P. 432 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43101-43200   *Has distinct variable group(s)

Type	Label	Description
Statement
21.33.39  Addenda for structure powers
Theorem	pwssplit4 43101*	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 43102	Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM)
Theorem	pwslnmlem0 43103	Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s ∅)    ⇒   ⊢ (𝑊 ∈ LMod → 𝑌 ∈ LNoeM)
Theorem	pwslnmlem1 43104*	First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s {𝑖})    ⇒   ⊢ (𝑊 ∈ LNoeM → 𝑌 ∈ LNoeM)
Theorem	pwslnmlem2 43105	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 43106	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)
21.33.40  Every set admits a group structure iff choice
Theorem	unxpwdom3 43107*	Weaker version of unxpwdom 9629 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 43108*	The pw2f1o 9117 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 43109*	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 43110	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 43111	Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
⊢ (𝐺 ≃𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
Theorem	imasgim 43112	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 43113	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 43114	The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
⊢ (𝑆 ∈ 𝑉 → (har‘𝑆) ≠ ∅)
Theorem	numinfctb 43115	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 43116	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 43117	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 43118	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 43119	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 43120	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 ∖ {∅}))
21.33.41  Noetherian rings and left modules II
Syntax	clnr 43121	Extend class notation with the class of left Noetherian rings.
class LNoeR
Definition	df-lnr 43122	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 43123	Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
Theorem	lnrring 43124	Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)
Theorem	lnrlnm 43125	Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)
Theorem	islnr2 43126*	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 43127	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 43128*	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 43129	Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝑅 ∈ LPIR → 𝑅 ∈ LNoeR)
Theorem	lnrfrlm 43130	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 43131	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 43132	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)
21.33.42  Hilbert's Basis Theorem
Syntax	cldgis 43133	The leading ideal sequence used in the Hilbert Basis Theorem.
class ldgIdlSeq
Definition	df-ldgis 43134*	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 43142. (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 43135*	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 43136	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 43137	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 43138	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 43139	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 43140*	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 43141*	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 43142	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)
21.33.43  Additional material on polynomials [DEPRECATED]
Syntax	cmnc 43143	Extend class notation with the class of monic polynomials.
class Monic
Syntax	cplylt 43144	Extend class notation with the class of limited-degree polynomials.
class Poly<
Definition	df-mnc 43145*	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 43146*	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 43147	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 43148	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 43149	A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆))
Theorem	mnccoe 43150	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 43151	A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝)
21.33.44  Degree and minimal polynomial of algebraic numbers
Syntax	cdgraa 43152	Extend class notation to include the degree function for algebraic numbers.
class degAA
Syntax	cmpaa 43153	Extend class notation to include the minimal polynomial for an algebraic number.
class minPolyAA
Definition	df-dgraa 43154*	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 43155*	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 43156*	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 43157*	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 43158	Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ)
Theorem	dgraaf 43159	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 43160	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 43161	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 43162*	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 43163*	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 43164	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 43165	Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ))
Theorem	mpaadgr 43166	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 43167	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 43168	Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1)
21.33.45  Algebraic integers I
Syntax	citgo 43169	Extend class notation with the integral-over predicate.
class IntgOver
Syntax	cza 43170	Extend class notation with the class of algebraic integers.
class ℤ
Definition	df-itgo 43171*	A complex number is said to be integral over a subset if it is the root of a monic polynomial with coefficients from the subset. This definition is typically not used for fields but it works there, see aaitgo 43174. This definition could work for subsets of an arbitrary ring with a more general definition of polynomials. TODO: use Monic. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
Definition	df-za 43172	Define an algebraic integer as a complex number which is the root of a monic integer polynomial. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ ℤ = (IntgOver‘ℤ)
Theorem	itgoval 43173*	Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
Theorem	aaitgo 43174	The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝔸 = (IntgOver‘ℚ)
Theorem	itgoss 43175	An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))
Theorem	itgocn 43176	All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (IntgOver‘𝑆) ⊆ ℂ
Theorem	cnsrexpcl 43177	Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑋↑𝑌) ∈ 𝑆)
Theorem	fsumcnsrcl 43178*	Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)
Theorem	cnsrplycl 43179	Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld))    &   ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → (𝑃‘𝑋) ∈ 𝑆)
Theorem	rgspnid 43180	The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑆 = ((RingSpan‘𝑅)‘𝐴))    ⇒   ⊢ (𝜑 → 𝑆 = 𝐴)
Theorem	rngunsnply 43181*	Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ (𝜑 → 𝐵 ∈ (SubRing‘ℂfld))    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑆 = ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))    ⇒   ⊢ (𝜑 → (𝑉 ∈ 𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋)))
Theorem	flcidc 43182*	Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.)
⊢ (𝜑 → 𝐹 = (𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0)))    &   ⊢ (𝜑 → 𝑆 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑆) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑖 ∈ 𝑆 ((𝐹‘𝑖) · 𝐵) = ⦋𝐾 / 𝑖⦌𝐵)
21.33.46  Endomorphism algebra
Syntax	cmend 43183	Syntax for module endomorphism algebra.
class MEndo
Definition	df-mend 43184*	Define the endomorphism algebra of a module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ MEndo = (𝑚 ∈ V ↦ ⦋(𝑚 LMHom 𝑚) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 ∘f (+g‘𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦 ∈ 𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠 ‘𝑚)𝑦))⟩}))
Theorem	algstr 43185	Lemma to shorten proofs of algbase 43186 through algvsca 43190. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ 𝐴 Struct ⟨1, 6⟩
Theorem	algbase 43186	The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐴))
Theorem	algaddg 43187	The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝐴))
Theorem	algmulr 43188	The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( × ∈ 𝑉 → × = (.r‘𝐴))
Theorem	algsca 43189	The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝑆 = (Scalar‘𝐴))
Theorem	algvsca 43190	The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( · ∈ 𝑉 → · = ( ·𝑠 ‘𝐴))
Theorem	mendval 43191*	Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐵 = (𝑀 LMHom 𝑀)    &   ⊢ + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))    &   ⊢ × = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ · = (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))    ⇒   ⊢ (𝑀 ∈ 𝑋 → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
Theorem	mendbas 43192	Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    ⇒   ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
Theorem	mendplusgfval 43193*	Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))
Theorem	mendplusg 43194	A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ + = (+g‘𝑀)    &   ⊢ ✚ = (+g‘𝐴)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌))
Theorem	mendmulrfval 43195*	Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (.r‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))
Theorem	mendmulr 43196	A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ · = (.r‘𝐴)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑋 ∘ 𝑌))
Theorem	mendsca 43197	The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    ⇒   ⊢ 𝑆 = (Scalar‘𝐴)
Theorem	mendvscafval 43198*	Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐸 = (Base‘𝑀)    ⇒   ⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))
Theorem	mendvsca 43199	A specific scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐸 = (Base‘𝑀)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = ((𝐸 × {𝑋}) ∘f · 𝑌))
Theorem	mendring 43200	The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    ⇒   ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)