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Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcgmdl 30601 The set of models in a grammatical formal system.
class mGMdl
 
Syntaxcmitp 30602 The interpretation function of the model.
class mItp
 
Syntaxcmfitp 30603 The evaluation function derived from the interpretation.
class mFromItp
 
Definitiondf-muv 30604 Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mUV = Slot 7
 
Definitiondf-mfsh 30605 Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mFresh = Slot 8
 
Definitiondf-mevl 30606 Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mEval = Slot 9
 
Definitiondf-mvl 30607* 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 30608* 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 30609* 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 30610* 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 30611* 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 30612* 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 30613* 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 30614* 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 30615 Integral subring of a ring.
class IntgRing
 
Syntaxccpms 30616 Completion of a metric space.
class cplMetSp
 
Syntaxchlb 30617 Embeddings for a direct limit.
class HomLimB
 
Syntaxchlim 30618 Direct limit structure.
class HomLim
 
Syntaxcpfl 30619 Polynomial extension field.
class polyFld
 
Syntaxcsf1 30620 Splitting field for a single polynomial (auxiliary).
class splitFld1
 
Syntaxcsf 30621 Splitting field for a finite set of polynomials.
class splitFld
 
Syntaxcpsl 30622 Splitting field for a sequence of polynomials.
class polySplitLim
 
Definitiondf-irng 30623* 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 30624* 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 30625* 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 30626* 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 30627* 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 30628* 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 30629* 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 30630* 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 30631 Integral elements of a ring.
class ZRing
 
Syntaxcgf 30632 Galois finite field.
class GF
 
Syntaxcgfo 30633 Galois limit field.
class GF
 
Syntaxceqp 30634 Equivalence relation for df-qp 30645.
class ~Qp
 
Syntaxcrqp 30635 Equivalence relation representatives for df-qp 30645.
class /Qp
 
Syntaxcqp 30636 The set of 𝑝-adic rational numbers.
class Qp
 
Syntaxczp 30637 The set of 𝑝-adic integers. (Not to be confused with czn 19579.)
class Zp
 
Syntaxcqpa 30638 Algebraic completion of the 𝑝-adic rational numbers.
class _Qp
 
Syntaxccp 30639 Metric completion of _Qp.
class Cp
 
Definitiondf-zrng 30640 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 30641* 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 30642* 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 30643* 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 30644* 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 30645* 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.)
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↑-sup((𝑓 “ (ℤ ∖ {0})), ℝ, < ))))))
 
Definitiondf-zp 30646 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 30647* 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 30648 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 30649 Practice problem 1. Clues: 5p4e9 10925 3p2e5 10918 eqtri 2536 oveq1i 6441. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
((3 + 2) + 4) = 9
 
Theoremproblem2 30650 Practice problem 2. Clues: oveq12i 6443 adddiri 9810 add4i 10014 mulcli 9804 recni 9811 2re 10848 3eqtri 2540 10re 11260 5re 10857 1re 9798 4re 10855 eqcomi 2523 5p4e9 10925 oveq1i 6441 df-3 10838. (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 30651 Practice problem 2. Clues: oveq12i 6443 adddiri 9810 add4i 10014 mulcli 9804 recni 9811 2re 10848 3eqtri 2540 10re 11260 5re 10857 1re 9798 4re 10855 eqcomi 2523 5p4e9 10925 oveq1i 6441 df-3 10838. (Contributed by Filip Cernatescu, 16-Mar-2019.) Obsolete version of problem2 30650 as of 9-Sep-2021. (Proof modification is discouraged.) (New usage is discouraged.)
(((2 · 10) + 5) + ((1 · 10) + 4)) = ((3 · 10) + 9)
 
Theoremproblem3 30652 Practice problem 3. Clues: eqcomi 2523 eqtri 2536 subaddrii 10124 recni 9811 4re 10855 3re 10852 1re 9798 df-4 10839 addcomi 9981. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
𝐴 ∈ ℂ    &   (𝐴 + 3) = 4       𝐴 = 1
 
Theoremproblem4 30653 Practice problem 4. Clues: pm3.2i 469 eqcomi 2523 eqtri 2536 subaddrii 10124 recni 9811 7re 10861 6re 10859 ax-1cn 9753 df-7 10842 ax-mp 5 oveq1i 6441 3cn 10853 2cn 10849 df-3 10838 mulid2i 9802 subdiri 10233 mp3an 1415 mulcli 9804 subadd23 10047 oveq2i 6442 oveq12i 6443 3t2e6 10937 mulcomi 9805 subcli 10111 biimpri 216 subadd2i 10123. (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 30654 Practice problem 5. Clues: 3brtr3i 4510 mpbi 218 breqtri 4506 ltaddsubi 10341 remulcli 9813 2re 10848 3re 10852 9re 10865 eqcomi 2523 mvlladdi 10053 3cn 6cn 10860 eqtr3i 2538 6p3e9 10928 addcomi 9981 ltdiv1ii 10706 6re 10859 nngt0i 10812 2nn 10943 divcan3i 10523 recni 9811 2cn 10849 2ne0 10871 mpbir 219 eqtri 2536 mulcomi 9805 3t2e6 10937 divmuli 10531. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
𝐴 ∈ ℝ    &   ((2 · 𝐴) + 3) < 9       𝐴 < 3
 
Theoremquad3 30655 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 30656* 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 30657* ((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 30658* ((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 30659* 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 30660 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 30661* 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 30662 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 30663 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 30664 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 30665 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)))
 
Theoremabs2sqlt 30666 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)))
 
Theoremabs2difi 30667 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵))
 
Theoremabs2difabsi 30668 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴𝐵))
 
20.8  Mathbox for Scott Fenton
 
20.8.1  ZFC Axioms in primitive form
 
Theoremaxextprim 30669 ax-ext 2494 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧))
 
Theoremaxrepprim 30670 ax-rep 4597 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥)))
 
Theoremaxunprim 30671 ax-un 6728 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦𝑥 → ¬ 𝑥𝑧) → 𝑦𝑥)
 
Theoremaxpowprim 30672 ax-pow 4668 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
(∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) → 𝑥 = 𝑦)
 
Theoremaxregprim 30673 ax-reg 8260 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
(𝑥𝑦 → ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
 
Theoremaxinfprim 30674 ax-inf 8298 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.)
¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
 
Theoremaxacprim 30675 ax-ac 9044 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)
¬ ∀𝑥 ¬ ∀𝑦𝑧(∀𝑥 ¬ (𝑦𝑧 → ¬ 𝑧𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦𝑧 → (𝑧𝑤 → (𝑦𝑤 → ¬ 𝑤𝑥))))))
 
20.8.2  Untangled classes
 
Theoremuntelirr 30676* We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 30777). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.)
(∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
 
Theoremuntuni 30677* The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)
(∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥)
 
Theoremuntsucf 30678* If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑦𝐴       (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦𝑦)
 
Theoremunt0 30679 The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥 ∈ ∅ ¬ 𝑥𝑥
 
Theoremuntint 30680* If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
(∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑦 𝐴 ¬ 𝑦𝑦)
 
Theoremefrunt 30681* If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
( E Fr 𝐴 → ∀𝑥𝐴 ¬ 𝑥𝑥)
 
Theoremuntangtr 30682* A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)
(Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
 
20.8.3  Extra propositional calculus theorems
 
Theorem3orel1 30683 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
𝜑 → ((𝜑𝜓𝜒) → (𝜓𝜒)))
 
Theorem3orel2 30684 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝜓 → ((𝜑𝜓𝜒) → (𝜑𝜒)))
 
Theorem3orel3 30685 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
𝜒 → ((𝜑𝜓𝜒) → (𝜑𝜓)))
 
Theorem3pm3.2ni 30686 Triple negated disjunction introduction. (Contributed by Scott Fenton, 20-Apr-2011.)
¬ 𝜑    &    ¬ 𝜓    &    ¬ 𝜒        ¬ (𝜑𝜓𝜒)
 
Theorem3jaodd 30687 Double deduction form of 3jaoi 1382. (Contributed by Scott Fenton, 20-Apr-2011.)
(𝜑 → (𝜓 → (𝜒𝜂)))    &   (𝜑 → (𝜓 → (𝜃𝜂)))    &   (𝜑 → (𝜓 → (𝜏𝜂)))       (𝜑 → (𝜓 → ((𝜒𝜃𝜏) → 𝜂)))
 
Theorem3orit 30688 Closed form of 3ori 1379. (Contributed by Scott Fenton, 20-Apr-2011.)
((𝜑𝜓𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))
 
Theorembiimpexp 30689 A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑𝜓) → ((𝜓𝜑) → 𝜒)))
 
Theorem3orel13 30690 Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.)
((¬ 𝜑 ∧ ¬ 𝜒) → ((𝜑𝜓𝜒) → 𝜓))
 
20.8.4  Misc. Useful Theorems
 
Theoremnepss 30691 Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
(𝐴𝐵 ↔ ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵))
 
Theorem3ccased 30692 Triple disjunction form of ccased 984. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝜑 → ((𝜒𝜂) → 𝜓))    &   (𝜑 → ((𝜒𝜁) → 𝜓))    &   (𝜑 → ((𝜒𝜎) → 𝜓))    &   (𝜑 → ((𝜃𝜂) → 𝜓))    &   (𝜑 → ((𝜃𝜁) → 𝜓))    &   (𝜑 → ((𝜃𝜎) → 𝜓))    &   (𝜑 → ((𝜏𝜂) → 𝜓))    &   (𝜑 → ((𝜏𝜁) → 𝜓))    &   (𝜑 → ((𝜏𝜎) → 𝜓))       (𝜑 → (((𝜒𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜓))
 
Theoremdfso3 30693* Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
(𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
 
Theorembrtpid1 30694 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵
 
Theorembrtpid2 30695 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵
 
Theorembrtpid3 30696 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵
 
Theoremceqsrexv2 30697* Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
 
Theoremiota5f 30698* A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
𝑥𝜑    &   𝑥𝐴    &   ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
 
Theoremceqsralv2 30699* Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
 
20.8.5  Properties of real and complex numbers
 
Theoremsqdivzi 30700 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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