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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | efsep 11701* | Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) & β’ π = (π + 1) & β’ π β β0 & β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (expβπ΄) = (π΅ + Ξ£π β (β€β₯βπ)(πΉβπ))) & β’ (π β (π΅ + ((π΄βπ) / (!βπ))) = π·) β β’ (π β (expβπ΄) = (π· + Ξ£π β (β€β₯βπ)(πΉβπ))) | ||
Theorem | effsumlt 11702* | The partial sums of the series expansion of the exponential function at a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) & β’ (π β π΄ β β+) & β’ (π β π β β0) β β’ (π β (seq0( + , πΉ)βπ) < (expβπ΄)) | ||
Theorem | eft0val 11703 | The value of the first term of the series expansion of the exponential function is 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ (π΄ β β β ((π΄β0) / (!β0)) = 1) | ||
Theorem | ef4p 11704* | Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) β β’ (π΄ β β β (expβπ΄) = ((((1 + π΄) + ((π΄β2) / 2)) + ((π΄β3) / 6)) + Ξ£π β (β€β₯β4)(πΉβπ))) | ||
Theorem | efgt1p2 11705 | The exponential of a positive real number is greater than the sum of the first three terms of the series expansion. (Contributed by Mario Carneiro, 15-Sep-2014.) |
β’ (π΄ β β+ β ((1 + π΄) + ((π΄β2) / 2)) < (expβπ΄)) | ||
Theorem | efgt1p 11706 | The exponential of a positive real number is greater than 1 plus that number. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β+ β (1 + π΄) < (expβπ΄)) | ||
Theorem | efgt1 11707 | The exponential of a positive real number is greater than 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β+ β 1 < (expβπ΄)) | ||
Theorem | efltim 11708 | The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ < π΅ β (expβπ΄) < (expβπ΅))) | ||
Theorem | reef11 11709 | The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.) |
β’ ((π΄ β β β§ π΅ β β) β ((expβπ΄) = (expβπ΅) β π΄ = π΅)) | ||
Theorem | reeff1 11710 | The exponential function maps real arguments one-to-one to positive reals. (Contributed by Steve Rodriguez, 25-Aug-2007.) (Revised by Mario Carneiro, 10-Nov-2013.) |
β’ (exp βΎ β):ββ1-1ββ+ | ||
Theorem | eflegeo 11711 | The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.) |
β’ (π β π΄ β β) & β’ (π β 0 β€ π΄) & β’ (π β π΄ < 1) β β’ (π β (expβπ΄) β€ (1 / (1 β π΄))) | ||
Theorem | sinval 11712 | Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.) |
β’ (π΄ β β β (sinβπ΄) = (((expβ(i Β· π΄)) β (expβ(-i Β· π΄))) / (2 Β· i))) | ||
Theorem | cosval 11713 | Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.) |
β’ (π΄ β β β (cosβπ΄) = (((expβ(i Β· π΄)) + (expβ(-i Β· π΄))) / 2)) | ||
Theorem | sinf 11714 | Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ sin:ββΆβ | ||
Theorem | cosf 11715 | Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ cos:ββΆβ | ||
Theorem | sincl 11716 | Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β β (sinβπ΄) β β) | ||
Theorem | coscl 11717 | Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β β (cosβπ΄) β β) | ||
Theorem | tanvalap 11718 | Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.) |
β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβπ΄) = ((sinβπ΄) / (cosβπ΄))) | ||
Theorem | tanclap 11719 | The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.) |
β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβπ΄) β β) | ||
Theorem | sincld 11720 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (sinβπ΄) β β) | ||
Theorem | coscld 11721 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (cosβπ΄) β β) | ||
Theorem | tanclapd 11722 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.) |
β’ (π β π΄ β β) & β’ (π β (cosβπ΄) # 0) β β’ (π β (tanβπ΄) β β) | ||
Theorem | tanval2ap 11723 | Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.) |
β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβπ΄) = (((expβ(i Β· π΄)) β (expβ(-i Β· π΄))) / (i Β· ((expβ(i Β· π΄)) + (expβ(-i Β· π΄)))))) | ||
Theorem | tanval3ap 11724 | Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.) |
β’ ((π΄ β β β§ ((expβ(2 Β· (i Β· π΄))) + 1) # 0) β (tanβπ΄) = (((expβ(2 Β· (i Β· π΄))) β 1) / (i Β· ((expβ(2 Β· (i Β· π΄))) + 1)))) | ||
Theorem | resinval 11725 | The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (sinβπ΄) = (ββ(expβ(i Β· π΄)))) | ||
Theorem | recosval 11726 | The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (cosβπ΄) = (ββ(expβ(i Β· π΄)))) | ||
Theorem | efi4p 11727* | Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ πΉ = (π β β0 β¦ (((i Β· π΄)βπ) / (!βπ))) β β’ (π΄ β β β (expβ(i Β· π΄)) = (((1 β ((π΄β2) / 2)) + (i Β· (π΄ β ((π΄β3) / 6)))) + Ξ£π β (β€β₯β4)(πΉβπ))) | ||
Theorem | resin4p 11728* | Separate out the first four terms of the infinite series expansion of the sine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ πΉ = (π β β0 β¦ (((i Β· π΄)βπ) / (!βπ))) β β’ (π΄ β β β (sinβπ΄) = ((π΄ β ((π΄β3) / 6)) + (ββΞ£π β (β€β₯β4)(πΉβπ)))) | ||
Theorem | recos4p 11729* | Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ πΉ = (π β β0 β¦ (((i Β· π΄)βπ) / (!βπ))) β β’ (π΄ β β β (cosβπ΄) = ((1 β ((π΄β2) / 2)) + (ββΞ£π β (β€β₯β4)(πΉβπ)))) | ||
Theorem | resincl 11730 | The sine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (sinβπ΄) β β) | ||
Theorem | recoscl 11731 | The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (cosβπ΄) β β) | ||
Theorem | retanclap 11732 | The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβπ΄) β β) | ||
Theorem | resincld 11733 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (sinβπ΄) β β) | ||
Theorem | recoscld 11734 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (cosβπ΄) β β) | ||
Theorem | retanclapd 11735 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) & β’ (π β (cosβπ΄) # 0) β β’ (π β (tanβπ΄) β β) | ||
Theorem | sinneg 11736 | The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (sinβ-π΄) = -(sinβπ΄)) | ||
Theorem | cosneg 11737 | The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (cosβ-π΄) = (cosβπ΄)) | ||
Theorem | tannegap 11738 | The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.) |
β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβ-π΄) = -(tanβπ΄)) | ||
Theorem | sin0 11739 | Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.) |
β’ (sinβ0) = 0 | ||
Theorem | cos0 11740 | Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.) |
β’ (cosβ0) = 1 | ||
Theorem | tan0 11741 | The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.) |
β’ (tanβ0) = 0 | ||
Theorem | efival 11742 | The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (expβ(i Β· π΄)) = ((cosβπ΄) + (i Β· (sinβπ΄)))) | ||
Theorem | efmival 11743 | The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.) |
β’ (π΄ β β β (expβ(-i Β· π΄)) = ((cosβπ΄) β (i Β· (sinβπ΄)))) | ||
Theorem | efeul 11744 | Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.) |
β’ (π΄ β β β (expβπ΄) = ((expβ(ββπ΄)) Β· ((cosβ(ββπ΄)) + (i Β· (sinβ(ββπ΄)))))) | ||
Theorem | efieq 11745 | The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β ((expβ(i Β· π΄)) = (expβ(i Β· π΅)) β ((cosβπ΄) = (cosβπ΅) β§ (sinβπ΄) = (sinβπ΅)))) | ||
Theorem | sinadd 11746 | Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ ((π΄ β β β§ π΅ β β) β (sinβ(π΄ + π΅)) = (((sinβπ΄) Β· (cosβπ΅)) + ((cosβπ΄) Β· (sinβπ΅)))) | ||
Theorem | cosadd 11747 | Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ ((π΄ β β β§ π΅ β β) β (cosβ(π΄ + π΅)) = (((cosβπ΄) Β· (cosβπ΅)) β ((sinβπ΄) Β· (sinβπ΅)))) | ||
Theorem | tanaddaplem 11748 | A useful intermediate step in tanaddap 11749 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.) |
β’ (((π΄ β β β§ π΅ β β) β§ ((cosβπ΄) # 0 β§ (cosβπ΅) # 0)) β ((cosβ(π΄ + π΅)) # 0 β ((tanβπ΄) Β· (tanβπ΅)) # 1)) | ||
Theorem | tanaddap 11749 | Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (((π΄ β β β§ π΅ β β) β§ ((cosβπ΄) # 0 β§ (cosβπ΅) # 0 β§ (cosβ(π΄ + π΅)) # 0)) β (tanβ(π΄ + π΅)) = (((tanβπ΄) + (tanβπ΅)) / (1 β ((tanβπ΄) Β· (tanβπ΅))))) | ||
Theorem | sinsub 11750 | Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β (sinβ(π΄ β π΅)) = (((sinβπ΄) Β· (cosβπ΅)) β ((cosβπ΄) Β· (sinβπ΅)))) | ||
Theorem | cossub 11751 | Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β (cosβ(π΄ β π΅)) = (((cosβπ΄) Β· (cosβπ΅)) + ((sinβπ΄) Β· (sinβπ΅)))) | ||
Theorem | addsin 11752 | Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β ((sinβπ΄) + (sinβπ΅)) = (2 Β· ((sinβ((π΄ + π΅) / 2)) Β· (cosβ((π΄ β π΅) / 2))))) | ||
Theorem | subsin 11753 | Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β ((sinβπ΄) β (sinβπ΅)) = (2 Β· ((cosβ((π΄ + π΅) / 2)) Β· (sinβ((π΄ β π΅) / 2))))) | ||
Theorem | sinmul 11754 | Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11747 and cossub 11751. (Contributed by David A. Wheeler, 26-May-2015.) |
β’ ((π΄ β β β§ π΅ β β) β ((sinβπ΄) Β· (sinβπ΅)) = (((cosβ(π΄ β π΅)) β (cosβ(π΄ + π΅))) / 2)) | ||
Theorem | cosmul 11755 | Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11747 and cossub 11751. (Contributed by David A. Wheeler, 26-May-2015.) |
β’ ((π΄ β β β§ π΅ β β) β ((cosβπ΄) Β· (cosβπ΅)) = (((cosβ(π΄ β π΅)) + (cosβ(π΄ + π΅))) / 2)) | ||
Theorem | addcos 11756 | Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β ((cosβπ΄) + (cosβπ΅)) = (2 Β· ((cosβ((π΄ + π΅) / 2)) Β· (cosβ((π΄ β π΅) / 2))))) | ||
Theorem | subcos 11757 | Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.) |
β’ ((π΄ β β β§ π΅ β β) β ((cosβπ΅) β (cosβπ΄)) = (2 Β· ((sinβ((π΄ + π΅) / 2)) Β· (sinβ((π΄ β π΅) / 2))))) | ||
Theorem | sincossq 11758 | Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.) |
β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) | ||
Theorem | sin2t 11759 | Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.) |
β’ (π΄ β β β (sinβ(2 Β· π΄)) = (2 Β· ((sinβπ΄) Β· (cosβπ΄)))) | ||
Theorem | cos2t 11760 | Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.) |
β’ (π΄ β β β (cosβ(2 Β· π΄)) = ((2 Β· ((cosβπ΄)β2)) β 1)) | ||
Theorem | cos2tsin 11761 | Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.) |
β’ (π΄ β β β (cosβ(2 Β· π΄)) = (1 β (2 Β· ((sinβπ΄)β2)))) | ||
Theorem | sinbnd 11762 | The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.) |
β’ (π΄ β β β (-1 β€ (sinβπ΄) β§ (sinβπ΄) β€ 1)) | ||
Theorem | cosbnd 11763 | The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.) |
β’ (π΄ β β β (-1 β€ (cosβπ΄) β§ (cosβπ΄) β€ 1)) | ||
Theorem | sinbnd2 11764 | The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.) |
β’ (π΄ β β β (sinβπ΄) β (-1[,]1)) | ||
Theorem | cosbnd2 11765 | The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.) |
β’ (π΄ β β β (cosβπ΄) β (-1[,]1)) | ||
Theorem | ef01bndlem 11766* | Lemma for sin01bnd 11767 and cos01bnd 11768. (Contributed by Paul Chapman, 19-Jan-2008.) |
β’ πΉ = (π β β0 β¦ (((i Β· π΄)βπ) / (!βπ))) β β’ (π΄ β (0(,]1) β (absβΞ£π β (β€β₯β4)(πΉβπ)) < ((π΄β4) / 6)) | ||
Theorem | sin01bnd 11767 | Bounds on the sine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β (0(,]1) β ((π΄ β ((π΄β3) / 3)) < (sinβπ΄) β§ (sinβπ΄) < π΄)) | ||
Theorem | cos01bnd 11768 | Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β (0(,]1) β ((1 β (2 Β· ((π΄β2) / 3))) < (cosβπ΄) β§ (cosβπ΄) < (1 β ((π΄β2) / 3)))) | ||
Theorem | cos1bnd 11769 | Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
β’ ((1 / 3) < (cosβ1) β§ (cosβ1) < (2 / 3)) | ||
Theorem | cos2bnd 11770 | Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
β’ (-(7 / 9) < (cosβ2) β§ (cosβ2) < -(1 / 9)) | ||
Theorem | sin01gt0 11771 | The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Wolf Lammen, 25-Sep-2020.) |
β’ (π΄ β (0(,]1) β 0 < (sinβπ΄)) | ||
Theorem | cos01gt0 11772 | The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) |
β’ (π΄ β (0(,]1) β 0 < (cosβπ΄)) | ||
Theorem | sin02gt0 11773 | The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) |
β’ (π΄ β (0(,]2) β 0 < (sinβπ΄)) | ||
Theorem | sincos1sgn 11774 | The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
β’ (0 < (sinβ1) β§ 0 < (cosβ1)) | ||
Theorem | sincos2sgn 11775 | The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
β’ (0 < (sinβ2) β§ (cosβ2) < 0) | ||
Theorem | sin4lt0 11776 | The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.) |
β’ (sinβ4) < 0 | ||
Theorem | cos12dec 11777 | Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.) |
β’ ((π΄ β (1[,]2) β§ π΅ β (1[,]2) β§ π΄ < π΅) β (cosβπ΅) < (cosβπ΄)) | ||
Theorem | absefi 11778 | The absolute value of the exponential of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.) |
β’ (π΄ β β β (absβ(expβ(i Β· π΄))) = 1) | ||
Theorem | absef 11779 | The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.) |
β’ (π΄ β β β (absβ(expβπ΄)) = (expβ(ββπ΄))) | ||
Theorem | absefib 11780 | A complex number is real iff the exponential of its product with i has absolute value one. (Contributed by NM, 21-Aug-2008.) |
β’ (π΄ β β β (π΄ β β β (absβ(expβ(i Β· π΄))) = 1)) | ||
Theorem | efieq1re 11781 | A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.) |
β’ ((π΄ β β β§ (expβ(i Β· π΄)) = 1) β π΄ β β) | ||
Theorem | demoivre 11782 | De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. See also demoivreALT 11783 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.) |
β’ ((π΄ β β β§ π β β€) β (((cosβπ΄) + (i Β· (sinβπ΄)))βπ) = ((cosβ(π Β· π΄)) + (i Β· (sinβ(π Β· π΄))))) | ||
Theorem | demoivreALT 11783 | Alternate proof of demoivre 11782. It is longer but does not use the exponential function. This is Metamath 100 proof #17. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π β β0) β (((cosβπ΄) + (i Β· (sinβπ΄)))βπ) = ((cosβ(π Β· π΄)) + (i Β· (sinβ(π Β· π΄))))) | ||
Syntax | ctau 11784 | Extend class notation to include the constant tau, Ο = 6.28318.... |
class Ο | ||
Definition | df-tau 11785 | Define the circle constant tau, Ο = 6.28318..., which is the smallest positive real number whose cosine is one. Various notations have been used or proposed for this number including Ο, a three-legged variant of Ο, or 2Ο. Note the difference between this constant Ο and the formula variable π. Following our convention, the constant is displayed in upright font while the variable is in italic font; furthermore, the colors are different. (Contributed by Jim Kingdon, 9-Apr-2018.) (Revised by AV, 1-Oct-2020.) |
β’ Ο = inf((β+ β© (β‘cos β {1})), β, < ) | ||
Theorem | eirraplem 11786* | Lemma for eirrap 11787. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.) |
β’ πΉ = (π β β0 β¦ (1 / (!βπ))) & β’ (π β π β β€) & β’ (π β π β β) β β’ (π β e # (π / π)) | ||
Theorem | eirrap 11787 | e is irrational. That is, for any rational number, e is apart from it. In the absence of excluded middle, we can distinguish between this and saying that e is not rational, which is eirr 11788. (Contributed by Jim Kingdon, 6-Jan-2023.) |
β’ (π β β β e # π) | ||
Theorem | eirr 11788 | e is not rational. In the absence of excluded middle, we can distinguish between this and saying that e is irrational in the sense of being apart from any rational number, which is eirrap 11787. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.) |
β’ e β β | ||
Theorem | egt2lt3 11789 | Euler's constant e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.) |
β’ (2 < e β§ e < 3) | ||
Theorem | epos 11790 | Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.) |
β’ 0 < e | ||
Theorem | epr 11791 | Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.) |
β’ e β β+ | ||
Theorem | ene0 11792 | e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.) |
β’ e β 0 | ||
Theorem | eap0 11793 | e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.) |
β’ e # 0 | ||
Theorem | ene1 11794 | e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.) |
β’ e β 1 | ||
Theorem | eap1 11795 | e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.) |
β’ e # 1 | ||
This part introduces elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory. | ||
Syntax | cdvds 11796 | Extend the definition of a class to include the divides relation. See df-dvds 11797. |
class β₯ | ||
Definition | df-dvds 11797* | Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
β’ β₯ = {β¨π₯, π¦β© β£ ((π₯ β β€ β§ π¦ β β€) β§ βπ β β€ (π Β· π₯) = π¦)} | ||
Theorem | divides 11798* | Define the divides relation. π β₯ π means π divides into π with no remainder. For example, 3 β₯ 6 (ex-dvds 14567). As proven in dvdsval3 11800, π β₯ π β (π mod π) = 0. See divides 11798 and dvdsval2 11799 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.) |
β’ ((π β β€ β§ π β β€) β (π β₯ π β βπ β β€ (π Β· π) = π)) | ||
Theorem | dvdsval2 11799 | One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.) |
β’ ((π β β€ β§ π β 0 β§ π β β€) β (π β₯ π β (π / π) β β€)) | ||
Theorem | dvdsval3 11800 | One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.) |
β’ ((π β β β§ π β β€) β (π β₯ π β (π mod π) = 0)) |
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