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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)))
 
20.5.16  p-adic number fields
 
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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