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Type | Label | Description |
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Statement | ||
Theorem | aomclem4 40001* | Lemma for dfac11 40006. Limit case. Patch together well-orderings constructed so far using fnwe2 39997 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ 𝐹 = {〈𝑎, 𝑏〉 ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} & ⊢ (𝜑 → dom 𝑧 ∈ On) & ⊢ (𝜑 → dom 𝑧 = ∪ dom 𝑧) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) ⇒ ⊢ (𝜑 → 𝐹 We (𝑅1‘dom 𝑧)) | ||
Theorem | aomclem5 40002* | Lemma for dfac11 40006. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ 𝐵 = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) & ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) & ⊢ 𝐸 = {〈𝑎, 𝑏〉 ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} & ⊢ 𝐹 = {〈𝑎, 𝑏〉 ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} & ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) & ⊢ (𝜑 → dom 𝑧 ∈ On) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧)) | ||
Theorem | aomclem6 40003* | Lemma for dfac11 40006. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ 𝐵 = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) & ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) & ⊢ 𝐸 = {〈𝑎, 𝑏〉 ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} & ⊢ 𝐹 = {〈𝑎, 𝑏〉 ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} & ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) & ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺)) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → (𝐻‘𝐴) We (𝑅1‘𝐴)) | ||
Theorem | aomclem7 40004* | Lemma for dfac11 40006. (𝑅1‘𝐴) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ 𝐵 = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) & ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) & ⊢ 𝐸 = {〈𝑎, 𝑏〉 ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} & ⊢ 𝐹 = {〈𝑎, 𝑏〉 ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} & ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) & ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺)) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴)) | ||
Theorem | aomclem8 40005* | Lemma for dfac11 40006. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴)) | ||
Theorem | dfac11 40006* |
The right-hand side of this theorem (compare with ac4 9886),
sometimes
known as the "axiom of multiple choice", is a choice
equivalent.
Curiously, this statement cannot be proved without ax-reg 9040, despite
not mentioning the cumulative hierarchy in any way as most consequences
of regularity do.
This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it. A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well-ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))) | ||
Theorem | kelac1 40007* | Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐽 ∈ Top) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 ∈ (Clsd‘𝐽)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵:𝑆–1-1-onto→𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑈 ∈ ∪ 𝐽) & ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Comp) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅) | ||
Theorem | kelac2lem 40008 | Lemma for kelac2 40009 and dfac21 40010: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp) | ||
Theorem | kelac2 40009* | Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅) & ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))) ∈ Comp) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅) | ||
Theorem | dfac21 40010 | Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.) |
⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp)) | ||
Syntax | clfig 40011 | Extend class notation with the class of finitely generated left modules. |
class LFinGen | ||
Definition | df-lfig 40012 | Define the class of finitely generated left modules. Finite generation of subspaces can be intepreted using ↾s. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))} | ||
Theorem | islmodfg 40013* | Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) | ||
Theorem | islssfg 40014* | Property of a finitely generated left (sub)module. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝑈(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝑈))) | ||
Theorem | islssfg2 40015* | Property of a finitely generated left (sub)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑁‘𝑏) = 𝑈)) | ||
Theorem | islssfgi 40016 | Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s (𝑁‘𝐵)) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝑋 ∈ LFinGen) | ||
Theorem | fglmod 40017 | Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) | ||
Theorem | lsmfgcl 40018 | The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐷 = (𝑊 ↾s 𝐴) & ⊢ 𝐸 = (𝑊 ↾s 𝐵) & ⊢ 𝐹 = (𝑊 ↾s (𝐴 ⊕ 𝐵)) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ LFinGen) & ⊢ (𝜑 → 𝐸 ∈ LFinGen) ⇒ ⊢ (𝜑 → 𝐹 ∈ LFinGen) | ||
Syntax | clnm 40019 | Extend class notation with the class of Noetherian left modules. |
class LNoeM | ||
Definition | df-lnm 40020* | A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen} | ||
Theorem | islnm 40021* | Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
⊢ 𝑆 = (LSubSp‘𝑀) ⇒ ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) | ||
Theorem | islnm2 40022* | Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (LSubSp‘𝑀) & ⊢ 𝑁 = (LSpan‘𝑀) ⇒ ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) | ||
Theorem | lnmlmod 40023 | A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) | ||
Theorem | lnmlssfg 40024 | A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑀) & ⊢ 𝑅 = (𝑀 ↾s 𝑈) ⇒ ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen) | ||
Theorem | lnmlsslnm 40025 | All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑀) & ⊢ 𝑅 = (𝑀 ↾s 𝑈) ⇒ ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LNoeM) | ||
Theorem | lnmfg 40026 | A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen) | ||
Theorem | kercvrlsm 40027 | The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑈 = (LSubSp‘𝑆) & ⊢ ⊕ = (LSSum‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → (𝐹 “ 𝐷) = ran 𝐹) ⇒ ⊢ (𝜑 → (𝐾 ⊕ 𝐷) = 𝐵) | ||
Theorem | lmhmfgima 40028 | A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑇 ↾s (𝐹 “ 𝐴)) & ⊢ 𝑋 = (𝑆 ↾s 𝐴) & ⊢ 𝑈 = (LSubSp‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ LFinGen) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) ⇒ ⊢ (𝜑 → 𝑌 ∈ LFinGen) | ||
Theorem | lnmepi 40029 | Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵) → 𝑇 ∈ LNoeM) | ||
Theorem | lmhmfgsplit 40030 | If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 0 = (0g‘𝑇) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝑈 = (𝑆 ↾s 𝐾) & ⊢ 𝑉 = (𝑇 ↾s ran 𝐹) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen) | ||
Theorem | lmhmlnmsplit 40031 | If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.) |
⊢ 0 = (0g‘𝑇) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝑈 = (𝑆 ↾s 𝐾) & ⊢ 𝑉 = (𝑇 ↾s ran 𝐹) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM) | ||
Theorem | lnmlmic 40032 | Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM)) | ||
Theorem | pwssplit4 40033* | Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝐸 = (𝑅 ↑s (𝐴 ∪ 𝐵)) & ⊢ 𝐺 = (Base‘𝐸) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐾 = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })} & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ↾ 𝐵)) & ⊢ 𝐶 = (𝑅 ↑s 𝐴) & ⊢ 𝐷 = (𝑅 ↑s 𝐵) & ⊢ 𝐿 = (𝐸 ↾s 𝐾) ⇒ ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷)) | ||
Theorem | filnm 40034 | Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM) | ||
Theorem | pwslnmlem0 40035 | Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑊 ↑s ∅) ⇒ ⊢ (𝑊 ∈ LMod → 𝑌 ∈ LNoeM) | ||
Theorem | pwslnmlem1 40036* | First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑊 ↑s {𝑖}) ⇒ ⊢ (𝑊 ∈ LNoeM → 𝑌 ∈ LNoeM) | ||
Theorem | pwslnmlem2 40037 | A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑋 = (𝑊 ↑s 𝐴) & ⊢ 𝑌 = (𝑊 ↑s 𝐵) & ⊢ 𝑍 = (𝑊 ↑s (𝐴 ∪ 𝐵)) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → 𝑋 ∈ LNoeM) & ⊢ (𝜑 → 𝑌 ∈ LNoeM) ⇒ ⊢ (𝜑 → 𝑍 ∈ LNoeM) | ||
Theorem | pwslnm 40038 | Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑊 ↑s 𝐼) ⇒ ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM) | ||
Theorem | unxpwdom3 40039* | Weaker version of unxpwdom 9037 where a function is required only to be cancellative, not an injection. 𝐷 and 𝐵 are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into 𝐴, each row must hit an element of 𝐵; by column injectivity, each row can be identified in at least one way by the 𝐵 element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐷 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐷) → (𝑎 + 𝑏) ∈ (𝐴 ∪ 𝐵)) & ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑏 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐)) & ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ (𝑎 ∈ 𝐶 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎)) & ⊢ (𝜑 → ¬ 𝐷 ≼ 𝐴) ⇒ ⊢ (𝜑 → 𝐶 ≼* (𝐷 × 𝐵)) | ||
Theorem | pwfi2f1o 40040* | The pw2f1o 8605 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.) |
⊢ 𝑆 = {𝑦 ∈ (2o ↑m 𝐴) ∣ 𝑦 finSupp ∅} & ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o})) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)) | ||
Theorem | pwfi2en 40041* | Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.) |
⊢ 𝑆 = {𝑦 ∈ (2o ↑m 𝐴) ∣ 𝑦 finSupp ∅} ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝑆 ≈ (𝒫 𝐴 ∩ Fin)) | ||
Theorem | frlmpwfi 40042 | Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Proof shortened by AV, 14-Jun-2020.) |
⊢ 𝑅 = (ℤ/nℤ‘2) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin)) | ||
Theorem | gicabl 40043 | Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) |
⊢ (𝐺 ≃𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel)) | ||
Theorem | imasgim 40044 | A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈)) | ||
Theorem | isnumbasgrplem1 40045 | A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Abel ∧ 𝐶 ≈ 𝐵) → 𝐶 ∈ (Base “ Abel)) | ||
Theorem | harn0 40046 | The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) |
⊢ (𝑆 ∈ 𝑉 → (har‘𝑆) ≠ ∅) | ||
Theorem | numinfctb 40047 | A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) |
⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) | ||
Theorem | isnumbasgrplem2 40048 | If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card) | ||
Theorem | isnumbasgrplem3 40049 | Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.) |
⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel)) | ||
Theorem | isnumbasabl 40050 | A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) | ||
Theorem | isnumbasgrp 40051 | A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp)) | ||
Theorem | dfacbasgrp 40052 | A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
⊢ (CHOICE ↔ (Base “ Grp) = (V ∖ {∅})) | ||
Syntax | clnr 40053 | Extend class notation with the class of left Noetherian rings. |
class LNoeR | ||
Definition | df-lnr 40054 | A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM} | ||
Theorem | islnr 40055 | Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM)) | ||
Theorem | lnrring 40056 | Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝐴 ∈ LNoeR → 𝐴 ∈ Ring) | ||
Theorem | lnrlnm 40057 | Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM) | ||
Theorem | islnr2 40058* | Property of being a left-Noetherian ring in terms of finite generation of ideals (the usual "pure ring theory" definition). (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 𝑁 = (RSpan‘𝑅) ⇒ ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ ∀𝑖 ∈ 𝑈 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) | ||
Theorem | islnr3 40059 | Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ 𝑈 ∈ (NoeACS‘𝐵))) | ||
Theorem | lnr2i 40060* | Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 𝑁 = (RSpan‘𝑅) ⇒ ⊢ ((𝑅 ∈ LNoeR ∧ 𝐼 ∈ 𝑈) → ∃𝑔 ∈ (𝒫 𝐼 ∩ Fin)𝐼 = (𝑁‘𝑔)) | ||
Theorem | lpirlnr 40061 | Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝑅 ∈ LPIR → 𝑅 ∈ LNoeR) | ||
Theorem | lnrfrlm 40062 | Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ LNoeR ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM) | ||
Theorem | lnrfg 40063 | Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
⊢ 𝑆 = (Scalar‘𝑀) ⇒ ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM) | ||
Theorem | lnrfgtr 40064 | A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑈 = (LSubSp‘𝑀) & ⊢ 𝑁 = (𝑀 ↾s 𝑃) ⇒ ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR ∧ 𝑃 ∈ 𝑈) → 𝑁 ∈ LFinGen) | ||
Syntax | cldgis 40065 | The leading ideal sequence used in the Hilbert Basis Theorem. |
class ldgIdlSeq | ||
Definition | df-ldgis 40066* | Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree- 𝑥 elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 40074. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))) | ||
Theorem | hbtlem1 40067* | Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝑆 = (ldgIdlSeq‘𝑅) & ⊢ 𝐷 = ( deg1 ‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))}) | ||
Theorem | hbtlem2 40068 | Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝑆 = (ldgIdlSeq‘𝑅) & ⊢ 𝑇 = (LIdeal‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ 𝑇) | ||
Theorem | hbtlem7 40069 | Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝑆 = (ldgIdlSeq‘𝑅) & ⊢ 𝑇 = (LIdeal‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼):ℕ0⟶𝑇) | ||
Theorem | hbtlem4 40070 | The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝑆 = (ldgIdlSeq‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ ℕ0) & ⊢ (𝜑 → 𝑌 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) ⇒ ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐼)‘𝑌)) | ||
Theorem | hbtlem3 40071 | The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝑆 = (ldgIdlSeq‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝐽 ∈ 𝑈) & ⊢ (𝜑 → 𝐼 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋)) | ||
Theorem | hbtlem5 40072* | The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝑆 = (ldgIdlSeq‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝐽 ∈ 𝑈) & ⊢ (𝜑 → 𝐼 ⊆ 𝐽) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥)) ⇒ ⊢ (𝜑 → 𝐼 = 𝐽) | ||
Theorem | hbtlem6 40073* | There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝑆 = (ldgIdlSeq‘𝑅) & ⊢ 𝑁 = (RSpan‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ LNoeR) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ ℕ0) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)) | ||
Theorem | hbt 40074 | The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR) | ||
Syntax | cmnc 40075 | Extend class notation with the class of monic polynomials. |
class Monic | ||
Syntax | cplylt 40076 | Extend class notatin with the class of limited-degree polynomials. |
class Poly< | ||
Definition | df-mnc 40077* | Define the class of monic polynomials. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}) | ||
Definition | df-plylt 40078* | Define the class of limited-degree polynomials. (Contributed by Stefan O'Rear, 8-Dec-2014.) |
⊢ Poly< = (𝑠 ∈ 𝒫 ℂ, 𝑥 ∈ ℕ0 ↦ {𝑝 ∈ (Poly‘𝑠) ∣ (𝑝 = 0𝑝 ∨ (deg‘𝑝) < 𝑥)}) | ||
Theorem | dgrsub2 40079 | Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ 𝑁 = (deg‘𝐹) ⇒ ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁) | ||
Theorem | elmnc 40080 | Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1)) | ||
Theorem | mncply 40081 | A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆)) | ||
Theorem | mnccoe 40082 | A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝑃 ∈ ( Monic ‘𝑆) → ((coeff‘𝑃)‘(deg‘𝑃)) = 1) | ||
Theorem | mncn0 40083 | A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝) | ||
Syntax | cdgraa 40084 | Extend class notation to include the degree function for algebraic numbers. |
class degAA | ||
Syntax | cmpaa 40085 | Extend class notation to include the minimal polynomial for an algebraic number. |
class minPolyAA | ||
Definition | df-dgraa 40086* | Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.) |
⊢ degAA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑥) = 0)}, ℝ, < )) | ||
Definition | df-mpaa 40087* | Define the minimal polynomial of an algebraic number as the unique monic polynomial which achieves the minimum of degAA. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ minPolyAA = (𝑥 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑥) ∧ (𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑥)) = 1))) | ||
Theorem | dgraaval 40088* | Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.) |
⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < )) | ||
Theorem | dgraalem 40089* | Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.) |
⊢ (𝐴 ∈ 𝔸 → ((degAA‘𝐴) ∈ ℕ ∧ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0))) | ||
Theorem | dgraacl 40090 | Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ) | ||
Theorem | dgraaf 40091 | Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.) |
⊢ degAA:𝔸⟶ℕ | ||
Theorem | dgraaub 40092 | Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.) |
⊢ (((𝑃 ∈ (Poly‘ℚ) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → (degAA‘𝐴) ≤ (deg‘𝑃)) | ||
Theorem | dgraa0p 40093 | A rational polynomial of degree less than an algebraic number cannot be zero at that number unless it is the zero polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ ((𝐴 ∈ 𝔸 ∧ 𝑃 ∈ (Poly‘ℚ) ∧ (deg‘𝑃) < (degAA‘𝐴)) → ((𝑃‘𝐴) = 0 ↔ 𝑃 = 0𝑝)) | ||
Theorem | mpaaeu 40094* | An algebraic number has exactly one monic polynomial of the least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) | ||
Theorem | mpaaval 40095* | Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))) | ||
Theorem | mpaalem 40096 | Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (degAA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1))) | ||
Theorem | mpaacl 40097 | Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ)) | ||
Theorem | mpaadgr 40098 | Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → (deg‘(minPolyAA‘𝐴)) = (degAA‘𝐴)) | ||
Theorem | mpaaroot 40099 | The minimal polynomial of an algebraic number has the number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴)‘𝐴) = 0) | ||
Theorem | mpaamn 40100 | Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1) |
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