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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | crngmxidl 33601 | 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 33602 | 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 33603 | 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 33604 | 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 33605* | The set 𝑃 used in the proof of ssmxidl 33606 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 33606* | 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 33607 | 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 33608 | 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 33609 | 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 | drngmxidlr 33610 | If a ring's only maximal ideal is the zero ideal, it is a division ring. See also drngmxidl 33609. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (MaxIdeal‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 = {{ 0 }}) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | krull 33611* | 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 33612* | 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 | krullndrng 33613* | Krull's theorem for non-division-rings: Existence of a nonzero maximal ideal. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑚 ≠ { 0 }) | ||
| Theorem | opprabs 33614 | 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 33615 | 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 33616 | 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 33617 | 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 33618 | 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 33619 | 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 33620 | 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 33621 | 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 33622 | 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 33623 | 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 33624 | 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 33625 | 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 33626* | Lemma for qsdrngi 33627. (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 33627 | 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 33628 | Lemma for qsdrng 33629. (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 33629 | 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 33630 | 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 33631 | 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)) | ||
| Theorem | drnglring 33632 | A division ring is a local ring. (Contributed by Thierry Arnoux, 2-Jun-2026.) |
| ⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝐹 ∈ LRing) | ||
| Theorem | dflring2 33633* | Alternate definition of a local ring. (Contributed by Thierry Arnoux, 2-Jun-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ − = (-g‘𝑅) ⇒ ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝑈 ∨ ( 1 − 𝑥) ∈ 𝑈))) | ||
| Theorem | dflringlem 33634 | Lemma for dflring3 33637. If a ring 𝑅 has a single maximal ideal 𝑀, then any element 𝑋 outside of 𝑀 is a unit. (Contributed by Thierry Arnoux, 2-Jun-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → (MaxIdeal‘𝑅) = {𝑀}) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑀)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑈) | ||
| Theorem | dflringlem2 33635 | Lemma for dflring3 33637. In a commutative local ring 𝑅, the set (𝐵 ∖ 𝑈) of non-units is an ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ LRing) ⇒ ⊢ (𝜑 → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) | ||
| Theorem | dflringlem3 33636 | Lemma for dflring3 33637. In a commutative local ring 𝑅, the set (𝐵 ∖ 𝑈) of non-units is a maximal ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ LRing) ⇒ ⊢ (𝜑 → (𝐵 ∖ 𝑈) ∈ (MaxIdeal‘𝑅)) | ||
| Theorem | dflring3 33637 | Alternate definition of a local ring: local rings have a single maximal ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.) |
| ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1o)) | ||
| Theorem | dflring4 33638 | Alternate definition of a local ring: the set (𝐵 ∖ 𝑈) of non-units is an ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))) | ||
| Theorem | fldlring 33639 | A field is a local ring. (Contributed by Thierry Arnoux, 3-Jun-2026.) (Proof modification is discouraged.) |
| ⊢ (𝜑 → 𝐹 ∈ Field) ⇒ ⊢ (𝜑 → 𝐹 ∈ LRing) | ||
| Syntax | cidlsrg 33640 | Extend class notation with the semiring of ideals of a ring. |
| class IDLsrg | ||
| Definition | df-idlsrg 33641* | 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 33642 | 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 33643* | Lemma for idlsrgbas 33644 through idlsrgtset 33648. (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 33644 | 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 33645 | 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 33646 | 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 33647* | 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 33648* | 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 33649 | 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 33650 | 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 33651 | 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 33652 | 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 33653 | 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 33654 | The ideals of a ring form a monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Mnd) | ||
| Theorem | idlsrgcmnd 33655 | The ideals of a ring form a commutative monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑆 ∈ CMnd) | ||
| Theorem | rprmval 33656* | 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 33657* | 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 | rprmcl 33658 | A ring prime is an element of the base set. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐵) | ||
| Theorem | rprmdvds 33659 | If a ring prime 𝑄 divides a product 𝑋 · 𝑌, then it divides either 𝑋 or 𝑌. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∥ (𝑋 · 𝑌)) ⇒ ⊢ (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)) | ||
| Theorem | rprmnz 33660 | A ring prime is nonzero. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝑄 ≠ 0 ) | ||
| Theorem | rprmnunit 33661 | A ring prime is not a unit. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → ¬ 𝑄 ∈ 𝑈) | ||
| Theorem | rsprprmprmidl 33662 | In a commutative ring, ideals generated by prime elements are prime ideals. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑃 ∈ (RPrime‘𝑅)) ⇒ ⊢ (𝜑 → (𝐾‘{𝑃}) ∈ (PrmIdeal‘𝑅)) | ||
| Theorem | rsprprmprmidlb 33663 | An ideal generated by a single element is a prime iff that element is prime. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑃 ↔ (𝐾‘{𝑋}) ∈ (PrmIdeal‘𝑅))) | ||
| Theorem | rprmndvdsr1 33664 | A ring prime element does not divide the ring multiplicative identity. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 1 = (1r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → ¬ 𝑄 ∥ 1 ) | ||
| Theorem | rprmasso 33665 | In an integral domain, the associate of a prime is a prime. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∥ 𝑌) & ⊢ (𝜑 → 𝑌 ∥ 𝑋) ⇒ ⊢ (𝜑 → 𝑌 ∈ 𝑃) | ||
| Theorem | rprmasso2 33666 | In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∥ 𝑌) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝑌 ∥ 𝑋) | ||
| Theorem | rprmasso3 33667* | In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∥ 𝑌) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ 𝑈 (𝑡 · 𝑋) = 𝑌) | ||
| Theorem | unitmulrprm 33668 | A ring unit multiplied by a ring prime is a ring prime. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐼 · 𝑄) ∈ 𝑃) | ||
| Theorem | rprmndvdsru 33669 | A ring prime element does not divide any ring unit. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) ⇒ ⊢ (𝜑 → ¬ 𝑄 ∥ 𝑇) | ||
| Theorem | rprmirredlem 33670 | Lemma for rprmirred 33671. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑄 ≠ 0 ) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑈)) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 = (𝑋 · 𝑌)) & ⊢ (𝜑 → 𝑄 ∥ 𝑋) ⇒ ⊢ (𝜑 → 𝑌 ∈ 𝑈) | ||
| Theorem | rprmirred 33671 | In an integral domain, ring primes are irreducible. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝐼 = (Irred‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → 𝑄 ∈ 𝐼) | ||
| Theorem | rprmirredb 33672 | In a principal ideal domain, the converse of rprmirred 33671 holds, i.e. irreducible elements are prime. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝐼 = (Irred‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ PID) ⇒ ⊢ (𝜑 → 𝐼 = 𝑃) | ||
| Theorem | rprmdvdspow 33673 | If a prime element divides a ring "power", it divides the term. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝑀) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑄 ∥ (𝑁 ↑ 𝑋)) ⇒ ⊢ (𝜑 → 𝑄 ∥ 𝑋) | ||
| Theorem | rprmdvdsprod 33674* | If a prime element 𝑄 divides a product, then it divides one term. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 finSupp 1 ) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ (𝜑 → 𝑄 ∥ (𝑀 Σg 𝐹)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐹 supp 1 )𝑄 ∥ (𝐹‘𝑥)) | ||
| Theorem | 1arithidomlem1 33675* | Lemma for 1arithidom 33677. (Contributed by Thierry Arnoux, 30-May-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐽 = (0..^(♯‘𝐹)) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑃) & ⊢ (𝜑 → 𝐺 ∈ Word 𝑃) & ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺)) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → ∀𝑔 ∈ Word 𝑃(∃𝑘 ∈ 𝑈 (𝑀 Σg 𝐹) = (𝑘 · (𝑀 Σg 𝑔)) → ∃𝑤∃𝑢 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑤:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ 𝑔 = (𝑢 ∘f · (𝐹 ∘ 𝑤))))) & ⊢ (𝜑 → 𝐻 ∈ Word 𝑃) & ⊢ (𝜑 → ∃𝑘 ∈ 𝑈 (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑘 · (𝑀 Σg 𝐻))) & ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐻))) & ⊢ (𝜑 → 𝑄(∥r‘𝑅)(𝐻‘𝐾)) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) & ⊢ (𝜑 → (𝑇 · 𝑄) = (𝐻‘𝐾)) & ⊢ (𝜑 → 𝑆:(0..^(♯‘𝐻))–1-1-onto→(0..^(♯‘𝐻))) & ⊢ (𝜑 → (𝐻 ∘ 𝑆) = (((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) ++ ⟨“(𝐻‘𝐾)”⟩)) & ⊢ (𝜑 → 𝑁 ∈ 𝑈) & ⊢ (𝜑 → (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑁 · (𝑀 Σg 𝐻))) ⇒ ⊢ (𝜑 → ∃𝑐∃𝑑 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑐:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ ((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) = (𝑑 ∘f · (𝐹 ∘ 𝑐)))) | ||
| Theorem | 1arithidomlem2 33676* | Lemma for 1arithidom 33677: induction step. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐽 = (0..^(♯‘𝐹)) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑃) & ⊢ (𝜑 → 𝐺 ∈ Word 𝑃) & ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺)) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → ∀𝑔 ∈ Word 𝑃(∃𝑘 ∈ 𝑈 (𝑀 Σg 𝐹) = (𝑘 · (𝑀 Σg 𝑔)) → ∃𝑤∃𝑢 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑤:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ 𝑔 = (𝑢 ∘f · (𝐹 ∘ 𝑤))))) & ⊢ (𝜑 → 𝐻 ∈ Word 𝑃) & ⊢ (𝜑 → ∃𝑘 ∈ 𝑈 (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑘 · (𝑀 Σg 𝐻))) & ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐻))) & ⊢ (𝜑 → 𝑄(∥r‘𝑅)(𝐻‘𝐾)) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) & ⊢ (𝜑 → (𝑇 · 𝑄) = (𝐻‘𝐾)) & ⊢ (𝜑 → 𝑆:(0..^(♯‘𝐻))–1-1-onto→(0..^(♯‘𝐻))) & ⊢ (𝜑 → (𝐻 ∘ 𝑆) = (((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) ++ ⟨“(𝐻‘𝐾)”⟩)) & ⊢ (𝜑 → 𝑁 ∈ 𝑈) & ⊢ (𝜑 → (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑁 · (𝑀 Σg 𝐻))) & ⊢ (𝜑 → 𝐷 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))) & ⊢ (𝜑 → 𝐶:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹))) & ⊢ (𝜑 → ((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) = (𝐷 ∘f · (𝐹 ∘ 𝐶))) ⇒ ⊢ (𝜑 → (((𝐶 ++ ⟨“(♯‘𝐹)”⟩) ∘ ◡𝑆):(0..^(♯‘(𝐹 ++ ⟨“𝑄”⟩)))–1-1-onto→(0..^(♯‘(𝐹 ++ ⟨“𝑄”⟩))) ∧ 𝐻 = (((𝐷 ++ ⟨“𝑇”⟩) ∘ ◡𝑆) ∘f · ((𝐹 ++ ⟨“𝑄”⟩) ∘ ((𝐶 ++ ⟨“(♯‘𝐹)”⟩) ∘ ◡𝑆))))) | ||
| Theorem | 1arithidom 33677* | Uniqueness of prime factorizations in an integral domain 𝑅. Given two equal products 𝐹 and 𝐺 of prime elements, 𝐹 and 𝐺 are equal up to a renumbering 𝑤 and a multiplication by units 𝑢. See also 1arith 16935. Chapter VII, Paragraph 3, Section 3, Proposition 2 of [BourbakiCAlg2], p. 228. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐽 = (0..^(♯‘𝐹)) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑃) & ⊢ (𝜑 → 𝐺 ∈ Word 𝑃) & ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺)) ⇒ ⊢ (𝜑 → ∃𝑤∃𝑢 ∈ (𝑈 ↑m 𝐽)(𝑤:𝐽–1-1-onto→𝐽 ∧ 𝐺 = (𝑢 ∘f · (𝐹 ∘ 𝑤)))) | ||
| Syntax | cufd 33678 | Class of unique factorization domains. |
| class UFD | ||
| Definition | df-ufd 33679* | Define the class of unique factorization domains. A unique factorization domain (UFD for short), is an integral domain such that every nonzero 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.) Exclude the 0 prime ideal. (Revised by Thierry Arnoux, 9-May-2025.) Exclude the 0 ring. (Revised by Thierry Arnoux, 14-Jun-2025.) |
| ⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅} | ||
| Theorem | isufd 33680* | The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝐼 = (PrmIdeal‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖 ∩ 𝑃) ≠ ∅)) | ||
| Theorem | ufdprmidl 33681* | In a unique factorization domain 𝑅, a nonzero prime ideal 𝐽 contains a prime element 𝑝. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐼 = (PrmIdeal‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ UFD) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐽 ≠ { 0 }) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽) | ||
| Theorem | ufdidom 33682 | A nonzero unique factorization domain is an integral domain. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ UFD) ⇒ ⊢ (𝜑 → 𝑅 ∈ IDomn) | ||
| Theorem | pidufd 33683 | Every principal ideal domain is a unique factorization domain. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ PID) ⇒ ⊢ (𝜑 → 𝑅 ∈ UFD) | ||
| Theorem | 1arithufdlem1 33684* | Lemma for 1arithufd 33688. The set 𝑆 of elements which can be written as a product of primes is not empty. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ UFD) & ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing) & ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)} ⇒ ⊢ (𝜑 → 𝑆 ≠ ∅) | ||
| Theorem | 1arithufdlem2 33685* | Lemma for 1arithufd 33688. The set 𝑆 of elements which can be written as a product of primes is multiplicatively closed. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ UFD) & ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing) & ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)} & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑆) | ||
| Theorem | 1arithufdlem3 33686* | Lemma for 1arithufd 33688. If a product (𝑌 · 𝑋) can be written as a product of primes, with 𝑋 non-unit, nonzero, so can 𝑋. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ UFD) & ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing) & ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)} & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑌 · 𝑋) ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑆) | ||
| Theorem | 1arithufdlem4 33687* | Lemma for 1arithufd 33688. Nonzero ring, non-field case. Those trivial cases are handled in the final proof. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ UFD) & ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing) & ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)} & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑆) | ||
| Theorem | 1arithufd 33688* | Existence of a factorization into irreducible elements in a unique factorization domain. Any nonzero, non-unit element 𝑋 of a UFD 𝑅 can be written as a product of primes 𝑓. As shown in 1arithidom 33677, that factorization is unique, up to the order of the factors and multiplication by units. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ UFD) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓)) | ||
| Theorem | dfufd2lem 33689 | Lemma for dfufd2 33690. (Contributed by Thierry Arnoux, 6-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐼 ∈ (PrmIdeal‘𝑅)) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑃) & ⊢ (𝜑 → (𝑀 Σg 𝐹) ∈ 𝐼) & ⊢ (𝜑 → (𝑀 Σg 𝐹) ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐼 ∩ 𝑃) ≠ ∅) | ||
| Theorem | dfufd2 33690* | Alternative definition of unique factorization domain (UFD). This is often the textbook definition. Chapter VII, Paragraph 3, Section 3, Proposition 2 of [BourbakiCAlg2], p. 228. (Contributed by Thierry Arnoux, 6-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑥 ∈ ((𝐵 ∖ 𝑈) ∖ { 0 })∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓))) | ||
| Theorem | zringidom 33691 | The ring of integers is an integral domain. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ ℤring ∈ IDomn | ||
| Theorem | zringpid 33692 | The ring of integers is a principal ideal domain. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ ℤring ∈ PID | ||
| Theorem | dfprm3 33693 | The (positive) prime elements of the integer ring are the prime numbers. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ ℙ = (ℕ ∩ (RPrime‘ℤring)) | ||
| Theorem | zringfrac 33694* | The field of fractions of the ring of integers is isomorphic to the field of the rational numbers. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ ∼ = (ℤring ~RL (ℤ ∖ {0})) & ⊢ 𝐹 = (𝑞 ∈ ℚ ↦ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ) ⇒ ⊢ 𝐹 ∈ (𝑄 RingIso ( Frac ‘ℤring)) | ||
| Theorem | assaassd 33695 | Left-associative property of an associative algebra, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ × = (.r‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌))) | ||
| Theorem | assaassrd 33696 | Right-associative property of an associative algebra, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ × = (.r‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))) | ||
| Theorem | 0ringmon1p 33697 | 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 33698 | 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 33699 | 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 | evls1fn 33700 | 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) | ||
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