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Type | Label | Description |
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Statement | ||
Theorem | prmidl2 33001* | A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 37405 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅)) | ||
Theorem | idlmulssprm 33002 | Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 ⊆ 𝑃 or 𝐽 ⊆ 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.) |
⊢ × = (LSSum‘(mulGrp‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃) ⇒ ⊢ (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)) | ||
Theorem | pridln1 33003 | A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ¬ 1 ∈ 𝐼) | ||
Theorem | prmidlidl 33004 | A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅)) | ||
Theorem | prmidlssidl 33005 | Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅)) | ||
Theorem | lidlnsg 33006 | An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.) |
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅)) | ||
Theorem | cringm4 33007 | Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊))) | ||
Theorem | isprmidlc 33008* | The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃))))) | ||
Theorem | prmidlc 33009 | Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ 𝐵 ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼 ∈ 𝑃 ∨ 𝐽 ∈ 𝑃)) | ||
Theorem | 0ringprmidl 33010 | The trivial ring does not have any prime ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (PrmIdeal‘𝑅) = ∅) | ||
Theorem | prmidl0 33011 | The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.) |
⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn) | ||
Theorem | rhmpreimaprmidl 33012 | The preimage of a prime ideal by a ring homomorphism is a prime ideal. (Contributed by Thierry Arnoux, 29-Jun-2024.) |
⊢ 𝑃 = (PrmIdeal‘𝑅) ⇒ ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃) | ||
Theorem | qsidomlem1 33013 | If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅)) | ||
Theorem | qsidomlem2 33014 | A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn) | ||
Theorem | qsidom 33015 | An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅))) | ||
Theorem | qsnzr 33016 | A quotient of a non-zero ring by a proper ideal is a non-zero ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) & ⊢ (𝜑 → 𝐼 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝑄 ∈ NzRing) | ||
Syntax | cmxidl 33017 | Extend class notation with the class of maximal ideals. |
class MaxIdeal | ||
Definition | df-mxidl 33018* | Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))}) | ||
Theorem | mxidlval 33019* | The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))}) | ||
Theorem | ismxidl 33020* | The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))) | ||
Theorem | mxidlidl 33021 | A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅)) | ||
Theorem | mxidlnr 33022 | A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵) | ||
Theorem | mxidlmax 33023 | A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)) | ||
Theorem | mxidln1 33024 | One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1 ∈ 𝑀) | ||
Theorem | mxidlnzr 33025 | A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing) | ||
Theorem | mxidlmaxv 33026 | An ideal 𝐼 strictly containing a maximal ideal 𝑀 is the whole ring 𝐵. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ⊆ 𝐼) & ⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀)) ⇒ ⊢ (𝜑 → 𝐼 = 𝐵) | ||
Theorem | crngmxidl 33027 | In a commutative ring, maximal ideals of the opposite ring coincide with maximal ideals. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
⊢ 𝑀 = (MaxIdeal‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑀 = (MaxIdeal‘𝑂)) | ||
Theorem | mxidlprm 33028 | Every maximal ideal is prime. Statement in [Lang] p. 92. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
⊢ × = (LSSum‘(mulGrp‘𝑅)) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅)) | ||
Theorem | mxidlirredi 33029 | In an integral domain, the generator of a maximal ideal is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (𝐾‘{𝑋}) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅)) | ||
Theorem | mxidlirred 33030 | In a principal ideal domain, maximal ideals are exactly the ideals generated by irreducible elements. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (𝐾‘{𝑋}) & ⊢ (𝜑 → 𝑅 ∈ PID) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅))) | ||
Theorem | ssmxidllem 33031* | The set 𝑃 used in the proof of ssmxidl 33032 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ≠ 𝐵) & ⊢ (𝜑 → 𝑍 ⊆ 𝑃) & ⊢ (𝜑 → 𝑍 ≠ ∅) & ⊢ (𝜑 → [⊊] Or 𝑍) ⇒ ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃) | ||
Theorem | ssmxidl 33032* | Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼 ⊆ 𝑚) | ||
Theorem | drnglidl1ne0 33033 | In a nonzero ring, the zero ideal is different of the unit ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → 𝐵 ≠ { 0 }) | ||
Theorem | drng0mxidl 33034 | In a division ring, the zero ideal is a maximal ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅)) | ||
Theorem | drngmxidl 33035 | The zero ideal is the only ideal of a division ring. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }}) | ||
Theorem | krull 33036* | Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
⊢ (𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)) | ||
Theorem | mxidlnzrb 33037* | A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))) | ||
Theorem | opprabs 33038 | The opposite ring of the opposite ring is the original ring. Note the conditions on this theorem, which makes it unpractical in case we only have e.g. 𝑅 ∈ Ring as a premise. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝑅) & ⊢ (𝜑 → (.r‘ndx) ∈ dom 𝑅) & ⊢ (𝜑 → · Fn (𝐵 × 𝐵)) ⇒ ⊢ (𝜑 → 𝑅 = (oppr‘𝑂)) | ||
Theorem | oppreqg 33039 | Group coset equivalence relation for the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼)) | ||
Theorem | opprnsg 33040 | Normal subgroups of the opposite ring are the same as the original normal subgroups. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂) | ||
Theorem | opprlidlabs 33041 | The ideals of the opposite ring's opposite ring are the ideals of the original ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂))) | ||
Theorem | oppr2idl 33042 | Two sided ideal of the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂)) | ||
Theorem | opprmxidlabs 33043 | The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂))) | ||
Theorem | opprqusbas 33044 | The base of the quotient of the opposite ring is the same as the base of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) | ||
Theorem | opprqusplusg 33045 | The group operation of the quotient of the opposite ring is the same as the group operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) & ⊢ 𝐸 = (Base‘𝑄) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ (𝜑 → 𝑌 ∈ 𝐸) ⇒ ⊢ (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌)) | ||
Theorem | opprqus0g 33046 | The group identity element of the quotient of the opposite ring is the same as the group identity element of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) ⇒ ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))) | ||
Theorem | opprqusmulr 33047 | The multiplication operation of the quotient of the opposite ring is the same as the multiplication operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) & ⊢ 𝐸 = (Base‘𝑄) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ (𝜑 → 𝑌 ∈ 𝐸) ⇒ ⊢ (𝜑 → (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌)) | ||
Theorem | opprqus1r 33048 | The ring unity of the quotient of the opposite ring is the same as the ring unity of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) ⇒ ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))) | ||
Theorem | opprqusdrng 33049 | The quotient of the opposite ring is a division ring iff the opposite of the quotient ring is. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) ⇒ ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing)) | ||
Theorem | qsdrngilem 33050* | Lemma for qsdrngi 33051. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀) ⇒ ⊢ (𝜑 → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄)) | ||
Theorem | qsdrngi 33051 | A quotient by a maximal left and maximal right ideal is a division ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂)) ⇒ ⊢ (𝜑 → 𝑄 ∈ DivRing) | ||
Theorem | qsdrnglem2 33052 | Lemma for qsdrng 33053. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ DivRing) & ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 ∈ (𝐽 ∖ 𝑀)) ⇒ ⊢ (𝜑 → 𝐽 = 𝐵) | ||
Theorem | qsdrng 33053 | An ideal 𝑀 is both left and right maximal if and only if the factor ring 𝑄 is a division ring. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅)) ⇒ ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂)))) | ||
Theorem | qsfld 33054 | An ideal 𝑀 in the commutative ring 𝑅 is maximal if and only if the factor ring 𝑄 is a field. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝑄 ∈ Field ↔ 𝑀 ∈ (MaxIdeal‘𝑅))) | ||
Theorem | mxidlprmALT 33055 | Every maximal ideal is prime - alternative proof. (Contributed by Thierry Arnoux, 15-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝑀 ∈ (PrmIdeal‘𝑅)) | ||
Syntax | cidlsrg 33056 | Extend class notation with the semiring of ideals of a ring. |
class IDLsrg | ||
Definition | df-idlsrg 33057* | Define a structure for the ideals of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
⊢ IDLsrg = (𝑟 ∈ V ↦ ⦋(LIdeal‘𝑟) / 𝑏⦌({〈(Base‘ndx), 𝑏〉, 〈(+g‘ndx), (LSSum‘𝑟)〉, 〈(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}〉})) | ||
Theorem | idlsrgstr 33058 | A constructed semiring of ideals is a structure. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉}) ⇒ ⊢ 𝑊 Struct 〈1, ;10〉 | ||
Theorem | idlsrgval 33059* | Lemma for idlsrgbas 33060 through idlsrgtset 33064. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ ⊕ = (LSSum‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ⊗ = (LSSum‘𝐺) ⇒ ⊢ (𝑅 ∈ 𝑉 → (IDLsrg‘𝑅) = ({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), ⊕ 〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉})) | ||
Theorem | idlsrgbas 33060 | Base of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝐼 = (Base‘𝑆)) | ||
Theorem | idlsrgplusg 33061 | Additive operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ ⊕ = (LSSum‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆)) | ||
Theorem | idlsrg0g 33062 | The zero ideal is the additive identity of the semiring of ideals. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → { 0 } = (0g‘𝑆)) | ||
Theorem | idlsrgmulr 33063* | Multiplicative operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ⊗ = (LSSum‘𝐺) ⇒ ⊢ (𝑅 ∈ 𝑉 → (𝑖 ∈ 𝐵, 𝑗 ∈ 𝐵 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗))) = (.r‘𝑆)) | ||
Theorem | idlsrgtset 33064* | Topology component of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝐽 = (TopSet‘𝑆)) | ||
Theorem | idlsrgmulrval 33065 | Value of the ring multiplication for the ideals of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ · = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼 · 𝐽))) | ||
Theorem | idlsrgmulrcl 33066 | Ideals of a ring 𝑅 are closed under multiplication. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ∈ 𝐵) | ||
Theorem | idlsrgmulrss1 33067 | In a commutative ring, the product of two ideals is a subset of the first one. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐼) | ||
Theorem | idlsrgmulrss2 33068 | The product of two ideals is a subset of the second one. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐽) | ||
Theorem | idlsrgmulrssin 33069 | In a commutative ring, the product of two ideals is a subset of their intersection. (Contributed by Thierry Arnoux, 17-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ (𝐼 ∩ 𝐽)) | ||
Theorem | idlsrgmnd 33070 | The ideals of a ring form a monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Mnd) | ||
Theorem | idlsrgcmnd 33071 | The ideals of a ring form a commutative monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝑆 = (IDLsrg‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑆 ∈ CMnd) | ||
Syntax | cufd 33072 | Class of unique factorization domains. |
class UFD | ||
Definition | df-ufd 33073* | Define the class of unique factorization domains. A unique factorization domain (UFD for short), is a commutative ring with an absolute value (from abvtriv 20680 this is equivalent to being a domain) such that every prime ideal contains a prime element (this is a characterization due to Irving Kaplansky). A UFD is sometimes also called a "factorial ring" following the terminology of Bourbaki. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)} | ||
Theorem | isufd 33074* | The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐼 = (PrmIdeal‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) ⇒ ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅))) | ||
Theorem | rprmval 33075* | The prime elements of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}) | ||
Theorem | isrprm 33076* | Property for 𝑃 to be a prime element in the ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦))))) | ||
Theorem | asclmulg 33077 | Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ↑ = (.g‘𝑊) & ⊢ ∗ = (.g‘𝐹) ⇒ ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘(𝑁 ∗ 𝑋)) = (𝑁 ↑ (𝐴‘𝑋))) | ||
Theorem | 0ringmon1p 33078 | There are no monic polynomials over a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.) |
⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → (♯‘𝐵) = 1) ⇒ ⊢ (𝜑 → 𝑀 = ∅) | ||
Theorem | fply1 33079 | Conditions for a function to be a univariate polynomial. (Contributed by Thierry Arnoux, 19-Aug-2023.) |
⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (Base‘(Poly1‘𝑅)) & ⊢ (𝜑 → 𝐹:(ℕ0 ↑m 1o)⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝑃) | ||
Theorem | ply1lvec 33080 | In a division ring, the univariate polynomials form a vector space. (Contributed by Thierry Arnoux, 19-Feb-2025.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑃 ∈ LVec) | ||
Theorem | ply1scleq 33081 | Equality of a constant polynomial is the same as equality of the constant term. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐴‘𝐸) = (𝐴‘𝐹) ↔ 𝐸 = 𝐹)) | ||
Theorem | evls1fn 33082 | Functionality of the subring polynomial evaluation. (Contributed by Thierry Arnoux, 9-Feb-2025.) |
⊢ 𝑂 = (𝑅 evalSub1 𝑆) & ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆)) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → 𝑂 Fn 𝑈) | ||
Theorem | evls1dm 33083 | The domain of the subring polynomial evaluation function. (Contributed by Thierry Arnoux, 9-Feb-2025.) |
⊢ 𝑂 = (𝑅 evalSub1 𝑆) & ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆)) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → dom 𝑂 = 𝑈) | ||
Theorem | evls1fvf 33084 | The subring evaluation function for a univariate polynomial as a function, with domain and codomain. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
⊢ 𝑂 = (𝑅 evalSub1 𝑆) & ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆)) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑂‘𝑄):𝐵⟶𝐵) | ||
Theorem | evls1scafv 33085 | Value of the univariate polynomial evaluation for scalars. (Contributed by Thierry Arnoux, 21-Jan-2025.) |
⊢ 𝑄 = (𝑆 evalSub1 𝑅) & ⊢ 𝑊 = (Poly1‘𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝑅) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐶) = 𝑋) | ||
Theorem | evls1expd 33086 | Univariate polynomial evaluation builder for an exponential. See also evl1expd 22185. (Contributed by Thierry Arnoux, 24-Jan-2025.) |
⊢ 𝑄 = (𝑆 evalSub1 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly1‘𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ ∧ = (.g‘(mulGrp‘𝑊)) & ⊢ ↑ = (.g‘(mulGrp‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑀))‘𝐶) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐶))) | ||
Theorem | evls1varpwval 33087 | Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 22202. (Contributed by Thierry Arnoux, 24-Jan-2025.) |
⊢ 𝑄 = (𝑆 evalSub1 𝑅) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝑊 = (Poly1‘𝑈) & ⊢ 𝑋 = (var1‘𝑈) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ ∧ = (.g‘(mulGrp‘𝑊)) & ⊢ ↑ = (.g‘(mulGrp‘𝑆)) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑋))‘𝐶) = (𝑁 ↑ 𝐶)) | ||
Theorem | evls1fpws 33088* | Evaluation of a univariate subring polynomial as a function in a power series. (Contributed by Thierry Arnoux, 23-Jan-2025.) |
⊢ 𝑄 = (𝑆 evalSub1 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly1‘𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ · = (.r‘𝑆) & ⊢ ↑ = (.g‘(mulGrp‘𝑆)) & ⊢ 𝐴 = (coe1‘𝑀) ⇒ ⊢ (𝜑 → (𝑄‘𝑀) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))))) | ||
Theorem | ressply1evl 33089 | Evaluation of a univariate subring polynomial is the same as the evaluation in the bigger ring. (Contributed by Thierry Arnoux, 23-Jan-2025.) |
⊢ 𝑄 = (𝑆 evalSub1 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly1‘𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐸 = (eval1‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) ⇒ ⊢ (𝜑 → 𝑄 = (𝐸 ↾ 𝐵)) | ||
Theorem | evls1addd 33090 | Univariate polynomial evaluation of a sum of polynomials. (Contributed by Thierry Arnoux, 8-Feb-2025.) |
⊢ 𝑄 = (𝑆 evalSub1 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly1‘𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ⨣ = (+g‘𝑊) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) + ((𝑄‘𝑁)‘𝐶))) | ||
Theorem | evls1muld 33091 | Univariate polynomial evaluation of a product of polynomials. (Contributed by Thierry Arnoux, 24-Jan-2025.) |
⊢ 𝑄 = (𝑆 evalSub1 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly1‘𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ × = (.r‘𝑊) & ⊢ · = (.r‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶))) | ||
Theorem | evls1vsca 33092 | Univariate polynomial evaluation of a scalar product of polynomials. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
⊢ 𝑄 = (𝑆 evalSub1 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly1‘𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ × = ( ·𝑠 ‘𝑊) & ⊢ · = (.r‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝑁 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐴 × 𝑁))‘𝐶) = (𝐴 · ((𝑄‘𝑁)‘𝐶))) | ||
Theorem | ressdeg1 33093 | The degree of a univariate polynomial in a structure restriction. (Contributed by Thierry Arnoux, 20-Jan-2025.) |
⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑈 = (Poly1‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝐷‘𝑃) = (( deg1 ‘𝐻)‘𝑃)) | ||
Theorem | ply1ascl0 33094 | The zero scalar as a polynomial. (Contributed by Thierry Arnoux, 20-Jan-2025.) |
⊢ 𝑊 = (Poly1‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝑂 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐴‘𝑂) = 0 ) | ||
Theorem | ply1ascl1 33095 | The multiplicative unit scalar as a univariate polynomial. (Contributed by Thierry Arnoux, 20-Jan-2025.) |
⊢ 𝑊 = (Poly1‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐼 = (1r‘𝑅) & ⊢ 1 = (1r‘𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐴‘𝐼) = 1 ) | ||
Theorem | ply1asclunit 33096 | A non-zero scalar polynomial is a unit. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐴‘𝑌) ∈ (Unit‘𝑃)) | ||
Theorem | deg1le0eq0 33097 | A polynomial with nonpositive degree is the zero polynomial iff its constant term is zero. Biconditional version of deg1scl 25970. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑂 = (0g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 0) ⇒ ⊢ (𝜑 → (𝐹 = 𝑂 ↔ ((coe1‘𝐹)‘0) = 0 )) | ||
Theorem | ressply10g 33098 | A restricted polynomial algebra has the same group identity (zero polynomial). (Contributed by Thierry Arnoux, 20-Jan-2025.) |
⊢ 𝑆 = (Poly1‘𝑅) & ⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝑈 = (Poly1‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) & ⊢ 𝑍 = (0g‘𝑆) ⇒ ⊢ (𝜑 → 𝑍 = (0g‘𝑈)) | ||
Theorem | ressply1mon1p 33099 | The monic polynomials of a restricted polynomial algebra. (Contributed by Thierry Arnoux, 21-Jan-2025.) |
⊢ 𝑆 = (Poly1‘𝑅) & ⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝑈 = (Poly1‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 𝑁 = (Monic1p‘𝐻) ⇒ ⊢ (𝜑 → 𝑁 = (𝐵 ∩ 𝑀)) | ||
Theorem | ressply1invg 33100 | An element of a restricted polynomial algebra has the same group inverse. (Contributed by Thierry Arnoux, 30-Jan-2025.) |
⊢ 𝑆 = (Poly1‘𝑅) & ⊢ 𝐻 = (𝑅 ↾s 𝑇) & ⊢ 𝑈 = (Poly1‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) & ⊢ 𝑃 = (𝑆 ↾s 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((invg‘𝑈)‘𝑋) = ((invg‘𝑃)‘𝑋)) |
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