | Value of the generating elements of the power series structure, expressed using the indicator function. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 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) & ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑋}) ⇒ ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 )) | ||
| Theorem | mplmulmvr 33938* | Multiply a polynomial 𝐹 with a variable 𝑋 (i.e. with a monic monomial). (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑋 = ((𝐼 mVar 𝑅)‘𝑌) & ⊢ 𝑀 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑌}) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) ⇒ ⊢ (𝜑 → (𝑋 · 𝐹) = (𝑏 ∈ 𝐷 ↦ if((𝑏‘𝑌) = 0, 0 , (𝐹‘(𝑏 ∘f − 𝐴))))) | ||
| Theorem | evlscaval 33939 | Polynomial evaluation for scalars. See evlsscaval 22288. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐿:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋) | ||
| Theorem | evlvarval 33940 | Polynomial evaluation builder for a variable. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑄 = (𝐼 eval 𝑆) & ⊢ 𝑃 = (𝐼 mPoly 𝑆) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ ∙ = (.r‘𝑃) & ⊢ · = (.r‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) & ⊢ 𝑉 = (𝐼 mVar 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑉‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝑉‘𝑋))‘𝐴) = (𝐴‘𝑋))) | ||
| Theorem | evlextv 33941 | Evaluating a variable-extended polynomial is the same as evaluating the polynomial in the original set of variables (in both cases, the additionial variable is ignored). (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝑂 = (𝐽 eval 𝑅) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (𝐼extendVars𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘((𝐸‘𝑌)‘𝐹))‘𝐴) = ((𝑂‘𝐹)‘(𝐴 ↾ 𝐽))) | ||
| Theorem | mplvrpmlem 33942* | Lemma for mplvrpmga 33944 and others. (Contributed by Thierry Arnoux, 11-Jan-2026.) |
| ⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ⇒ ⊢ (𝜑 → (𝑋 ∘ 𝐷) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) | ||
| Theorem | mplvrpmfgalem 33943* | Permuting variables in a multivariate polynomial conserves finite support. (Contributed by Thierry Arnoux, 10-Jan-2026.) |
| ⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑄𝐴𝐹) finSupp 0 ) | ||
| Theorem | mplvrpmga 33944* | The action of permuting variables in a multivariate polynomial is a group action. (Contributed by Thierry Arnoux, 10-Jan-2026.) |
| ⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀)) | ||
| Theorem | mplvrpmmhm 33945* | The action of permuting variables in a multivariate polynomial is a monoid homomorphism. (Contributed by Thierry Arnoux, 11-Jan-2026.) |
| ⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓)) & ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑊 MndHom 𝑊)) | ||
| Theorem | mplvrpmrhm 33946* | The action of permuting variables in a multivariate polynomial is a ring homomorphism. (Contributed by Thierry Arnoux, 15-Jan-2026.) |
| ⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓)) & ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑊 RingHom 𝑊)) | ||
| Theorem | psrgsum 33947* | Finite commutative sums of power series are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝑆 Σg 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦))))) | ||
| Theorem | psrmon 33948* | A monomial is a power series. (Contributed by Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵) | ||
| Theorem | psrmonmul 33949* | The product of two power series 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 Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ · = (.r‘𝑆) & ⊢ (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 | psrmonmul2 33950* | The product of two power series monomials adds the exponent vectors together. Here, the function 𝐺 is a monomial builder, which maps a bag of variables with the monic monomial with only those variables. (Contributed by Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ · = (.r‘𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) & ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 ))) ⇒ ⊢ (𝜑 → ((𝐺‘𝑋) · (𝐺‘𝑌)) = (𝐺‘(𝑋 ∘f + 𝑌))) | ||
| Theorem | psrmonprod 33951* | Finite product of bags of variables in a power series. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐷) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑆) & ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 ))) ⇒ ⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖)))))) | ||
| Theorem | mplgsum 33952* | Finite commutative sums of polynomials are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝑃 Σg 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦))))) | ||
| Theorem | mplmonprod 33953* | Finite product of monomials. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐷) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 ))) ⇒ ⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖)))))) | ||
| Syntax | csply 33954 | Extend class notation with the symmetric polynomials. |
| class SymPoly | ||
| Syntax | cesply 33955 | Extend class notation with the elementary symmetric polynomials. |
| class eSymPoly | ||
| Definition | df-sply 33956* | Define symmetric polynomials. See splyval 33958 for a more readable expression. (Contributed by Thierry Arnoux, 11-Jan-2026.) |
| ⊢ SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))) | ||
| Definition | df-esply 33957* | Define elementary symmetric polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘}))))) | ||
| Theorem | splyval 33958* | The symmetric polynomials for a given index 𝐼 of variables and base ring 𝑅. These are the fixed points of the action 𝐴 which permutes variables. (Contributed by Thierry Arnoux, 11-Jan-2026.) |
| ⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴)) | ||
| Theorem | splysubrg 33959* | The symmetric polynomials form a subring of the ring of polynomials. (Contributed by Thierry Arnoux, 15-Jan-2026.) |
| ⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅))) | ||
| Theorem | issply 33960* | Conditions for being a symmetric polynomial. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝑥 ∘ 𝑝)) = (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐼SymPoly𝑅)) | ||
| Theorem | esplyval 33961* | The elementary polynomials for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐼eSymPoly𝑅) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘}))))) | ||
| Theorem | esplyfval 33962* | The 𝐾-th elementary polynomial for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))) | ||
| Theorem | esplyfval0 33963 | The 0-th elementary symmetric polynomial is the constant 1. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝑈 = (1r‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘0) = 𝑈) | ||
| Theorem | esplyfval2 33964* | When 𝐾 is out-of-bounds, the 𝐾-th elementary symmetric polynomial is zero. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ (ℕ0 ∖ (0...(♯‘𝐼)))) & ⊢ 𝑍 = (0g‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = 𝑍) | ||
| Theorem | esplylem 33965* | Lemma for esplyfv 33969 and others. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷) | ||
| Theorem | esplympl 33966* | Elementary symmetric polynomials are polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ 𝑀) | ||
| Theorem | esplymhp 33967* | The 𝐾-th elementary symmetric polynomial is homogeneous of degree 𝐾. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ 𝐻 = (𝐼 mHomP 𝑅) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐻‘𝐾)) | ||
| Theorem | esplyfv1 33968* | Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹 where variables are not raised to a power. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼))) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → ran 𝐹 ⊆ {0, 1}) ⇒ ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 )) | ||
| Theorem | esplyfv 33969* | Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹: the coefficient is 1 for the bags of exactly 𝐾 variables, having exponent at most 1. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼))) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 )) | ||
| Theorem | esplysply 33970* | The 𝐾-th elementary symmetric polynomial is symmetric. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼))) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐼SymPoly𝑅)) | ||
| Theorem | esplyfval3 33971* | Alternate expression for the value of the 𝐾-th elementary symmetric polynomial. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 ))) | ||
| Theorem | esplyfval1 33972 | The first elementary symmetric polynomial is the sum of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐸‘1) = (𝑊 Σg 𝑉)) | ||
| Theorem | esplyfvaln 33973 | The last elementary symmetric polynomial is the product of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.) |
| ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ 𝑁 = (♯‘𝐼) & ⊢ 𝑀 = (mulGrp‘𝑊) ⇒ ⊢ (𝜑 → (𝐸‘𝑁) = (𝑀 Σg 𝑉)) | ||
| Theorem | esplyind 33974* | A recursive formula for the elementary symmetric polynomials. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ + = (+g‘𝑊) & ⊢ · = (.r‘𝑊) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐺 = ((𝐼extendVars𝑅)‘𝑌) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ 𝐸 = (𝐽eSymPoly𝑅) & ⊢ (𝜑 → 𝐾 ∈ (1...(♯‘𝐼))) & ⊢ 𝐶 = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (((𝑉‘𝑌) · (𝐺‘(𝐸‘(𝐾 − 1)))) + (𝐺‘(𝐸‘𝐾)))) | ||
| Theorem | esplyindfv 33975* | A recursive formula for the elementary symmetric polynomials, evaluated at a given set of points 𝑍. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ 𝐸 = (𝐽eSymPoly𝑅) & ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐽))) & ⊢ 𝐶 = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} & ⊢ 𝐹 = ((𝐼eSymPoly𝑅)‘(𝐾 + 1)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝑂 = (𝐽 eval 𝑅) & ⊢ + = (+g‘𝑅) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘𝐹)‘𝑍) = (((𝑍‘𝑌) · ((𝑂‘(𝐸‘𝐾))‘(𝑍 ↾ 𝐽))) + ((𝑂‘(𝐸‘(𝐾 + 1)))‘(𝑍 ↾ 𝐽)))) | ||
| Theorem | esplyfvn 33976 | Express the last elementary symmetric polynomial, evaluated at a given set of points 𝑍, in terms of the last elementary symmetric polynomial with one less variable. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝑂 = (𝐽 eval 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ 𝐹 = (𝐽eSymPoly𝑅) & ⊢ 𝐻 = (♯‘𝐼) & ⊢ 𝐾 = (♯‘𝐽) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐸‘𝐻))‘𝑍) = ((𝑍‘𝑌) · ((𝑂‘(𝐹‘𝐾))‘(𝑍 ↾ 𝐽)))) | ||
| Theorem | vietadeg1 33977* | The degree of a product of 𝐻 of linear polynomials of the form 𝑋 − 𝑍 is 𝐻. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑊 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ − = (-g‘𝑊) & ⊢ 𝑀 = (mulGrp‘𝑊) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ ↑ = (.g‘(mulGrp‘𝑅)) & ⊢ 𝐻 = (♯‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) & ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛))))) & ⊢ 𝐷 = (deg1‘𝑅) ⇒ ⊢ (𝜑 → (𝐷‘𝐹) = 𝐻) | ||
| Theorem | vietalem 33978* | Lemma for vieta 33979: induction step. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑊 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ − = (-g‘𝑊) & ⊢ 𝑀 = (mulGrp‘𝑊) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ ↑ = (.g‘(mulGrp‘𝑅)) & ⊢ 𝐻 = (♯‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) & ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛))))) & ⊢ (𝜑 → 𝐾 ∈ (0...𝐻)) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ (𝜑 → ∀𝑧 ∈ (𝐵 ↑m 𝐽)∀𝑘 ∈ (0...(♯‘𝐽))((coe1‘(𝑀 Σg (𝑛 ∈ 𝐽 ↦ (𝑋 − (𝐴‘(𝑧‘𝑛))))))‘((♯‘𝐽) − 𝑘)) = ((𝑘 ↑ (𝑁‘ 1 )) · (((𝐽 eval 𝑅)‘((𝐽eSymPoly𝑅)‘𝑘))‘𝑧))) & ⊢ (𝜑 → ((deg1‘𝑅)‘(𝑀 Σg (𝑛 ∈ 𝐽 ↦ (𝑋 − (𝐴‘((𝑍 ↾ 𝐽)‘𝑛)))))) = (♯‘𝐽)) ⇒ ⊢ (𝜑 → ((coe1‘𝐹)‘𝐾) = (((𝐻 − 𝐾) ↑ (𝑁‘ 1 )) · ((𝑄‘(𝐸‘(𝐻 − 𝐾)))‘𝑍))) | ||
| Theorem | vieta 33979* | Vieta's Formulas: Coefficients of a monic polynomial 𝐹 expressed as a product of linear polynomials of the form 𝑋 − 𝑍 can be expressed in terms of elementary symmetric polynomials. The formulas appear in Chapter 6 of [Lang], p. 190. Theorem vieta1 26484 is a special case for the complex numbers, for the case 𝐾 = 1. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑊 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ − = (-g‘𝑊) & ⊢ 𝑀 = (mulGrp‘𝑊) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ ↑ = (.g‘(mulGrp‘𝑅)) & ⊢ 𝐻 = (♯‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) & ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛))))) & ⊢ (𝜑 → 𝐾 ∈ (0...𝐻)) & ⊢ 𝐶 = (coe1‘𝐹) ⇒ ⊢ (𝜑 → (𝐶‘(𝐻 − 𝐾)) = ((𝐾 ↑ (𝑁‘ 1 )) · ((𝑄‘(𝐸‘𝐾))‘𝑍))) | ||
| Theorem | sra1r 33980 | The unity element of a subring algebra. (Contributed by Thierry Arnoux, 24-Jul-2023.) |
| ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 1 = (1r‘𝑊)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → 1 = (1r‘𝐴)) | ||
| Theorem | sradrng 33981 | Condition for a subring algebra to be a division ring. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ DivRing) | ||
| Theorem | sraidom 33982 | Condition for a subring algebra to be an integral domain. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑉 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ IDomn) | ||
| Theorem | srasubrg 33983 | A subring of the original structure is also a subring of the constructed subring algebra. (Contributed by Thierry Arnoux, 24-Jul-2023.) |
| ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑊)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐴)) | ||
| Theorem | sralvec 33984 | Given a sub division ring 𝐹 of a division ring 𝐸, 𝐸 may be considered as a vector space over 𝐹, which becomes the field of scalars. (Contributed by Thierry Arnoux, 24-May-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) ⇒ ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec) | ||
| Theorem | srafldlvec 33985 | Given a subfield 𝐹 of a field 𝐸, 𝐸 may be considered as a vector space over 𝐹, which becomes the field of scalars. (Contributed by Thierry Arnoux, 24-May-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) ⇒ ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec) | ||
| Theorem | resssra 33986 | The subring algebra of a restricted structure is the restriction of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝑆 = (𝑅 ↾s 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐶 ⊆ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵)) | ||
| Theorem | lsssra 33987 | A subring is a subspace of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑊 = ((subringAlg ‘𝑅)‘𝐶) & ⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝑆 = (𝑅 ↾s 𝐵) & ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝑆)) ⇒ ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑊)) | ||
| Theorem | srapwov 33988 | The "power" operation on a subring algebra. (Contributed by Thierry Arnoux, 10-Jan-2026.) |
| ⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆) & ⊢ (𝜑 → 𝑊 ∈ Ring) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.g‘(mulGrp‘𝑊)) = (.g‘(mulGrp‘𝐴))) | ||
| Theorem | drgext0g 33989 | The additive neutral element of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.) |
| ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) ⇒ ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐵)) | ||
| Theorem | drgextvsca 33990 | The scalar multiplication operation of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.) |
| ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) ⇒ ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵)) | ||
| Theorem | drgext0gsca 33991 | The additive neutral element of the scalar field of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.) |
| ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) ⇒ ⊢ (𝜑 → (0g‘𝐵) = (0g‘(Scalar‘𝐵))) | ||
| Theorem | drgextsubrg 33992 | The scalar field is a subring of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.) |
| ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐵)) | ||
| Theorem | drgextlsp 33993 | The scalar field is a subspace of a subring algebra. (Contributed by Thierry Arnoux, 17-Jul-2023.) |
| ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝐵)) | ||
| Theorem | drgextgsum 33994* | Group sum in a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.) |
| ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ 𝑌)) = (𝐵 Σg (𝑖 ∈ 𝑋 ↦ 𝑌))) | ||
| Theorem | lvecdimfi 33995 | Finite version of lvecdim 21292 which does not require the axiom of choice. The axiom of choice is used in acsmapd 18616, which is required in the infinite case. Suggested by Gérard Lang. (Contributed by Thierry Arnoux, 24-May-2023.) |
| ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑆 ∈ 𝐽) & ⊢ (𝜑 → 𝑇 ∈ 𝐽) & ⊢ (𝜑 → 𝑆 ∈ Fin) ⇒ ⊢ (𝜑 → 𝑆 ≈ 𝑇) | ||
| Theorem | exsslsb 33996* | Any finite generating set 𝑆 of a vector space 𝑊 contains a basis. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → (𝐾‘𝑆) = 𝐵) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝑠 ⊆ 𝑆) | ||
| Theorem | lbslelsp 33997 | The size of a basis 𝑋 of a vector space 𝑊 is less than the size of a generating set 𝑌. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝐽) & ⊢ (𝜑 → 𝑌 ⊆ 𝐵) & ⊢ (𝜑 → (𝐾‘𝑌) = 𝐵) ⇒ ⊢ (𝜑 → (♯‘𝑋) ≤ (♯‘𝑌)) | ||
| Syntax | cldim 33998 | Extend class notation with the dimension of a vector space. |
| class dim | ||
| Definition | df-dim 33999 | Define the dimension of a vector space as the cardinality of its bases. Note that by lvecdim 21292, all bases are equinumerous. (Contributed by Thierry Arnoux, 6-May-2023.) |
| ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓))) | ||
| Theorem | dimval 34000 | The dimension of a vector space 𝐹 is the cardinality of one of its bases. (Contributed by Thierry Arnoux, 6-May-2023.) |
| ⊢ 𝐽 = (LBasis‘𝐹) ⇒ ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = (♯‘𝑆)) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |