HomeHome Metamath Proof Explorer
Theorem List (p. 257 of 470)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-29646)
  Hilbert Space Explorer  Hilbert Space Explorer
(29647-31169)
  Users' Mathboxes  Users' Mathboxes
(31170-46966)
 

Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelqaalem1 25601* Lemma for elqaa 25604. The function 𝑁 represents the denominators of the rational coefficients 𝐡. By multiplying them all together to make 𝑅, we get a number big enough to clear all the denominators and make 𝑅 Β· 𝐹 an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐹 ∈ ((Polyβ€˜β„š) βˆ– {0𝑝}))    &   (πœ‘ β†’ (πΉβ€˜π΄) = 0)    &   π΅ = (coeffβ€˜πΉ)    &   π‘ = (π‘˜ ∈ β„•0 ↦ inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘˜) Β· 𝑛) ∈ β„€}, ℝ, < ))    &   π‘… = (seq0( Β· , 𝑁)β€˜(degβ€˜πΉ))    β‡’   ((πœ‘ ∧ 𝐾 ∈ β„•0) β†’ ((π‘β€˜πΎ) ∈ β„• ∧ ((π΅β€˜πΎ) Β· (π‘β€˜πΎ)) ∈ β„€))
 
Theoremelqaalem2 25602* Lemma for elqaa 25604. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐹 ∈ ((Polyβ€˜β„š) βˆ– {0𝑝}))    &   (πœ‘ β†’ (πΉβ€˜π΄) = 0)    &   π΅ = (coeffβ€˜πΉ)    &   π‘ = (π‘˜ ∈ β„•0 ↦ inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘˜) Β· 𝑛) ∈ β„€}, ℝ, < ))    &   π‘… = (seq0( Β· , 𝑁)β€˜(degβ€˜πΉ))    &   π‘ƒ = (π‘₯ ∈ V, 𝑦 ∈ V ↦ ((π‘₯ Β· 𝑦) mod (π‘β€˜πΎ)))    β‡’   ((πœ‘ ∧ 𝐾 ∈ (0...(degβ€˜πΉ))) β†’ (𝑅 mod (π‘β€˜πΎ)) = 0)
 
Theoremelqaalem3 25603* Lemma for elqaa 25604. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐹 ∈ ((Polyβ€˜β„š) βˆ– {0𝑝}))    &   (πœ‘ β†’ (πΉβ€˜π΄) = 0)    &   π΅ = (coeffβ€˜πΉ)    &   π‘ = (π‘˜ ∈ β„•0 ↦ inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘˜) Β· 𝑛) ∈ β„€}, ℝ, < ))    &   π‘… = (seq0( Β· , 𝑁)β€˜(degβ€˜πΉ))    β‡’   (πœ‘ β†’ 𝐴 ∈ 𝔸)
 
Theoremelqaa 25604* The set of numbers generated by the roots of polynomials in the rational numbers is the same as the set of algebraic numbers, which by elaa 25598 are defined only in terms of polynomials over the integers. (Contributed by Mario Carneiro, 23-Jul-2014.) (Proof shortened by AV, 3-Oct-2020.)
(𝐴 ∈ 𝔸 ↔ (𝐴 ∈ β„‚ ∧ βˆƒπ‘“ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})(π‘“β€˜π΄) = 0))
 
Theoremqaa 25605 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ β„š β†’ 𝐴 ∈ 𝔸)
 
Theoremqssaa 25606 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
β„š βŠ† 𝔸
 
Theoremiaa 25607 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
i ∈ 𝔸
 
Theoremaareccl 25608 The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴 ∈ 𝔸 ∧ 𝐴 β‰  0) β†’ (1 / 𝐴) ∈ 𝔸)
 
Theoremaacjcl 25609 The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝐴 ∈ 𝔸 β†’ (βˆ—β€˜π΄) ∈ 𝔸)
 
Theoremaannenlem1 25610* Lemma for aannen 25613. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐻 = (π‘Ž ∈ β„•0 ↦ {𝑏 ∈ β„‚ ∣ βˆƒπ‘ ∈ {𝑑 ∈ (Polyβ€˜β„€) ∣ (𝑑 β‰  0𝑝 ∧ (degβ€˜π‘‘) ≀ π‘Ž ∧ βˆ€π‘’ ∈ β„•0 (absβ€˜((coeffβ€˜π‘‘)β€˜π‘’)) ≀ π‘Ž)} (π‘β€˜π‘) = 0})    β‡’   (𝐴 ∈ β„•0 β†’ (π»β€˜π΄) ∈ Fin)
 
Theoremaannenlem2 25611* Lemma for aannen 25613. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐻 = (π‘Ž ∈ β„•0 ↦ {𝑏 ∈ β„‚ ∣ βˆƒπ‘ ∈ {𝑑 ∈ (Polyβ€˜β„€) ∣ (𝑑 β‰  0𝑝 ∧ (degβ€˜π‘‘) ≀ π‘Ž ∧ βˆ€π‘’ ∈ β„•0 (absβ€˜((coeffβ€˜π‘‘)β€˜π‘’)) ≀ π‘Ž)} (π‘β€˜π‘) = 0})    β‡’   π”Έ = βˆͺ ran 𝐻
 
Theoremaannenlem3 25612* The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐻 = (π‘Ž ∈ β„•0 ↦ {𝑏 ∈ β„‚ ∣ βˆƒπ‘ ∈ {𝑑 ∈ (Polyβ€˜β„€) ∣ (𝑑 β‰  0𝑝 ∧ (degβ€˜π‘‘) ≀ π‘Ž ∧ βˆ€π‘’ ∈ β„•0 (absβ€˜((coeffβ€˜π‘‘)β€˜π‘’)) ≀ π‘Ž)} (π‘β€˜π‘) = 0})    β‡’   π”Έ β‰ˆ β„•
 
Theoremaannen 25613 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝔸 β‰ˆ β„•
 
14.1.6  Liouville's approximation theorem
 
Theoremaalioulem1 25614 Lemma for aaliou 25620. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
(πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„€))    &   (πœ‘ β†’ 𝑋 ∈ β„€)    &   (πœ‘ β†’ π‘Œ ∈ β„•)    β‡’   (πœ‘ β†’ ((πΉβ€˜(𝑋 / π‘Œ)) Β· (π‘Œβ†‘(degβ€˜πΉ))) ∈ β„€)
 
Theoremaalioulem2 25615* Lemma for aaliou 25620. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Proof shortened by AV, 28-Sep-2020.)
𝑁 = (degβ€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„€))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘ ∈ β„€ βˆ€π‘ž ∈ β„• ((πΉβ€˜(𝑝 / π‘ž)) = 0 β†’ (𝐴 = (𝑝 / π‘ž) ∨ (π‘₯ / (π‘žβ†‘π‘)) ≀ (absβ€˜(𝐴 βˆ’ (𝑝 / π‘ž))))))
 
Theoremaalioulem3 25616* Lemma for aaliou 25620. (Contributed by Stefan O'Rear, 15-Nov-2014.)
𝑁 = (degβ€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„€))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (πΉβ€˜π΄) = 0)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘Ÿ ∈ ℝ ((absβ€˜(𝐴 βˆ’ π‘Ÿ)) ≀ 1 β†’ (π‘₯ Β· (absβ€˜(πΉβ€˜π‘Ÿ))) ≀ (absβ€˜(𝐴 βˆ’ π‘Ÿ))))
 
Theoremaalioulem4 25617* Lemma for aaliou 25620. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (degβ€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„€))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (πΉβ€˜π΄) = 0)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘ ∈ β„€ βˆ€π‘ž ∈ β„• (((πΉβ€˜(𝑝 / π‘ž)) β‰  0 ∧ (absβ€˜(𝐴 βˆ’ (𝑝 / π‘ž))) ≀ 1) β†’ (𝐴 = (𝑝 / π‘ž) ∨ (π‘₯ / (π‘žβ†‘π‘)) ≀ (absβ€˜(𝐴 βˆ’ (𝑝 / π‘ž))))))
 
Theoremaalioulem5 25618* Lemma for aaliou 25620. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (degβ€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„€))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (πΉβ€˜π΄) = 0)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘ ∈ β„€ βˆ€π‘ž ∈ β„• ((πΉβ€˜(𝑝 / π‘ž)) β‰  0 β†’ (𝐴 = (𝑝 / π‘ž) ∨ (π‘₯ / (π‘žβ†‘π‘)) ≀ (absβ€˜(𝐴 βˆ’ (𝑝 / π‘ž))))))
 
Theoremaalioulem6 25619* Lemma for aaliou 25620. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (degβ€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„€))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (πΉβ€˜π΄) = 0)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘ ∈ β„€ βˆ€π‘ž ∈ β„• (𝐴 = (𝑝 / π‘ž) ∨ (π‘₯ / (π‘žβ†‘π‘)) ≀ (absβ€˜(𝐴 βˆ’ (𝑝 / π‘ž)))))
 
Theoremaaliou 25620* Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial 𝐹 in integer coefficients, is not approximable beyond order 𝑁 = deg(𝐹) by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. This is Metamath 100 proof #18. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (degβ€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„€))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (πΉβ€˜π΄) = 0)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘ ∈ β„€ βˆ€π‘ž ∈ β„• (𝐴 = (𝑝 / π‘ž) ∨ (π‘₯ / (π‘žβ†‘π‘)) < (absβ€˜(𝐴 βˆ’ (𝑝 / π‘ž)))))
 
Theoremgeolim3 25621* Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π΅) < 1)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   πΉ = (π‘˜ ∈ (β„€β‰₯β€˜π΄) ↦ (𝐢 Β· (𝐡↑(π‘˜ βˆ’ 𝐴))))    β‡’   (πœ‘ β†’ seq𝐴( + , 𝐹) ⇝ (𝐢 / (1 βˆ’ 𝐡)))
 
Theoremaaliou2 25622* Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐴 ∈ (𝔸 ∩ ℝ) β†’ βˆƒπ‘˜ ∈ β„• βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘ ∈ β„€ βˆ€π‘ž ∈ β„• (𝐴 = (𝑝 / π‘ž) ∨ (π‘₯ / (π‘žβ†‘π‘˜)) < (absβ€˜(𝐴 βˆ’ (𝑝 / π‘ž)))))
 
Theoremaaliou2b 25623* Liouville's approximation theorem extended to complex 𝐴. (Contributed by Stefan O'Rear, 20-Nov-2014.)
(𝐴 ∈ 𝔸 β†’ βˆƒπ‘˜ ∈ β„• βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘ ∈ β„€ βˆ€π‘ž ∈ β„• (𝐴 = (𝑝 / π‘ž) ∨ (π‘₯ / (π‘žβ†‘π‘˜)) < (absβ€˜(𝐴 βˆ’ (𝑝 / π‘ž)))))
 
Theoremaaliou3lem1 25624* Lemma for aaliou3 25633. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (β„€β‰₯β€˜π΄) ↦ ((2↑-(!β€˜π΄)) Β· ((1 / 2)↑(𝑐 βˆ’ 𝐴))))    β‡’   ((𝐴 ∈ β„• ∧ 𝐡 ∈ (β„€β‰₯β€˜π΄)) β†’ (πΊβ€˜π΅) ∈ ℝ)
 
Theoremaaliou3lem2 25625* Lemma for aaliou3 25633. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (β„€β‰₯β€˜π΄) ↦ ((2↑-(!β€˜π΄)) Β· ((1 / 2)↑(𝑐 βˆ’ 𝐴))))    &   πΉ = (π‘Ž ∈ β„• ↦ (2↑-(!β€˜π‘Ž)))    β‡’   ((𝐴 ∈ β„• ∧ 𝐡 ∈ (β„€β‰₯β€˜π΄)) β†’ (πΉβ€˜π΅) ∈ (0(,](πΊβ€˜π΅)))
 
Theoremaaliou3lem3 25626* Lemma for aaliou3 25633. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (β„€β‰₯β€˜π΄) ↦ ((2↑-(!β€˜π΄)) Β· ((1 / 2)↑(𝑐 βˆ’ 𝐴))))    &   πΉ = (π‘Ž ∈ β„• ↦ (2↑-(!β€˜π‘Ž)))    β‡’   (𝐴 ∈ β„• β†’ (seq𝐴( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (β„€β‰₯β€˜π΄)(πΉβ€˜π‘) ∈ ℝ+ ∧ Σ𝑏 ∈ (β„€β‰₯β€˜π΄)(πΉβ€˜π‘) ≀ (2 Β· (2↑-(!β€˜π΄)))))
 
Theoremaaliou3lem8 25627* Lemma for aaliou3 25633. (Contributed by Stefan O'Rear, 20-Nov-2014.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ ℝ+) β†’ βˆƒπ‘₯ ∈ β„• (2 Β· (2↑-(!β€˜(π‘₯ + 1)))) ≀ (𝐡 / ((2↑(!β€˜π‘₯))↑𝐴)))
 
Theoremaaliou3lem4 25628* Lemma for aaliou3 25633. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (π‘Ž ∈ β„• ↦ (2↑-(!β€˜π‘Ž)))    &   πΏ = Σ𝑏 ∈ β„• (πΉβ€˜π‘)    &   π» = (𝑐 ∈ β„• ↦ Σ𝑏 ∈ (1...𝑐)(πΉβ€˜π‘))    β‡’   πΏ ∈ ℝ
 
Theoremaaliou3lem5 25629* Lemma for aaliou3 25633. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (π‘Ž ∈ β„• ↦ (2↑-(!β€˜π‘Ž)))    &   πΏ = Σ𝑏 ∈ β„• (πΉβ€˜π‘)    &   π» = (𝑐 ∈ β„• ↦ Σ𝑏 ∈ (1...𝑐)(πΉβ€˜π‘))    β‡’   (𝐴 ∈ β„• β†’ (π»β€˜π΄) ∈ ℝ)
 
Theoremaaliou3lem6 25630* Lemma for aaliou3 25633. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (π‘Ž ∈ β„• ↦ (2↑-(!β€˜π‘Ž)))    &   πΏ = Σ𝑏 ∈ β„• (πΉβ€˜π‘)    &   π» = (𝑐 ∈ β„• ↦ Σ𝑏 ∈ (1...𝑐)(πΉβ€˜π‘))    β‡’   (𝐴 ∈ β„• β†’ ((π»β€˜π΄) Β· (2↑(!β€˜π΄))) ∈ β„€)
 
Theoremaaliou3lem7 25631* Lemma for aaliou3 25633. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (π‘Ž ∈ β„• ↦ (2↑-(!β€˜π‘Ž)))    &   πΏ = Σ𝑏 ∈ β„• (πΉβ€˜π‘)    &   π» = (𝑐 ∈ β„• ↦ Σ𝑏 ∈ (1...𝑐)(πΉβ€˜π‘))    β‡’   (𝐴 ∈ β„• β†’ ((π»β€˜π΄) β‰  𝐿 ∧ (absβ€˜(𝐿 βˆ’ (π»β€˜π΄))) ≀ (2 Β· (2↑-(!β€˜(𝐴 + 1))))))
 
Theoremaaliou3lem9 25632* Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 20-Nov-2014.)
𝐹 = (π‘Ž ∈ β„• ↦ (2↑-(!β€˜π‘Ž)))    &   πΏ = Σ𝑏 ∈ β„• (πΉβ€˜π‘)    &   π» = (𝑐 ∈ β„• ↦ Σ𝑏 ∈ (1...𝑐)(πΉβ€˜π‘))    β‡’    Β¬ 𝐿 ∈ 𝔸
 
Theoremaaliou3 25633 Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 23-Nov-2014.)
Ξ£π‘˜ ∈ β„• (2↑-(!β€˜π‘˜)) βˆ‰ 𝔸
 
14.2  Sequences and series
 
14.2.1  Taylor polynomials and Taylor's theorem
 
Syntaxctayl 25634 Taylor polynomial of a function.
class Tayl
 
Syntaxcana 25635 The class of analytic functions.
class Ana
 
Definitiondf-tayl 25636* Define the Taylor polynomial or Taylor series of a function. TODO-AV: 𝑛 ∈ (β„•0 βˆͺ {+∞}) should be replaced by 𝑛 ∈ β„•0*. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl = (𝑠 ∈ {ℝ, β„‚}, 𝑓 ∈ (β„‚ ↑pm 𝑠) ↦ (𝑛 ∈ (β„•0 βˆͺ {+∞}), π‘Ž ∈ ∩ π‘˜ ∈ ((0[,]𝑛) ∩ β„€)dom ((𝑠 D𝑛 𝑓)β€˜π‘˜) ↦ βˆͺ π‘₯ ∈ β„‚ ({π‘₯} Γ— (β„‚fld tsums (π‘˜ ∈ ((0[,]𝑛) ∩ β„€) ↦ (((((𝑠 D𝑛 𝑓)β€˜π‘˜)β€˜π‘Ž) / (!β€˜π‘˜)) Β· ((π‘₯ βˆ’ π‘Ž)β†‘π‘˜)))))))
 
Definitiondf-ana 25637* Define the set of analytic functions, which are functions such that the Taylor series of the function at each point converges to the function in some neighborhood of the point. (Contributed by Mario Carneiro, 31-Dec-2016.)
Ana = (𝑠 ∈ {ℝ, β„‚} ↦ {𝑓 ∈ (β„‚ ↑pm 𝑠) ∣ βˆ€π‘₯ ∈ dom 𝑓 π‘₯ ∈ ((intβ€˜((TopOpenβ€˜β„‚fld) β†Ύt 𝑠))β€˜dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)π‘₯)))})
 
Theoremtaylfvallem1 25638* Lemma for taylfval 25640. (Contributed by Mario Carneiro, 30-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ (𝑁 ∈ β„•0 ∨ 𝑁 = +∞))    &   ((πœ‘ ∧ π‘˜ ∈ ((0[,]𝑁) ∩ β„€)) β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘˜))    β‡’   (((πœ‘ ∧ 𝑋 ∈ β„‚) ∧ π‘˜ ∈ ((0[,]𝑁) ∩ β„€)) β†’ (((((𝑆 D𝑛 𝐹)β€˜π‘˜)β€˜π΅) / (!β€˜π‘˜)) Β· ((𝑋 βˆ’ 𝐡)β†‘π‘˜)) ∈ β„‚)
 
Theoremtaylfvallem 25639* Lemma for taylfval 25640. (Contributed by Mario Carneiro, 30-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ (𝑁 ∈ β„•0 ∨ 𝑁 = +∞))    &   ((πœ‘ ∧ π‘˜ ∈ ((0[,]𝑁) ∩ β„€)) β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘˜))    β‡’   ((πœ‘ ∧ 𝑋 ∈ β„‚) β†’ (β„‚fld tsums (π‘˜ ∈ ((0[,]𝑁) ∩ β„€) ↦ (((((𝑆 D𝑛 𝐹)β€˜π‘˜)β€˜π΅) / (!β€˜π‘˜)) Β· ((𝑋 βˆ’ 𝐡)β†‘π‘˜)))) βŠ† β„‚)
 
Theoremtaylfval 25640* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 𝑆 is the base set with respect to evaluate the derivatives (generally ℝ or β„‚), 𝐹 is the function we are approximating, at point 𝐡, to order 𝑁. The result is a polynomial function of π‘₯.

This "extended" version of taylpfval 25646 additionally handles the case 𝑁 = +∞, in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)

(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ (𝑁 ∈ β„•0 ∨ 𝑁 = +∞))    &   ((πœ‘ ∧ π‘˜ ∈ ((0[,]𝑁) ∩ β„€)) β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘˜))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    β‡’   (πœ‘ β†’ 𝑇 = βˆͺ π‘₯ ∈ β„‚ ({π‘₯} Γ— (β„‚fld tsums (π‘˜ ∈ ((0[,]𝑁) ∩ β„€) ↦ (((((𝑆 D𝑛 𝐹)β€˜π‘˜)β€˜π΅) / (!β€˜π‘˜)) Β· ((π‘₯ βˆ’ 𝐡)β†‘π‘˜))))))
 
Theoremeltayl 25641* Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ (𝑁 ∈ β„•0 ∨ 𝑁 = +∞))    &   ((πœ‘ ∧ π‘˜ ∈ ((0[,]𝑁) ∩ β„€)) β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘˜))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    β‡’   (πœ‘ β†’ (π‘‹π‘‡π‘Œ ↔ (𝑋 ∈ β„‚ ∧ π‘Œ ∈ (β„‚fld tsums (π‘˜ ∈ ((0[,]𝑁) ∩ β„€) ↦ (((((𝑆 D𝑛 𝐹)β€˜π‘˜)β€˜π΅) / (!β€˜π‘˜)) Β· ((𝑋 βˆ’ 𝐡)β†‘π‘˜)))))))
 
Theoremtaylf 25642* The Taylor series defines a function on a subset of the complex numbers. (Contributed by Mario Carneiro, 30-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ (𝑁 ∈ β„•0 ∨ 𝑁 = +∞))    &   ((πœ‘ ∧ π‘˜ ∈ ((0[,]𝑁) ∩ β„€)) β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘˜))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    β‡’   (πœ‘ β†’ 𝑇:dom π‘‡βŸΆβ„‚)
 
Theoremtayl0 25643* The Taylor series is always defined at the basepoint, with value equal to the value of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ (𝑁 ∈ β„•0 ∨ 𝑁 = +∞))    &   ((πœ‘ ∧ π‘˜ ∈ ((0[,]𝑁) ∩ β„€)) β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘˜))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    β‡’   (πœ‘ β†’ (𝐡 ∈ dom 𝑇 ∧ (π‘‡β€˜π΅) = (πΉβ€˜π΅)))
 
Theoremtaylplem1 25644* Lemma for taylpfval 25646 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘))    β‡’   ((πœ‘ ∧ π‘˜ ∈ ((0[,]𝑁) ∩ β„€)) β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘˜))
 
Theoremtaylplem2 25645* Lemma for taylpfval 25646 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘))    β‡’   (((πœ‘ ∧ 𝑋 ∈ β„‚) ∧ π‘˜ ∈ (0...𝑁)) β†’ (((((𝑆 D𝑛 𝐹)β€˜π‘˜)β€˜π΅) / (!β€˜π‘˜)) Β· ((𝑋 βˆ’ 𝐡)β†‘π‘˜)) ∈ β„‚)
 
Theoremtaylpfval 25646* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 𝑆 is the base set with respect to evaluate the derivatives (generally ℝ or β„‚), 𝐹 is the function we are approximating, at point 𝐡, to order 𝑁. The result is a polynomial function of π‘₯. (Contributed by Mario Carneiro, 31-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    β‡’   (πœ‘ β†’ 𝑇 = (π‘₯ ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)β€˜π‘˜)β€˜π΅) / (!β€˜π‘˜)) Β· ((π‘₯ βˆ’ 𝐡)β†‘π‘˜))))
 
Theoremtaylpf 25647 The Taylor polynomial is a function on the complex numbers (even if the base set of the original function is the reals). (Contributed by Mario Carneiro, 31-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    β‡’   (πœ‘ β†’ 𝑇:β„‚βŸΆβ„‚)
 
Theoremtaylpval 25648* Value of the Taylor polynomial. (Contributed by Mario Carneiro, 31-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    β‡’   (πœ‘ β†’ (π‘‡β€˜π‘‹) = Ξ£π‘˜ ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)β€˜π‘˜)β€˜π΅) / (!β€˜π‘˜)) Β· ((𝑋 βˆ’ 𝐡)β†‘π‘˜)))
 
Theoremtaylply2 25649* The Taylor polynomial is a polynomial of degree (at most) 𝑁. This version of taylply 25650 shows that the coefficients of 𝑇 are in a subring of the complex numbers. (Contributed by Mario Carneiro, 1-Jan-2017.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    &   (πœ‘ β†’ 𝐷 ∈ (SubRingβ€˜β„‚fld))    &   (πœ‘ β†’ 𝐡 ∈ 𝐷)    &   ((πœ‘ ∧ π‘˜ ∈ (0...𝑁)) β†’ ((((𝑆 D𝑛 𝐹)β€˜π‘˜)β€˜π΅) / (!β€˜π‘˜)) ∈ 𝐷)    β‡’   (πœ‘ β†’ (𝑇 ∈ (Polyβ€˜π·) ∧ (degβ€˜π‘‡) ≀ 𝑁))
 
Theoremtaylply 25650 The Taylor polynomial is a polynomial of degree (at most) 𝑁. (Contributed by Mario Carneiro, 31-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    β‡’   (πœ‘ β†’ (𝑇 ∈ (Polyβ€˜β„‚) ∧ (degβ€˜π‘‡) ≀ 𝑁))
 
Theoremdvtaylp 25651 The derivative of the Taylor polynomial is the Taylor polynomial of the derivative of the function. (Contributed by Mario Carneiro, 31-Dec-2016.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜(𝑁 + 1)))    β‡’   (πœ‘ β†’ (β„‚ D ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐡)) = (𝑁(𝑆 Tayl (𝑆 D 𝐹))𝐡))
 
Theoremdvntaylp 25652 The 𝑀-th derivative of the Taylor polynomial is the Taylor polynomial of the 𝑀-th derivative of the function. (Contributed by Mario Carneiro, 1-Jan-2017.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜(𝑁 + 𝑀)))    β‡’   (πœ‘ β†’ ((β„‚ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐡))β€˜π‘€) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)β€˜π‘€))𝐡))
 
Theoremdvntaylp0 25653 The first 𝑁 derivatives of the Taylor polynomial at 𝐡 match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ 𝑀 ∈ (0...𝑁))    &   (πœ‘ β†’ 𝐡 ∈ dom ((𝑆 D𝑛 𝐹)β€˜π‘))    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    β‡’   (πœ‘ β†’ (((β„‚ D𝑛 𝑇)β€˜π‘€)β€˜π΅) = (((𝑆 D𝑛 𝐹)β€˜π‘€)β€˜π΅))
 
Theoremtaylthlem1 25654* Lemma for taylth 25656. This is the main part of Taylor's theorem, except for the induction step, which is supposed to be proven using L'HΓ΄pital's rule. However, since our proof of L'HΓ΄pital assumes that 𝑆 = ℝ, we can only do this part generically, and for taylth 25656 itself we must restrict to ℝ. (Contributed by Mario Carneiro, 1-Jan-2017.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   (πœ‘ β†’ dom ((𝑆 D𝑛 𝐹)β€˜π‘) = 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   π‘‡ = (𝑁(𝑆 Tayl 𝐹)𝐡)    &   π‘… = (π‘₯ ∈ (𝐴 βˆ– {𝐡}) ↦ (((πΉβ€˜π‘₯) βˆ’ (π‘‡β€˜π‘₯)) / ((π‘₯ βˆ’ 𝐡)↑𝑁)))    &   ((πœ‘ ∧ (𝑛 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 βˆ– {𝐡}) ↦ (((((𝑆 D𝑛 𝐹)β€˜(𝑁 βˆ’ 𝑛))β€˜π‘¦) βˆ’ (((β„‚ D𝑛 𝑇)β€˜(𝑁 βˆ’ 𝑛))β€˜π‘¦)) / ((𝑦 βˆ’ 𝐡)↑𝑛))) limβ„‚ 𝐡))) β†’ 0 ∈ ((π‘₯ ∈ (𝐴 βˆ– {𝐡}) ↦ (((((𝑆 D𝑛 𝐹)β€˜(𝑁 βˆ’ (𝑛 + 1)))β€˜π‘₯) βˆ’ (((β„‚ D𝑛 𝑇)β€˜(𝑁 βˆ’ (𝑛 + 1)))β€˜π‘₯)) / ((π‘₯ βˆ’ 𝐡)↑(𝑛 + 1)))) limβ„‚ 𝐡))    β‡’   (πœ‘ β†’ 0 ∈ (𝑅 limβ„‚ 𝐡))
 
Theoremtaylthlem2 25655* Lemma for taylth 25656. (Contributed by Mario Carneiro, 1-Jan-2017.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ dom ((ℝ D𝑛 𝐹)β€˜π‘) = 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   π‘‡ = (𝑁(ℝ Tayl 𝐹)𝐡)    &   (πœ‘ β†’ 𝑀 ∈ (1..^𝑁))    &   (πœ‘ β†’ 0 ∈ ((π‘₯ ∈ (𝐴 βˆ– {𝐡}) ↦ (((((ℝ D𝑛 𝐹)β€˜(𝑁 βˆ’ 𝑀))β€˜π‘₯) βˆ’ (((β„‚ D𝑛 𝑇)β€˜(𝑁 βˆ’ 𝑀))β€˜π‘₯)) / ((π‘₯ βˆ’ 𝐡)↑𝑀))) limβ„‚ 𝐡))    β‡’   (πœ‘ β†’ 0 ∈ ((π‘₯ ∈ (𝐴 βˆ– {𝐡}) ↦ (((((ℝ D𝑛 𝐹)β€˜(𝑁 βˆ’ (𝑀 + 1)))β€˜π‘₯) βˆ’ (((β„‚ D𝑛 𝑇)β€˜(𝑁 βˆ’ (𝑀 + 1)))β€˜π‘₯)) / ((π‘₯ βˆ’ 𝐡)↑(𝑀 + 1)))) limβ„‚ 𝐡))
 
Theoremtaylth 25656* Taylor's theorem. The Taylor polynomial of a 𝑁-times differentiable function is such that the error term goes to zero faster than (π‘₯ βˆ’ 𝐡)↑𝑁. This is Metamath 100 proof #35. (Contributed by Mario Carneiro, 1-Jan-2017.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ dom ((ℝ D𝑛 𝐹)β€˜π‘) = 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   π‘‡ = (𝑁(ℝ Tayl 𝐹)𝐡)    &   π‘… = (π‘₯ ∈ (𝐴 βˆ– {𝐡}) ↦ (((πΉβ€˜π‘₯) βˆ’ (π‘‡β€˜π‘₯)) / ((π‘₯ βˆ’ 𝐡)↑𝑁)))    β‡’   (πœ‘ β†’ 0 ∈ (𝑅 limβ„‚ 𝐡))
 
14.2.2  Uniform convergence
 
Syntaxculm 25657 Extend class notation to include the uniform convergence predicate.
class ⇝𝑒
 
Definitiondf-ulm 25658* Define the uniform convergence of a sequence of functions. Here 𝐹(β‡π‘’β€˜π‘†)𝐺 if 𝐹 is a sequence of functions 𝐹(𝑛), 𝑛 ∈ β„• defined on 𝑆 and 𝐺 is a function on 𝑆, and for every 0 < π‘₯ there is a 𝑗 such that the functions 𝐹(π‘˜) for 𝑗 ≀ π‘˜ are all uniformly within π‘₯ of 𝐺 on the domain 𝑆. Compare with df-clim 15305. (Contributed by Mario Carneiro, 26-Feb-2015.)
⇝𝑒 = (𝑠 ∈ V ↦ {βŸ¨π‘“, π‘¦βŸ© ∣ βˆƒπ‘› ∈ β„€ (𝑓:(β„€β‰₯β€˜π‘›)⟢(β„‚ ↑m 𝑠) ∧ 𝑦:π‘ βŸΆβ„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘›)βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘§ ∈ 𝑠 (absβ€˜(((π‘“β€˜π‘˜)β€˜π‘§) βˆ’ (π‘¦β€˜π‘§))) < π‘₯)})
 
Theoremulmrel 25659 The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
Rel (β‡π‘’β€˜π‘†)
 
Theoremulmscl 25660 Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(β‡π‘’β€˜π‘†)𝐺 β†’ 𝑆 ∈ V)
 
Theoremulmval 25661* Express the predicate: The sequence of functions 𝐹 converges uniformly to 𝐺 on 𝑆. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝑆 ∈ 𝑉 β†’ (𝐹(β‡π‘’β€˜π‘†)𝐺 ↔ βˆƒπ‘› ∈ β„€ (𝐹:(β„€β‰₯β€˜π‘›)⟢(β„‚ ↑m 𝑆) ∧ 𝐺:π‘†βŸΆβ„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘›)βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘§ ∈ 𝑆 (absβ€˜(((πΉβ€˜π‘˜)β€˜π‘§) βˆ’ (πΊβ€˜π‘§))) < π‘₯)))
 
Theoremulmcl 25662 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(β‡π‘’β€˜π‘†)𝐺 β†’ 𝐺:π‘†βŸΆβ„‚)
 
Theoremulmf 25663* Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(β‡π‘’β€˜π‘†)𝐺 β†’ βˆƒπ‘› ∈ β„€ 𝐹:(β„€β‰₯β€˜π‘›)⟢(β„‚ ↑m 𝑆))
 
Theoremulmpm 25664 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(β‡π‘’β€˜π‘†)𝐺 β†’ 𝐹 ∈ ((β„‚ ↑m 𝑆) ↑pm β„€))
 
Theoremulmf2 25665 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)
((𝐹 Fn 𝑍 ∧ 𝐹(β‡π‘’β€˜π‘†)𝐺) β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))
 
Theoremulm2 25666* Simplify ulmval 25661 when 𝐹 and 𝐺 are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   ((πœ‘ ∧ (π‘˜ ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) β†’ ((πΉβ€˜π‘˜)β€˜π‘§) = 𝐡)    &   ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ (πΊβ€˜π‘§) = 𝐴)    &   (πœ‘ β†’ 𝐺:π‘†βŸΆβ„‚)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐹(β‡π‘’β€˜π‘†)𝐺 ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘§ ∈ 𝑆 (absβ€˜(𝐡 βˆ’ 𝐴)) < π‘₯))
 
Theoremulmi 25667* The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   ((πœ‘ ∧ (π‘˜ ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) β†’ ((πΉβ€˜π‘˜)β€˜π‘§) = 𝐡)    &   ((πœ‘ ∧ 𝑧 ∈ 𝑆) β†’ (πΊβ€˜π‘§) = 𝐴)    &   (πœ‘ β†’ 𝐹(β‡π‘’β€˜π‘†)𝐺)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    β‡’   (πœ‘ β†’ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘§ ∈ 𝑆 (absβ€˜(𝐡 βˆ’ 𝐴)) < 𝐢)
 
Theoremulmclm 25668* A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐻 ∈ π‘Š)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((πΉβ€˜π‘˜)β€˜π΄) = (π»β€˜π‘˜))    &   (πœ‘ β†’ 𝐹(β‡π‘’β€˜π‘†)𝐺)    β‡’   (πœ‘ β†’ 𝐻 ⇝ (πΊβ€˜π΄))
 
Theoremulmres 25669 A sequence of functions converges iff the tail of the sequence converges (for any finite cutoff). (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   π‘Š = (β„€β‰₯β€˜π‘)    &   (πœ‘ β†’ 𝑁 ∈ 𝑍)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    β‡’   (πœ‘ β†’ (𝐹(β‡π‘’β€˜π‘†)𝐺 ↔ (𝐹 β†Ύ π‘Š)(β‡π‘’β€˜π‘†)𝐺))
 
Theoremulmshftlem 25670* Lemma for ulmshft 25671. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   π‘Š = (β„€β‰₯β€˜(𝑀 + 𝐾))    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   (πœ‘ β†’ 𝐻 = (𝑛 ∈ π‘Š ↦ (πΉβ€˜(𝑛 βˆ’ 𝐾))))    β‡’   (πœ‘ β†’ (𝐹(β‡π‘’β€˜π‘†)𝐺 β†’ 𝐻(β‡π‘’β€˜π‘†)𝐺))
 
Theoremulmshft 25671* A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   π‘Š = (β„€β‰₯β€˜(𝑀 + 𝐾))    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   (πœ‘ β†’ 𝐻 = (𝑛 ∈ π‘Š ↦ (πΉβ€˜(𝑛 βˆ’ 𝐾))))    β‡’   (πœ‘ β†’ (𝐹(β‡π‘’β€˜π‘†)𝐺 ↔ 𝐻(β‡π‘’β€˜π‘†)𝐺))
 
Theoremulm0 25672 Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   (πœ‘ β†’ 𝐺:π‘†βŸΆβ„‚)    β‡’   ((πœ‘ ∧ 𝑆 = βˆ…) β†’ 𝐹(β‡π‘’β€˜π‘†)𝐺)
 
Theoremulmuni 25673 A sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.)
((𝐹(β‡π‘’β€˜π‘†)𝐺 ∧ 𝐹(β‡π‘’β€˜π‘†)𝐻) β†’ 𝐺 = 𝐻)
 
Theoremulmdm 25674 Two ways to express that a function has a limit. (The expression ((β‡π‘’β€˜π‘†)β€˜πΉ) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 5-Jul-2017.)
(𝐹 ∈ dom (β‡π‘’β€˜π‘†) ↔ 𝐹(β‡π‘’β€˜π‘†)((β‡π‘’β€˜π‘†)β€˜πΉ))
 
Theoremulmcaulem 25675* Lemma for ulmcau 25676 and ulmcau2 25677: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 15175. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    β‡’   (πœ‘ β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘§ ∈ 𝑆 (absβ€˜(((πΉβ€˜π‘˜)β€˜π‘§) βˆ’ ((πΉβ€˜π‘—)β€˜π‘§))) < π‘₯ ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘š ∈ (β„€β‰₯β€˜π‘˜)βˆ€π‘§ ∈ 𝑆 (absβ€˜(((πΉβ€˜π‘˜)β€˜π‘§) βˆ’ ((πΉβ€˜π‘š)β€˜π‘§))) < π‘₯))
 
Theoremulmcau 25676* A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 0 < π‘₯ there is a 𝑗 such that for all 𝑗 ≀ π‘˜ the functions 𝐹(π‘˜) and 𝐹(𝑗) are uniformly within π‘₯ of each other on 𝑆. This is the four-quantifier version, see ulmcau2 25677 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    β‡’   (πœ‘ β†’ (𝐹 ∈ dom (β‡π‘’β€˜π‘†) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘§ ∈ 𝑆 (absβ€˜(((πΉβ€˜π‘˜)β€˜π‘§) βˆ’ ((πΉβ€˜π‘—)β€˜π‘§))) < π‘₯))
 
Theoremulmcau2 25677* A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 0 < π‘₯ there is a 𝑗 such that for all 𝑗 ≀ π‘˜, π‘š the functions 𝐹(π‘˜) and 𝐹(π‘š) are uniformly within π‘₯ of each other on 𝑆. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    β‡’   (πœ‘ β†’ (𝐹 ∈ dom (β‡π‘’β€˜π‘†) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘š ∈ (β„€β‰₯β€˜π‘˜)βˆ€π‘§ ∈ 𝑆 (absβ€˜(((πΉβ€˜π‘˜)β€˜π‘§) βˆ’ ((πΉβ€˜π‘š)β€˜π‘§))) < π‘₯))
 
Theoremulmss 25678* A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑇 βŠ† 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑍) β†’ 𝐴 ∈ π‘Š)    &   (πœ‘ β†’ (π‘₯ ∈ 𝑍 ↦ 𝐴)(β‡π‘’β€˜π‘†)𝐺)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑍 ↦ (𝐴 β†Ύ 𝑇))(β‡π‘’β€˜π‘‡)(𝐺 β†Ύ 𝑇))
 
Theoremulmbdd 25679* A uniform limit of bounded functions is bounded. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ 𝑆 (absβ€˜((πΉβ€˜π‘˜)β€˜π‘§)) ≀ π‘₯)    &   (πœ‘ β†’ 𝐹(β‡π‘’β€˜π‘†)𝐺)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ 𝑆 (absβ€˜(πΊβ€˜π‘§)) ≀ π‘₯)
 
Theoremulmcn 25680 A uniform limit of continuous functions is continuous. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(𝑆–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐹(β‡π‘’β€˜π‘†)𝐺)    β‡’   (πœ‘ β†’ 𝐺 ∈ (𝑆–cnβ†’β„‚))
 
Theoremulmdvlem1 25681* Lemma for ulmdv 25684. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑋))    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   ((πœ‘ ∧ 𝑧 ∈ 𝑋) β†’ (π‘˜ ∈ 𝑍 ↦ ((πΉβ€˜π‘˜)β€˜π‘§)) ⇝ (πΊβ€˜π‘§))    &   (πœ‘ β†’ (π‘˜ ∈ 𝑍 ↦ (𝑆 D (πΉβ€˜π‘˜)))(β‡π‘’β€˜π‘‹)𝐻)    &   ((πœ‘ ∧ πœ“) β†’ 𝐢 ∈ 𝑋)    &   ((πœ‘ ∧ πœ“) β†’ 𝑅 ∈ ℝ+)    &   ((πœ‘ ∧ πœ“) β†’ π‘ˆ ∈ ℝ+)    &   ((πœ‘ ∧ πœ“) β†’ π‘Š ∈ ℝ+)    &   ((πœ‘ ∧ πœ“) β†’ π‘ˆ < π‘Š)    &   ((πœ‘ ∧ πœ“) β†’ (𝐢(ballβ€˜((abs ∘ βˆ’ ) β†Ύ (𝑆 Γ— 𝑆)))π‘ˆ) βŠ† 𝑋)    &   ((πœ‘ ∧ πœ“) β†’ (absβ€˜(π‘Œ βˆ’ 𝐢)) < π‘ˆ)    &   ((πœ‘ ∧ πœ“) β†’ 𝑁 ∈ 𝑍)    &   ((πœ‘ ∧ πœ“) β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘₯ ∈ 𝑋 (absβ€˜(((𝑆 D (πΉβ€˜π‘))β€˜π‘₯) βˆ’ ((𝑆 D (πΉβ€˜π‘š))β€˜π‘₯))) < ((𝑅 / 2) / 2))    &   ((πœ‘ ∧ πœ“) β†’ (absβ€˜(((𝑆 D (πΉβ€˜π‘))β€˜πΆ) βˆ’ (π»β€˜πΆ))) < (𝑅 / 2))    &   ((πœ‘ ∧ πœ“) β†’ π‘Œ ∈ 𝑋)    &   ((πœ‘ ∧ πœ“) β†’ π‘Œ β‰  𝐢)    &   ((πœ‘ ∧ πœ“) β†’ ((absβ€˜(π‘Œ βˆ’ 𝐢)) < π‘Š β†’ (absβ€˜(((((πΉβ€˜π‘)β€˜π‘Œ) βˆ’ ((πΉβ€˜π‘)β€˜πΆ)) / (π‘Œ βˆ’ 𝐢)) βˆ’ ((𝑆 D (πΉβ€˜π‘))β€˜πΆ))) < ((𝑅 / 2) / 2)))    β‡’   ((πœ‘ ∧ πœ“) β†’ (absβ€˜((((πΊβ€˜π‘Œ) βˆ’ (πΊβ€˜πΆ)) / (π‘Œ βˆ’ 𝐢)) βˆ’ (π»β€˜πΆ))) < 𝑅)
 
Theoremulmdvlem2 25682* Lemma for ulmdv 25684. (Contributed by Mario Carneiro, 8-May-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑋))    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   ((πœ‘ ∧ 𝑧 ∈ 𝑋) β†’ (π‘˜ ∈ 𝑍 ↦ ((πΉβ€˜π‘˜)β€˜π‘§)) ⇝ (πΊβ€˜π‘§))    &   (πœ‘ β†’ (π‘˜ ∈ 𝑍 ↦ (𝑆 D (πΉβ€˜π‘˜)))(β‡π‘’β€˜π‘‹)𝐻)    β‡’   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ dom (𝑆 D (πΉβ€˜π‘˜)) = 𝑋)
 
Theoremulmdvlem3 25683* Lemma for ulmdv 25684. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑋))    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   ((πœ‘ ∧ 𝑧 ∈ 𝑋) β†’ (π‘˜ ∈ 𝑍 ↦ ((πΉβ€˜π‘˜)β€˜π‘§)) ⇝ (πΊβ€˜π‘§))    &   (πœ‘ β†’ (π‘˜ ∈ 𝑍 ↦ (𝑆 D (πΉβ€˜π‘˜)))(β‡π‘’β€˜π‘‹)𝐻)    β‡’   ((πœ‘ ∧ 𝑧 ∈ 𝑋) β†’ 𝑧(𝑆 D 𝐺)(π»β€˜π‘§))
 
Theoremulmdv 25684* If 𝐹 is a sequence of differentiable functions on 𝑋 which converge pointwise to 𝐺, and the derivatives of 𝐹(𝑛) converge uniformly to 𝐻, then 𝐺 is differentiable with derivative 𝐻. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑋))    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   ((πœ‘ ∧ 𝑧 ∈ 𝑋) β†’ (π‘˜ ∈ 𝑍 ↦ ((πΉβ€˜π‘˜)β€˜π‘§)) ⇝ (πΊβ€˜π‘§))    &   (πœ‘ β†’ (π‘˜ ∈ 𝑍 ↦ (𝑆 D (πΉβ€˜π‘˜)))(β‡π‘’β€˜π‘‹)𝐻)    β‡’   (πœ‘ β†’ (𝑆 D 𝐺) = 𝐻)
 
Theoremmtest 25685* The Weierstrass M-test. If 𝐹 is a sequence of functions which are uniformly bounded by the convergent sequence 𝑀(π‘˜), then the series generated by the sequence 𝐹 converges uniformly. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (π‘€β€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ (π‘˜ ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) β†’ (absβ€˜((πΉβ€˜π‘˜)β€˜π‘§)) ≀ (π‘€β€˜π‘˜))    &   (πœ‘ β†’ seq𝑁( + , 𝑀) ∈ dom ⇝ )    β‡’   (πœ‘ β†’ seq𝑁( ∘f + , 𝐹) ∈ dom (β‡π‘’β€˜π‘†))
 
Theoremmtestbdd 25686* Given the hypotheses of the Weierstrass M-test, the convergent function of the sequence is uniformly bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)
𝑍 = (β„€β‰₯β€˜π‘)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘βŸΆ(β„‚ ↑m 𝑆))    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (π‘€β€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ (π‘˜ ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) β†’ (absβ€˜((πΉβ€˜π‘˜)β€˜π‘§)) ≀ (π‘€β€˜π‘˜))    &   (πœ‘ β†’ seq𝑁( + , 𝑀) ∈ dom ⇝ )    &   (πœ‘ β†’ seq𝑁( ∘f + , 𝐹)(β‡π‘’β€˜π‘†)𝑇)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ 𝑆 (absβ€˜(π‘‡β€˜π‘§)) ≀ π‘₯)
 
Theoremmbfulm 25687 A uniform limit of measurable functions is measurable. (This is just a corollary of the fact that a pointwise limit of measurable functions is measurable, see mbflim 24954.) (Contributed by Mario Carneiro, 18-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆMblFn)    &   (πœ‘ β†’ 𝐹(β‡π‘’β€˜π‘†)𝐺)    β‡’   (πœ‘ β†’ 𝐺 ∈ MblFn)
 
Theoremiblulm 25688 A uniform limit of integrable functions is integrable. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆπΏ1)    &   (πœ‘ β†’ 𝐹(β‡π‘’β€˜π‘†)𝐺)    &   (πœ‘ β†’ (volβ€˜π‘†) ∈ ℝ)    β‡’   (πœ‘ β†’ 𝐺 ∈ 𝐿1)
 
Theoremitgulm 25689* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆπΏ1)    &   (πœ‘ β†’ 𝐹(β‡π‘’β€˜π‘†)𝐺)    &   (πœ‘ β†’ (volβ€˜π‘†) ∈ ℝ)    β‡’   (πœ‘ β†’ (π‘˜ ∈ 𝑍 ↦ βˆ«π‘†((πΉβ€˜π‘˜)β€˜π‘₯) dπ‘₯) ⇝ βˆ«π‘†(πΊβ€˜π‘₯) dπ‘₯)
 
Theoremitgulm2 25690* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (π‘₯ ∈ 𝑆 ↦ 𝐴) ∈ 𝐿1)    &   (πœ‘ β†’ (π‘˜ ∈ 𝑍 ↦ (π‘₯ ∈ 𝑆 ↦ 𝐴))(β‡π‘’β€˜π‘†)(π‘₯ ∈ 𝑆 ↦ 𝐡))    &   (πœ‘ β†’ (volβ€˜π‘†) ∈ ℝ)    β‡’   (πœ‘ β†’ ((π‘₯ ∈ 𝑆 ↦ 𝐡) ∈ 𝐿1 ∧ (π‘˜ ∈ 𝑍 ↦ βˆ«π‘†π΄ dπ‘₯) ⇝ βˆ«π‘†π΅ dπ‘₯))
 
14.2.3  Power series
 
Theorempserval 25691* Value of the function 𝐺 that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    β‡’   (𝑋 ∈ β„‚ β†’ (πΊβ€˜π‘‹) = (π‘š ∈ β„•0 ↦ ((π΄β€˜π‘š) Β· (π‘‹β†‘π‘š))))
 
Theorempserval2 25692* Value of the function 𝐺 that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    β‡’   ((𝑋 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ ((πΊβ€˜π‘‹)β€˜π‘) = ((π΄β€˜π‘) Β· (𝑋↑𝑁)))
 
Theorempsergf 25693* The sequence of terms in the infinite sequence defining a power series for fixed 𝑋. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    β‡’   (πœ‘ β†’ (πΊβ€˜π‘‹):β„•0βŸΆβ„‚)
 
Theoremradcnvlem1 25694* Lemma for radcnvlt1 25699, radcnvle 25701. If 𝑋 is a point closer to zero than π‘Œ and the power series converges at π‘Œ, then it converges absolutely at 𝑋, even if the terms in the sequence are multiplied by 𝑛. (Contributed by Mario Carneiro, 31-Mar-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    &   (πœ‘ β†’ π‘Œ ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π‘‹) < (absβ€˜π‘Œ))    &   (πœ‘ β†’ seq0( + , (πΊβ€˜π‘Œ)) ∈ dom ⇝ )    &   π» = (π‘š ∈ β„•0 ↦ (π‘š Β· (absβ€˜((πΊβ€˜π‘‹)β€˜π‘š))))    β‡’   (πœ‘ β†’ seq0( + , 𝐻) ∈ dom ⇝ )
 
Theoremradcnvlem2 25695* Lemma for radcnvlt1 25699, radcnvle 25701. If 𝑋 is a point closer to zero than π‘Œ and the power series converges at π‘Œ, then it converges absolutely at 𝑋. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    &   (πœ‘ β†’ π‘Œ ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π‘‹) < (absβ€˜π‘Œ))    &   (πœ‘ β†’ seq0( + , (πΊβ€˜π‘Œ)) ∈ dom ⇝ )    β‡’   (πœ‘ β†’ seq0( + , (abs ∘ (πΊβ€˜π‘‹))) ∈ dom ⇝ )
 
Theoremradcnvlem3 25696* Lemma for radcnvlt1 25699, radcnvle 25701. If 𝑋 is a point closer to zero than π‘Œ and the power series converges at π‘Œ, then it converges at 𝑋. (Contributed by Mario Carneiro, 31-Mar-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    &   (πœ‘ β†’ π‘Œ ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π‘‹) < (absβ€˜π‘Œ))    &   (πœ‘ β†’ seq0( + , (πΊβ€˜π‘Œ)) ∈ dom ⇝ )    β‡’   (πœ‘ β†’ seq0( + , (πΊβ€˜π‘‹)) ∈ dom ⇝ )
 
Theoremradcnv0 25697* Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    β‡’   (πœ‘ β†’ 0 ∈ {π‘Ÿ ∈ ℝ ∣ seq0( + , (πΊβ€˜π‘Ÿ)) ∈ dom ⇝ })
 
Theoremradcnvcl 25698* The radius of convergence 𝑅 of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (πΊβ€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    β‡’   (πœ‘ β†’ 𝑅 ∈ (0[,]+∞))
 
Theoremradcnvlt1 25699* If 𝑋 is within the open disk of radius 𝑅 centered at zero, then the infinite series converges absolutely at 𝑋, and also converges when the series is multiplied by 𝑛. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (πΊβ€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π‘‹) < 𝑅)    &   π» = (π‘š ∈ β„•0 ↦ (π‘š Β· (absβ€˜((πΊβ€˜π‘‹)β€˜π‘š))))    β‡’   (πœ‘ β†’ (seq0( + , 𝐻) ∈ dom ⇝ ∧ seq0( + , (abs ∘ (πΊβ€˜π‘‹))) ∈ dom ⇝ ))
 
Theoremradcnvlt2 25700* If 𝑋 is within the open disk of radius 𝑅 centered at zero, then the infinite series converges at 𝑋. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (πΊβ€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π‘‹) < 𝑅)    β‡’   (πœ‘ β†’ seq0( + , (πΊβ€˜π‘‹)) ∈ dom ⇝ )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46966
  Copyright terms: Public domain < Previous  Next >