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Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmthmsta 31601 A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑈 = (mThm‘𝑇)    &   𝑆 = (mPreSt‘𝑇)       𝑈𝑆
 
Theoremmppsthm 31602 A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐽 = (mPPSt‘𝑇)    &   𝑈 = (mThm‘𝑇)       𝐽𝑈
 
Theoremmthmblem 31603 Lemma for mthmb 31604. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑈 = (mThm‘𝑇)       ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))
 
Theoremmthmb 31604 If two statements have the same reduct then one is a theorem iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑈 = (mThm‘𝑇)       ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))
 
Theoremmthmpps 31605 Given a theorem, there is an explicitly definable witnessing provable pre-statement for the provability of the theorem. (However, this pre-statement requires infinitely many dv conditions, which is sometimes inconvenient.) (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝐽 = (mPPSt‘𝑇)    &   𝑈 = (mThm‘𝑇)    &   𝐷 = (mDV‘𝑇)    &   𝑉 = (mVars‘𝑇)    &   𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))    &   𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))       (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
 
Theoremmclsppslem 31606* The closure is closed under application of provable pre-statements. (Compare mclsax 31592.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐽 = (mPPSt‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   (𝜑𝑆 ∈ ran 𝐿)    &   ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))    &   ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))    &   ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)    &   (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))    &   (𝜑𝑠 ∈ ran 𝐿)    &   (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵)))    &   (𝜑 → ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀))       (𝜑 → (𝑠𝑝) ∈ (𝑆 “ (𝐾𝐶𝐵)))
 
Theoremmclspps 31607* The closure is closed under application of provable pre-statements. (Compare mclsax 31592.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐽 = (mPPSt‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   (𝜑𝑆 ∈ ran 𝐿)    &   ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))    &   ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))    &   ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)       (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
 
20.5.13  Grammatical formal systems
 
Syntaxcm0s 31608 Mapping expressions to statements.
class m0St
 
Syntaxcmsa 31609 The set of syntax axioms.
class mSA
 
Syntaxcmwgfs 31610 The set of weakly grammatical formal systems.
class mWGFS
 
Syntaxcmsy 31611 The syntax typecode function.
class mSyn
 
Syntaxcmesy 31612 The syntax typecode function for expressions.
class mESyn
 
Syntaxcmgfs 31613 The set of grammatical formal systems.
class mGFS
 
Syntaxcmtree 31614 The set of proof trees.
class mTree
 
Syntaxcmst 31615 The set of syntax trees.
class mST
 
Syntaxcmsax 31616 The indexing set for a syntax axiom.
class mSAX
 
Syntaxcmufs 31617 The set of unambiguous formal sytems.
class mUFS
 
Definitiondf-m0s 31618 Define a function mapping expressions to statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
 
Definitiondf-msa 31619* Define the set of syntax axioms. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1st𝑎) ∈ (mVT‘𝑡) ∧ Fun ((2nd𝑎) ↾ (mVR‘𝑡)))})
 
Definitiondf-mwgfs 31620* Define the set of weakly grammatical formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑𝑎((⟨𝑑, , 𝑎⟩ ∈ (mAx‘𝑡) ∧ (1st𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))}
 
Definitiondf-msyn 31621 Define the syntax typecode function. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSyn = Slot 6
 
Definitiondf-mtree 31622* Define the set of proof trees. (Contributed by Mario Carneiro, 14-Jul-2016.)
mTree = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑟 ∣ (∀𝑒 ∈ ran (mVH‘𝑡)𝑒𝑟⟨(m0St‘𝑒), ∅⟩ ∧ ∀𝑒 𝑒𝑟⟨((mStRed‘𝑡)‘⟨𝑑, , 𝑒⟩), ∅⟩ ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) → ({(𝑠𝑝)} × X𝑒 ∈ (𝑜 ∪ ((mVH‘𝑡) “ ((mVars‘𝑡) “ (𝑜 ∪ {𝑝}))))(𝑟 “ {(𝑠𝑒)})) ⊆ 𝑟)))}))
 
Definitiondf-mst 31623 Define the function mapping syntax expressions to syntax trees. (Contributed by Mario Carneiro, 14-Jul-2016.)
mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾ ((mEx‘𝑡) ↾ (mVT‘𝑡))))
 
Definitiondf-msax 31624* Define the indexing set for a syntax axiom's representation in a tree. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝))))
 
Definitiondf-mufs 31625 Define the set of unambiguous formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
 
20.5.14  Models of formal systems
 
Syntaxcmuv 31626 The universe of a model.
class mUV
 
Syntaxcmvl 31627 The set of valuations.
class mVL
 
Syntaxcmvsb 31628 Substitution for a valuation.
class mVSubst
 
Syntaxcmfsh 31629 The freshness relation of a model.
class mFresh
 
Syntaxcmfr 31630 The set of freshness relations.
class mFRel
 
Syntaxcmevl 31631 The evaluation function of a model.
class mEval
 
Syntaxcmdl 31632 The set of models.
class mMdl
 
Syntaxcusyn 31633 The syntax function applied to elements of the model.
class mUSyn
 
Syntaxcgmdl 31634 The set of models in a grammatical formal system.
class mGMdl
 
Syntaxcmitp 31635 The interpretation function of the model.
class mItp
 
Syntaxcmfitp 31636 The evaluation function derived from the interpretation.
class mFromItp
 
Definitiondf-muv 31637 Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mUV = Slot 7
 
Definitiondf-mfsh 31638 Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mFresh = Slot 8
 
Definitiondf-mevl 31639 Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mEval = Slot 9
 
Definitiondf-mvl 31640* Define the set of valuations. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)}))
 
Definitiondf-mvsb 31641* Define substitution applied to a valuation. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVSubst = (𝑡 ∈ V ↦ {⟨⟨𝑠, 𝑚⟩, 𝑥⟩ ∣ ((𝑠 ∈ ran (mSubst‘𝑡) ∧ 𝑚 ∈ (mVL‘𝑡)) ∧ ∀𝑣 ∈ (mVR‘𝑡)𝑚dom (mEval‘𝑡)(𝑠‘((mVH‘𝑡)‘𝑣)) ∧ 𝑥 = (𝑣 ∈ (mVR‘𝑡) ↦ (𝑚(mEval‘𝑡)(𝑠‘((mVH‘𝑡)‘𝑣)))))})
 
Definitiondf-mfrel 31642* Define the set of freshness relations. (Contributed by Mario Carneiro, 14-Jul-2016.)
mFRel = (𝑡 ∈ V ↦ {𝑟 ∈ 𝒫 ((mUV‘𝑡) × (mUV‘𝑡)) ∣ (𝑟 = 𝑟 ∧ ∀𝑐 ∈ (mVT‘𝑡)∀𝑤 ∈ (𝒫 (mUV‘𝑡) ∩ Fin)∃𝑣 ∈ ((mUV‘𝑡) “ {𝑐})𝑤 ⊆ (𝑟 “ {𝑣}))})
 
Definitiondf-mdl 31643* Define the set of models of a formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mMdl = {𝑡 ∈ mFS ∣ [(mUV‘𝑡) / 𝑢][(mEx‘𝑡) / 𝑥][(mVL‘𝑡) / 𝑣][(mEval‘𝑡) / 𝑛][(mFresh‘𝑡) / 𝑓]((𝑢 ⊆ ((mTC‘𝑡) × V) ∧ 𝑓 ∈ (mFRel‘𝑡) ∧ 𝑛 ∈ (𝑢pm (𝑣 × (mEx‘𝑡)))) ∧ ∀𝑚𝑣 ((∀𝑒𝑥 (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑢 “ {(1st𝑒)}) ∧ ∀𝑦 ∈ (mVR‘𝑡)⟨𝑚, ((mVH‘𝑡)‘𝑦)⟩𝑛(𝑚𝑦) ∧ ∀𝑑𝑎(⟨𝑑, , 𝑎⟩ ∈ (mAx‘𝑡) → ((∀𝑦𝑧(𝑦𝑑𝑧 → (𝑚𝑦)𝑓(𝑚𝑧)) ∧ ⊆ (dom 𝑛 “ {𝑚})) → 𝑚dom 𝑛 𝑎))) ∧ (∀𝑠 ∈ ran (mSubst‘𝑡)∀𝑒 ∈ (mEx‘𝑡)∀𝑦(⟨𝑠, 𝑚⟩(mVSubst‘𝑡)𝑦 → (𝑛 “ {⟨𝑚, (𝑠𝑒)⟩}) = (𝑛 “ {⟨𝑦, 𝑒⟩})) ∧ ∀𝑝𝑣𝑒𝑥 ((𝑚 ↾ ((mVars‘𝑡)‘𝑒)) = (𝑝 ↾ ((mVars‘𝑡)‘𝑒)) → (𝑛 “ {⟨𝑚, 𝑒⟩}) = (𝑛 “ {⟨𝑝, 𝑒⟩})) ∧ ∀𝑦𝑢𝑒𝑥 ((𝑚 “ ((mVars‘𝑡)‘𝑒)) ⊆ (𝑓 “ {𝑦}) → (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑓 “ {𝑦})))))}
 
Definitiondf-musyn 31644* Define the syntax typecode function for the model universe. (Contributed by Mario Carneiro, 14-Jul-2016.)
mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1st𝑣)), (2nd𝑣)⟩))
 
Definitiondf-gmdl 31645* Define the set of models of a grammatical formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mGMdl = {𝑡 ∈ (mGFS ∩ mMdl) ∣ (∀𝑐 ∈ (mTC‘𝑡)((mUV‘𝑡) “ {𝑐}) ⊆ ((mUV‘𝑡) “ {((mSyn‘𝑡)‘𝑐)}) ∧ ∀𝑣 ∈ (mUV‘𝑐)∀𝑤 ∈ (mUV‘𝑐)(𝑣(mFresh‘𝑡)𝑤𝑣(mFresh‘𝑡)((mUSyn‘𝑡)‘𝑤)) ∧ ∀𝑚 ∈ (mVL‘𝑡)∀𝑒 ∈ (mEx‘𝑡)((mEval‘𝑡) “ {⟨𝑚, 𝑒⟩}) = (((mEval‘𝑡) “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1st𝑒)})))}
 
Definitiondf-mitp 31646* Define the interpretation function for a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mItp = (𝑡 ∈ V ↦ (𝑎 ∈ (mSA‘𝑡) ↦ (𝑔X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)}) ↦ (℩𝑥𝑚 ∈ (mVL‘𝑡)(𝑔 = (𝑚 ↾ ((mVars‘𝑡)‘𝑎)) ∧ 𝑥 = (𝑚(mEval‘𝑡)𝑎))))))
 
Definitiondf-mfitp 31647* Define a function that produces the evaluation function, given the interpretation function for a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mFromItp = (𝑡 ∈ V ↦ (𝑓X𝑎 ∈ (mSA‘𝑡)(((mUV‘𝑡) “ {((1st𝑡)‘𝑎)}) ↑𝑚 X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)})) ↦ (𝑛 ∈ ((mUV‘𝑡) ↑pm ((mVL‘𝑡) × (mEx‘𝑡)))∀𝑚 ∈ (mVL‘𝑡)(∀𝑣 ∈ (mVR‘𝑡)⟨𝑚, ((mVH‘𝑡)‘𝑣)⟩𝑛(𝑚𝑣) ∧ ∀𝑒𝑎𝑔(𝑒(mST‘𝑡)⟨𝑎, 𝑔⟩ → ⟨𝑚, 𝑒𝑛(𝑓‘(𝑖 ∈ ((mVars‘𝑡)‘𝑎) ↦ (𝑚𝑛(𝑔‘((mVH‘𝑡)‘𝑖)))))) ∧ ∀𝑒 ∈ (mEx‘𝑡)(𝑛 “ {⟨𝑚, 𝑒⟩}) = ((𝑛 “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1st𝑒)}))))))
 
20.5.15  Splitting fields
 
Syntaxcitr 31648 Integral subring of a ring.
class IntgRing
 
Syntaxccpms 31649 Completion of a metric space.
class cplMetSp
 
Syntaxchlb 31650 Embeddings for a direct limit.
class HomLimB
 
Syntaxchlim 31651 Direct limit structure.
class HomLim
 
Syntaxcpfl 31652 Polynomial extension field.
class polyFld
 
Syntaxcsf1 31653 Splitting field for a single polynomial (auxiliary).
class splitFld1
 
Syntaxcsf 31654 Splitting field for a finite set of polynomials.
class splitFld
 
Syntaxcpsl 31655 Splitting field for a sequence of polynomials.
class polySplitLim
 
Definitiondf-irng 31656* Define the subring of elements of 𝑟 integral over 𝑠 in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(𝑓 “ {(0g𝑟)}))
 
Definitiondf-cplmet 31657* A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.)
cplMetSp = (𝑤 ∈ V ↦ ((𝑤s ℕ) ↾s (Cau‘(dist‘𝑤))) / 𝑟(Base‘𝑟) / 𝑣{⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ (𝑓 ↾ (ℤ𝑗)):(ℤ𝑗)⟶((𝑔𝑗)(ball‘(dist‘𝑤))𝑥))} / 𝑒((𝑟 /s 𝑒) sSet {⟨(dist‘ndx), {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝑣𝑞𝑣 ((𝑥 = [𝑝]𝑒𝑦 = [𝑞]𝑒) ∧ (𝑝𝑓 (dist‘𝑟)𝑞) ⇝ 𝑧)}⟩}))
 
Definitiondf-homlimb 31658* The input to this function is a sequence (on ) of homomorphisms 𝐹(𝑛):𝑅(𝑛)⟶𝑅(𝑛 + 1). The resulting structure is the direct limit of the direct system so defined. This function returns the pair 𝑆, 𝐺 where 𝑆 is the terminal object and 𝐺 is a sequence of functions such that 𝐺(𝑛):𝑅(𝑛)⟶𝑆 and 𝐺(𝑛) = 𝐹(𝑛) ∘ 𝐺(𝑛 + 1). (Contributed by Mario Carneiro, 2-Dec-2014.)
HomLimB = (𝑓 ∈ V ↦ 𝑛 ∈ ℕ ({𝑛} × dom (𝑓𝑛)) / 𝑣 {𝑠 ∣ (𝑠 Er 𝑣 ∧ (𝑥𝑣 ↦ ⟨((1st𝑥) + 1), ((𝑓‘(1st𝑥))‘(2nd𝑥))⟩) ⊆ 𝑠)} / 𝑒⟨(𝑣 / 𝑒), (𝑛 ∈ ℕ ↦ (𝑥 ∈ dom (𝑓𝑛) ↦ [⟨𝑛, 𝑥⟩]𝑒))⟩)
 
Definitiondf-homlim 31659* The input to this function is a sequence (on ) of structures 𝑅(𝑛) and homomorphisms 𝐹(𝑛):𝑅(𝑛)⟶𝑅(𝑛 + 1). The resulting structure is the direct limit of the direct system so defined, and maintains any structures that were present in the original objects. TODO: generalize to directed sets? (Contributed by Mario Carneiro, 2-Dec-2014.)
HomLim = (𝑟 ∈ V, 𝑓 ∈ V ↦ ( HomLimB ‘𝑓) / 𝑒(1st𝑒) / 𝑣(2nd𝑒) / 𝑔({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔𝑛), 𝑦 ∈ dom (𝑔𝑛) ↦ ⟨⟨((𝑔𝑛)‘𝑥), ((𝑔𝑛)‘𝑦)⟩, ((𝑔𝑛)‘(𝑥(+g‘(𝑟𝑛))𝑦))⟩)⟩, ⟨(.r‘ndx), 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔𝑛), 𝑦 ∈ dom (𝑔𝑛) ↦ ⟨⟨((𝑔𝑛)‘𝑥), ((𝑔𝑛)‘𝑦)⟩, ((𝑔𝑛)‘(𝑥(.r‘(𝑟𝑛))𝑦))⟩)⟩} ∪ {⟨(TopOpen‘ndx), {𝑠 ∈ 𝒫 𝑣 ∣ ∀𝑛 ∈ ℕ ((𝑔𝑛) “ 𝑠) ∈ (TopOpen‘(𝑟𝑛))}⟩, ⟨(dist‘ndx), 𝑛 ∈ ℕ ran (𝑥 ∈ dom ((𝑔𝑛)‘𝑛), 𝑦 ∈ dom ((𝑔𝑛)‘𝑛) ↦ ⟨⟨((𝑔𝑛)‘𝑥), ((𝑔𝑛)‘𝑦)⟩, (𝑥(dist‘(𝑟𝑛))𝑦)⟩)⟩, ⟨(le‘ndx), 𝑛 ∈ ℕ ((𝑔𝑛) ∘ ((le‘(𝑟𝑛)) ∘ (𝑔𝑛)))⟩}))
 
Definitiondf-plfl 31660* Define the field extension that augments a field with the root of the given irreducible polynomial, and extends the norm if one exists and the extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.)
polyFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ (Poly1𝑟) / 𝑠((RSpan‘𝑠)‘{𝑝}) / 𝑖(𝑧 ∈ (Base‘𝑟) ↦ [(𝑧( ·𝑠𝑠)(1r𝑠))](𝑠 ~QG 𝑖)) / 𝑓(𝑠 /s (𝑠 ~QG 𝑖)) / 𝑡((𝑡 toNrmGrp (𝑛 ∈ (AbsVal‘𝑡)(𝑛𝑓) = (norm‘𝑟))) sSet ⟨(le‘ndx), (𝑧 ∈ (Base‘𝑡) ↦ (𝑞𝑧 (𝑟 deg1 𝑞) < (𝑟 deg1 𝑝))) / 𝑔(𝑔 ∘ ((le‘𝑠) ∘ 𝑔))⟩), 𝑓⟩)
 
Definitiondf-sfl1 31661* Temporary construction for the splitting field of a polynomial. The inputs are a field 𝑟 and a polynomial 𝑝 that we want to split, along with a tuple 𝑗 in the same format as the output. The output is a tuple 𝑆, 𝐹 where 𝑆 is the splitting field and 𝐹 is an injective homomorphism from the original field 𝑟.

The function works by repeatedly finding the smallest monic irreducible factor, and extending the field by that factor using the polyFld construction. We keep track of a total order in each of the splitting fields so that we can pick an element definably without needing global choice. (Contributed by Mario Carneiro, 2-Dec-2014.)

splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ( mPoly ‘𝑠) / 𝑚{𝑔 ∈ ((Monic1p𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r𝑚)(𝑝𝑓) ∧ 1 < (𝑠 deg1 𝑔))} / 𝑏if(((𝑝𝑓) = (0g𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, (glb‘𝑏) / (𝑠 polyFld ) / 𝑡⟨(1st𝑡), (𝑓 ∘ (2nd𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟 deg1 𝑝))))))
 
Definitiondf-sfl 31662* Define the splitting field of a finite collection of polynomials, given a total ordered base field. The output is a tuple 𝑆, 𝐹 where 𝑆 is the totally ordered splitting field and 𝐹 is an injective homomorphism from the original field 𝑟. (Contributed by Mario Carneiro, 2-Dec-2014.)
splitFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥𝑓(𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝)))))
 
Definitiondf-psl 31663* Define the direct limit of an increasing sequence of fields produced by pasting together the splitting fields for each sequence of polynomials. That is, given a ring 𝑟, a strict order on 𝑟, and a sequence 𝑝:ℕ⟶(𝒫 𝑟 ∩ Fin) of finite sets of polynomials to split, we construct the direct limit system of field extensions by splitting one set at a time and passing the resulting construction to HomLim. (Contributed by Mario Carneiro, 2-Dec-2014.)
polySplitLim = (𝑟 ∈ V, 𝑝 ∈ ((𝒫 (Base‘𝑟) ∩ Fin) ↑𝑚 ℕ) ↦ (1st ∘ seq0((𝑔 ∈ V, 𝑞 ∈ V ↦ (1st𝑔) / 𝑒(1st𝑒) / 𝑠(𝑠 splitFld ran (𝑥𝑞 ↦ (𝑥 ∘ (2nd𝑔)))) / 𝑓𝑓, ((2nd𝑔) ∘ (2nd𝑓))⟩), (𝑝 ∪ {⟨0, ⟨⟨𝑟, ∅⟩, ( I ↾ (Base‘𝑟))⟩⟩}))) / 𝑓((1st ∘ (𝑓 shift 1)) HomLim (2nd𝑓)))
 
20.5.16  p-adic number fields
 
Syntaxczr 31664 Integral elements of a ring.
class ZRing
 
Syntaxcgf 31665 Galois finite field.
class GF
 
Syntaxcgfo 31666 Galois limit field.
class GF
 
Syntaxceqp 31667 Equivalence relation for df-qp 31679.
class ~Qp
 
Syntaxcrqp 31668 Equivalence relation representatives for df-qp 31679.
class /Qp
 
Syntaxcqp 31669 The set of 𝑝-adic rational numbers.
class Qp
 
SyntaxcqpOLD 31670 The set of 𝑝-adic rational numbers. (New usage is discouraged.)
class QpOLD
 
Syntaxczp 31671 The set of 𝑝-adic integers. (Not to be confused with czn 19899.)
class Zp
 
Syntaxcqpa 31672 Algebraic completion of the 𝑝-adic rational numbers.
class _Qp
 
Syntaxccp 31673 Metric completion of _Qp.
class Cp
 
Definitiondf-zrng 31674 Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟)))
 
Definitiondf-gf 31675* Define the Galois finite field of order 𝑝𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.)
GF = (𝑝 ∈ ℙ, 𝑛 ∈ ℕ ↦ (ℤ/nℤ‘𝑝) / 𝑟(1st ‘(𝑟 splitFld {(Poly1𝑟) / 𝑠(var1𝑟) / 𝑥(((𝑝𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g𝑠)𝑥)})))
 
Definitiondf-gfoo 31676* Define the Galois field of order 𝑝↑+∞, as a direct limit of the Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.)
GF = (𝑝 ∈ ℙ ↦ (ℤ/nℤ‘𝑝) / 𝑟(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {(Poly1𝑟) / 𝑠(var1𝑟) / 𝑥(((𝑝𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g𝑠)𝑥)})))
 
Definitiondf-eqp 31677* Define an equivalence relation on -indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum Σ𝑘𝑛𝑓(𝑘)(𝑝𝑘) is a multiple of 𝑝↑(𝑛 + 1) for every 𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.)
~Qp = (𝑝 ∈ ℙ ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (ℤ ↑𝑚 ℤ) ∧ ∀𝑛 ∈ ℤ Σ𝑘 ∈ (ℤ‘-𝑛)(((𝑓‘-𝑘) − (𝑔‘-𝑘)) / (𝑝↑(𝑘 + (𝑛 + 1)))) ∈ ℤ)})
 
Definitiondf-rqp 31678* There is a unique element of (ℤ ↑𝑚 (0...(𝑝 − 1))) ~Qp -equivalent to any element of (ℤ ↑𝑚 ℤ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.)
/Qp = (𝑝 ∈ ℙ ↦ (~Qp ∩ {𝑓 ∈ (ℤ ↑𝑚 ℤ) ∣ ∃𝑥 ∈ ran ℤ(𝑓 “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑦(𝑦 × (𝑦 ∩ (ℤ ↑𝑚 (0...(𝑝 − 1)))))))
 
Definitiondf-qp 31679* Define the 𝑝-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 10-Oct-2021.)
Qp = (𝑝 ∈ ℙ ↦ { ∈ (ℤ ↑𝑚 (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran ℤ( “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑏(({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ ((/Qp‘𝑝)‘(𝑓𝑓 + 𝑔)))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓𝑘) · (𝑔‘(𝑛𝑘))))))⟩} ∪ {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}⟩}) toNrmGrp (𝑓𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-inf((𝑓 “ (ℤ ∖ {0})), ℝ, < ))))))
 
Definitiondf-qpOLD 31680* Define the 𝑝-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.) Obsolete version of df-qp 31679 as of 10-Oct-2021. (New usage is discouraged.)
QpOLD = (𝑝 ∈ ℙ ↦ { ∈ (ℤ ↑𝑚 (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran ℤ( “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑏(({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ ((/Qp‘𝑝)‘(𝑓𝑓 + 𝑔)))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓𝑘) · (𝑔‘(𝑛𝑘))))))⟩} ∪ {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}⟩}) toNrmGrp (𝑓𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-sup((𝑓 “ (ℤ ∖ {0})), ℝ, < ))))))
 
Definitiondf-zp 31681 Define the 𝑝-adic integers, as a subset of the 𝑝-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
Zp = (ZRing ∘ Qp)
 
Definitiondf-qpa 31682* Define the completion of the 𝑝-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the 𝑛-th set the collection of polynomials with degree less than 𝑛 and with coefficients < (𝑝𝑛)). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial 𝑥↑(𝑝𝑛) − 𝑥, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)
_Qp = (𝑝 ∈ ℙ ↦ (Qp‘𝑝) / 𝑟(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈ (Poly1𝑟) ∣ ((𝑟 deg1 𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1𝑓)(𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))})))
 
Definitiondf-cp 31683 Define the metric completion of the algebraic completion of the 𝑝 -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
Cp = ( cplMetSp ∘ _Qp)
 
20.6  Mathbox for Filip Cernatescu

I hope someone will enjoy solving (proving) the simple equations, inequalities, and calculations from this mathbox. I have proved these problems (theorems) using the Milpgame proof assistant. (It can be downloaded from http://us.metamath.org/other/milpgame/milpgame.html.)

 
Theoremproblem1 31684 Practice problem 1. Clues: 5p4e9 11205 3p2e5 11198 eqtri 2673 oveq1i 6700. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
((3 + 2) + 4) = 9
 
Theoremproblem2 31685 Practice problem 2. Clues: oveq12i 6702 adddiri 10089 add4i 10298 mulcli 10083 recni 10090 2re 11128 3eqtri 2677 10re 11555 5re 11137 1re 10077 4re 11135 eqcomi 2660 5p4e9 11205 oveq1i 6700 df-3 11118. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, 9-Sep-2021.) (Proof modification is discouraged.)
(((2 · 10) + 5) + ((1 · 10) + 4)) = ((3 · 10) + 9)
 
Theoremproblem2OLD 31686 Practice problem 2. Clues: oveq12i 6702 adddiri 10089 add4i 10298 mulcli 10083 recni 10090 2re 11128 3eqtri 2677 10re 11555 5re 11137 1re 10077 4re 11135 eqcomi 2660 5p4e9 11205 oveq1i 6700 df-3 11118. (Contributed by Filip Cernatescu, 16-Mar-2019.) Obsolete version of problem2 31685 as of 9-Sep-2021. (Proof modification is discouraged.) (New usage is discouraged.)
(((2 · 10) + 5) + ((1 · 10) + 4)) = ((3 · 10) + 9)
 
Theoremproblem3 31687 Practice problem 3. Clues: eqcomi 2660 eqtri 2673 subaddrii 10408 recni 10090 4re 11135 3re 11132 1re 10077 df-4 11119 addcomi 10265. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
𝐴 ∈ ℂ    &   (𝐴 + 3) = 4       𝐴 = 1
 
Theoremproblem4 31688 Practice problem 4. Clues: pm3.2i 470 eqcomi 2660 eqtri 2673 subaddrii 10408 recni 10090 7re 11141 6re 11139 ax-1cn 10032 df-7 11122 ax-mp 5 oveq1i 6700 3cn 11133 2cn 11129 df-3 11118 mulid2i 10081 subdiri 10518 mp3an 1464 mulcli 10083 subadd23 10331 oveq2i 6701 oveq12i 6702 3t2e6 11217 mulcomi 10084 subcli 10395 biimpri 218 subadd2i 10407. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 3    &   ((3 · 𝐴) + (2 · 𝐵)) = 7       (𝐴 = 1 ∧ 𝐵 = 2)
 
Theoremproblem5 31689 Practice problem 5. Clues: 3brtr3i 4714 mpbi 220 breqtri 4710 ltaddsubi 10627 remulcli 10092 2re 11128 3re 11132 9re 11145 eqcomi 2660 mvlladdi 10337 3cn 6cn 11140 eqtr3i 2675 6p3e9 11208 addcomi 10265 ltdiv1ii 10991 6re 11139 nngt0i 11092 2nn 11223 divcan3i 10809 recni 10090 2cn 11129 2ne0 11151 mpbir 221 eqtri 2673 mulcomi 10084 3t2e6 11217 divmuli 10817. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
𝐴 ∈ ℝ    &   ((2 · 𝐴) + 3) < 9       𝐴 < 3
 
Theoremquad3 31690 Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.)
𝑋 ∈ ℂ    &   𝐴 ∈ ℂ    &   𝐴 ≠ 0    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0       (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)))
 
20.7  Mathbox for Paul Chapman
 
20.7.1  Real and complex numbers (cont.)
 
Theoremclimuzcnv 31691* Utility lemma to convert between 𝑚𝑘 and 𝑘 ∈ (ℤ𝑚) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)
(𝑚 ∈ ℕ → ((𝑘 ∈ (ℤ𝑚) → 𝜑) ↔ (𝑘 ∈ ℕ → (𝑚𝑘𝜑))))
 
Theoremsinccvglem 31692* ((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)
(𝜑𝐹:ℕ⟶(ℝ ∖ {0}))    &   (𝜑𝐹 ⇝ 0)    &   𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))    &   𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3)))    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (abs‘(𝐹𝑘)) < 1)       (𝜑 → (𝐺𝐹) ⇝ 1)
 
Theoremsinccvg 31693* ((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝐹 ⇝ 0) → ((𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) ∘ 𝐹) ⇝ 1)
 
Theoremcircum 31694* The circumference of a circle of radius 𝑅, defined as the limit as 𝑛 ⇝ +∞ of the perimeter of an inscribed n-sided isogons, is ((2 · π) · 𝑅). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
𝐴 = ((2 · π) / 𝑛)    &   𝑃 = (𝑛 ∈ ℕ ↦ ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2)))))    &   𝑅 ∈ ℝ       𝑃 ⇝ ((2 · π) · 𝑅)
 
20.7.2  Miscellaneous theorems
 
Theoremelfzm12 31695 Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)
(𝑁 ∈ ℕ → (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (1...𝑁)))
 
Theoremnn0seqcvg 31696* A strictly-decreasing nonnegative integer sequence with initial term 𝑁 reaches zero by the 𝑁 th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.)
𝐹:ℕ0⟶ℕ0    &   𝑁 = (𝐹‘0)    &   (𝑘 ∈ ℕ0 → ((𝐹‘(𝑘 + 1)) ≠ 0 → (𝐹‘(𝑘 + 1)) < (𝐹𝑘)))       (𝐹𝑁) = 0
 
Theoremlediv2aALT 31697 Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) → (𝐴𝐵 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
 
Theoremabs2sqlei 31698 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2))
 
Theoremabs2sqlti 31699 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2))
 
Theoremabs2sqle 31700 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2)))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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