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Theorem List for Metamath Proof Explorer - 31501-31600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcsf1 31501 Splitting field for a single polynomial (auxiliary).
class splitFld1

Syntaxcsf 31502 Splitting field for a finite set of polynomials.
class splitFld

Syntaxcpsl 31503 Splitting field for a sequence of polynomials.
class polySplitLim

Definitiondf-irng 31504* 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 31505* 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 31506* 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 31507* 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 31508* 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 31509* 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 31510* 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 31511* 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)))

Syntaxczr 31512 Integral elements of a ring.
class ZRing

Syntaxcgf 31513 Galois finite field.
class GF

Syntaxcgfo 31514 Galois limit field.
class GF

Syntaxceqp 31515 Equivalence relation for df-qp 31527.
class ~Qp

Syntaxcrqp 31516 Equivalence relation representatives for df-qp 31527.
class /Qp

Syntaxcqp 31517 The set of 𝑝-adic rational numbers.
class Qp

SyntaxcqpOLD 31518 The set of 𝑝-adic rational numbers. (New usage is discouraged.)
class QpOLD

Syntaxczp 31519 The set of 𝑝-adic integers. (Not to be confused with czn 19832.)
class Zp

Syntaxcqpa 31520 Algebraic completion of the 𝑝-adic rational numbers.
class _Qp

Syntaxccp 31521 Metric completion of _Qp.
class Cp

Definitiondf-zrng 31522 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 31523* 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 31524* 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 31525* 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 31526* 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 31527* 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 31528* 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 31527 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 31529 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 31530* 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 31531 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 31532 Practice problem 1. Clues: 5p4e9 11152 3p2e5 11145 eqtri 2642 oveq1i 6645. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
((3 + 2) + 4) = 9

Theoremproblem2 31533 Practice problem 2. Clues: oveq12i 6647 adddiri 10036 add4i 10245 mulcli 10030 recni 10037 2re 11075 3eqtri 2646 10re 11502 5re 11084 1re 10024 4re 11082 eqcomi 2629 5p4e9 11152 oveq1i 6645 df-3 11065. (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 31534 Practice problem 2. Clues: oveq12i 6647 adddiri 10036 add4i 10245 mulcli 10030 recni 10037 2re 11075 3eqtri 2646 10re 11502 5re 11084 1re 10024 4re 11082 eqcomi 2629 5p4e9 11152 oveq1i 6645 df-3 11065. (Contributed by Filip Cernatescu, 16-Mar-2019.) Obsolete version of problem2 31533 as of 9-Sep-2021. (Proof modification is discouraged.) (New usage is discouraged.)
(((2 · 10) + 5) + ((1 · 10) + 4)) = ((3 · 10) + 9)

Theoremproblem3 31535 Practice problem 3. Clues: eqcomi 2629 eqtri 2642 subaddrii 10355 recni 10037 4re 11082 3re 11079 1re 10024 df-4 11066 addcomi 10212. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
𝐴 ∈ ℂ    &   (𝐴 + 3) = 4       𝐴 = 1

Theoremproblem4 31536 Practice problem 4. Clues: pm3.2i 471 eqcomi 2629 eqtri 2642 subaddrii 10355 recni 10037 7re 11088 6re 11086 ax-1cn 9979 df-7 11069 ax-mp 5 oveq1i 6645 3cn 11080 2cn 11076 df-3 11065 mulid2i 10028 subdiri 10465 mp3an 1422 mulcli 10030 subadd23 10278 oveq2i 6646 oveq12i 6647 3t2e6 11164 mulcomi 10031 subcli 10342 biimpri 218 subadd2i 10354. (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 31537 Practice problem 5. Clues: 3brtr3i 4673 mpbi 220 breqtri 4669 ltaddsubi 10574 remulcli 10039 2re 11075 3re 11079 9re 11092 eqcomi 2629 mvlladdi 10284 3cn 6cn 11087 eqtr3i 2644 6p3e9 11155 addcomi 10212 ltdiv1ii 10938 6re 11086 nngt0i 11039 2nn 11170 divcan3i 10756 recni 10037 2cn 11076 2ne0 11098 mpbir 221 eqtri 2642 mulcomi 10031 3t2e6 11164 divmuli 10764. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
𝐴 ∈ ℝ    &   ((2 · 𝐴) + 3) < 9       𝐴 < 3

Theoremquad3 31538 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 31539* 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 31540* ((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 31541* ((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 31542* 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 31543 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 31544* 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 31545 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 31546 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 31547 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 31548 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 31549 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 31550 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵))

Theoremabs2difabsi 31551 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 31552 ax-ext 2600 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧))

Theoremaxrepprim 31553 ax-rep 4762 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 31554 ax-un 6934 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦𝑥 → ¬ 𝑥𝑧) → 𝑦𝑥)

Theoremaxpowprim 31555 ax-pow 4834 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 31556 ax-reg 8482 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
(𝑥𝑦 → ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Theoremaxinfprim 31557 ax-inf 8520 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 31558 ax-ac 9266 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 31559* We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 31671). 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 31560* 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 31561* 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 31562 The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥 ∈ ∅ ¬ 𝑥𝑥

Theoremuntint 31563* 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 31564* If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
( E Fr 𝐴 → ∀𝑥𝐴 ¬ 𝑥𝑥)

Theoremuntangtr 31565* 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 31566 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
𝜑 → ((𝜑𝜓𝜒) → (𝜓𝜒)))

Theorem3orel2 31567 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 31568 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
𝜒 → ((𝜑𝜓𝜒) → (𝜑𝜓)))

Theorem3pm3.2ni 31569 Triple negated disjunction introduction. (Contributed by Scott Fenton, 20-Apr-2011.)
¬ 𝜑    &    ¬ 𝜓    &    ¬ 𝜒        ¬ (𝜑𝜓𝜒)

Theorem3jaodd 31570 Double deduction form of 3jaoi 1389. (Contributed by Scott Fenton, 20-Apr-2011.)
(𝜑 → (𝜓 → (𝜒𝜂)))    &   (𝜑 → (𝜓 → (𝜃𝜂)))    &   (𝜑 → (𝜓 → (𝜏𝜂)))       (𝜑 → (𝜓 → ((𝜒𝜃𝜏) → 𝜂)))

Theorem3orit 31571 Closed form of 3ori 1386. (Contributed by Scott Fenton, 20-Apr-2011.)
((𝜑𝜓𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))

Theorembiimpexp 31572 A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑𝜓) → ((𝜓𝜑) → 𝜒)))

Theorem3orel13 31573 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 31574 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 31575 Triple disjunction form of ccased 987. (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 31576* Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
(𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))

Theorembrtpid1 31577 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵

Theorembrtpid2 31578 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵

Theorembrtpid3 31579 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵

Theoremceqsrexv2 31580* 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 31581* A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
𝑥𝜑    &   𝑥𝐴    &   ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)

Theoremceqsralv2 31582* Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))

Theoremdford5 31583 A class is ordinal iff it is a subclass of On and transitive. (Contributed by Scott Fenton, 21-Nov-2021.)
(Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴))

Theoremjath 31584 Closed form of ja 173. Proved using the completeness script. (Proof modification is discouraged.) (Contributed by Scott Fenton, 13-Dec-2021.)
((¬ 𝜑𝜒) → ((𝜓𝜒) → ((𝜑𝜓) → 𝜒)))

20.8.5  Properties of real and complex numbers

Theoremsqdivzi 31585 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)))

Theoremsubdivcomb1 31586 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) − 𝐵) / 𝐶) = (𝐴 − (𝐵 / 𝐶)))

Theoremsubdivcomb2 31587 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − (𝐶 · 𝐵)) / 𝐶) = ((𝐴 / 𝐶) − 𝐵))

Theoremsupfz 31588 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
(𝑁 ∈ (ℤ𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁)

Theoreminffz 31589 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.)
(𝑁 ∈ (ℤ𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀)

TheoreminffzOLD 31590 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) Obsolete version of inffz 31589 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(𝑁 ∈ (ℤ𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑀)

Theoremfz0n 31591 The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.)
(𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0))

Theoremshftvalg 31592 Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵𝐴)))

Theoremdivcnvlin 31593* Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵)))       (𝜑𝐹 ⇝ 1)

Theoremclimlec3 31594* Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ 𝐵)       (𝜑𝐴𝐵)

Theoremlogi 31595 Calculate the logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.)
(log‘i) = (i · (π / 2))

Theoremiexpire 31596 i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.)
(i↑𝑐i) ∈ ℝ

Theorembcneg1 31597 The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.)
(𝑁 ∈ ℕ0 → (𝑁C-1) = 0)

Theorembcm1nt 31598 The proportion of one bionmial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁𝐾))))

Theorembcprod 31599* A product identity for binomial coefficents. (Contributed by Scott Fenton, 23-Jun-2020.)
(𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁)))

Theorembccolsum 31600* A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.)
((𝑁 ∈ ℕ0𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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