P. 436 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43501-43600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fglmod 43501	Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)
Theorem	lsmfgcl 43502	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)
21.33.38  Noetherian left modules I
Syntax	clnm 43503	Extend class notation with the class of Noetherian left modules.
class LNoeM
Definition	df-lnm 43504*	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 43505*	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 43506*	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 43507	A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
Theorem	lnmlssfg 43508	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 43509	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 43510	A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)
Theorem	kercvrlsm 43511	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 43512	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 43513	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 43514	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 43515	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 43516	Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
21.33.39  Addenda for structure powers
Theorem	pwssplit4 43517*	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 43518	Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM)
Theorem	pwslnmlem0 43519	Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s ∅)    ⇒   ⊢ (𝑊 ∈ LMod → 𝑌 ∈ LNoeM)
Theorem	pwslnmlem1 43520*	First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s {𝑖})    ⇒   ⊢ (𝑊 ∈ LNoeM → 𝑌 ∈ LNoeM)
Theorem	pwslnmlem2 43521	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 43522	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 43523*	Weaker version of unxpwdom 9504 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 43524*	The pw2f1o 9020 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 43525*	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 43526	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 43527	Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
⊢ (𝐺 ≃𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
Theorem	imasgim 43528	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 43529	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 43530	The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
⊢ (𝑆 ∈ 𝑉 → (har‘𝑆) ≠ ∅)
Theorem	numinfctb 43531	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 43532	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 43533	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 43534	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 43535	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 43536	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 43537	Extend class notation with the class of left Noetherian rings.
class LNoeR
Definition	df-lnr 43538	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 43539	Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
Theorem	lnrring 43540	Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)
Theorem	lnrlnm 43541	Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)
Theorem	islnr2 43542*	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 43543	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 43544*	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 43545	Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝑅 ∈ LPIR → 𝑅 ∈ LNoeR)
Theorem	lnrfrlm 43546	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 43547	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 43548	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 43549	The leading ideal sequence used in the Hilbert Basis Theorem.
class ldgIdlSeq
Definition	df-ldgis 43550*	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 43558. (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 43551*	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 43552	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 43553	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 43554	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 43555	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 43556*	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 43557*	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 43558	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 43559	Extend class notation with the class of monic polynomials.
class Monic
Syntax	cplylt 43560	Extend class notation with the class of limited-degree polynomials.
class Poly<
Definition	df-mnc 43561*	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 43562*	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 43563	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 43564	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 43565	A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆))
Theorem	mnccoe 43566	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 43567	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 43568	Extend class notation to include the degree function for algebraic numbers.
class degAA
Syntax	cmpaa 43569	Extend class notation to include the minimal polynomial for an algebraic number.
class minPolyAA
Definition	df-dgraa 43570*	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 43571*	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 43572*	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 43573*	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 43574	Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ)
Theorem	dgraaf 43575	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 43576	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 43577	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 43578*	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 43579*	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 43580	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 43581	Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ))
Theorem	mpaadgr 43582	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 43583	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 43584	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 43585	Extend class notation with the integral-over predicate.
class IntgOver
Syntax	cza 43586	Extend class notation with the class of algebraic integers.
class ℤ
Definition	df-itgo 43587*	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 43590. 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 43588	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 43589*	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 43590	The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝔸 = (IntgOver‘ℚ)
Theorem	itgoss 43591	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 43592	All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (IntgOver‘𝑆) ⊆ ℂ
Theorem	cnsrexpcl 43593	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 43594*	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 43595	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 43596	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 43597*	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 43598*	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 43599	Syntax for module endomorphism algebra.
class MEndo
Definition	df-mend 43600*	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))⟩}))