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Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtanval2 16101 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (i Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
 
Theoremtanval3 16102 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ β„‚ ∧ ((expβ€˜(2 Β· (i Β· 𝐴))) + 1) β‰  0) β†’ (tanβ€˜π΄) = (((expβ€˜(2 Β· (i Β· 𝐴))) βˆ’ 1) / (i Β· ((expβ€˜(2 Β· (i Β· 𝐴))) + 1))))
 
Theoremresinval 16103 The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ β†’ (sinβ€˜π΄) = (β„‘β€˜(expβ€˜(i Β· 𝐴))))
 
Theoremrecosval 16104 The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ β†’ (cosβ€˜π΄) = (β„œβ€˜(expβ€˜(i Β· 𝐴))))
 
Theoremefi4p 16105* 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.)
𝐹 = (𝑛 ∈ β„•0 ↦ (((i Β· 𝐴)↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) = (((1 βˆ’ ((𝐴↑2) / 2)) + (i Β· (𝐴 βˆ’ ((𝐴↑3) / 6)))) + Ξ£π‘˜ ∈ (β„€β‰₯β€˜4)(πΉβ€˜π‘˜)))
 
Theoremresin4p 16106* 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.)
𝐹 = (𝑛 ∈ β„•0 ↦ (((i Β· 𝐴)↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ ℝ β†’ (sinβ€˜π΄) = ((𝐴 βˆ’ ((𝐴↑3) / 6)) + (β„‘β€˜Ξ£π‘˜ ∈ (β„€β‰₯β€˜4)(πΉβ€˜π‘˜))))
 
Theoremrecos4p 16107* 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.)
𝐹 = (𝑛 ∈ β„•0 ↦ (((i Β· 𝐴)↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ ℝ β†’ (cosβ€˜π΄) = ((1 βˆ’ ((𝐴↑2) / 2)) + (β„œβ€˜Ξ£π‘˜ ∈ (β„€β‰₯β€˜4)(πΉβ€˜π‘˜))))
 
Theoremresincl 16108 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ β†’ (sinβ€˜π΄) ∈ ℝ)
 
Theoremrecoscl 16109 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ β†’ (cosβ€˜π΄) ∈ ℝ)
 
Theoremretancl 16110 The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
((𝐴 ∈ ℝ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜π΄) ∈ ℝ)
 
Theoremresincld 16111 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (sinβ€˜π΄) ∈ ℝ)
 
Theoremrecoscld 16112 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (cosβ€˜π΄) ∈ ℝ)
 
Theoremretancld 16113 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (cosβ€˜π΄) β‰  0)    β‡’   (πœ‘ β†’ (tanβ€˜π΄) ∈ ℝ)
 
Theoremsinneg 16114 The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ β„‚ β†’ (sinβ€˜-𝐴) = -(sinβ€˜π΄))
 
Theoremcosneg 16115 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ β„‚ β†’ (cosβ€˜-𝐴) = (cosβ€˜π΄))
 
Theoremtanneg 16116 The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
 
Theoremsin0 16117 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
(sinβ€˜0) = 0
 
Theoremcos0 16118 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
(cosβ€˜0) = 1
 
Theoremtan0 16119 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
(tanβ€˜0) = 0
 
Theoremefival 16120 The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) = ((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))))
 
Theoremefmival 16121 The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.)
(𝐴 ∈ β„‚ β†’ (expβ€˜(-i Β· 𝐴)) = ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))))
 
Theoremsinhval 16122 Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ β„‚ β†’ ((sinβ€˜(i Β· 𝐴)) / i) = (((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2))
 
Theoremcoshval 16123 Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ β„‚ β†’ (cosβ€˜(i Β· 𝐴)) = (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2))
 
Theoremresinhcl 16124 The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ β†’ ((sinβ€˜(i Β· 𝐴)) / i) ∈ ℝ)
 
Theoremrpcoshcl 16125 The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ β†’ (cosβ€˜(i Β· 𝐴)) ∈ ℝ+)
 
Theoremrecoshcl 16126 The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ β†’ (cosβ€˜(i Β· 𝐴)) ∈ ℝ)
 
Theoremretanhcl 16127 The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ β†’ ((tanβ€˜(i Β· 𝐴)) / i) ∈ ℝ)
 
Theoremtanhlt1 16128 The hyperbolic tangent of a real number is upper bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ β†’ ((tanβ€˜(i Β· 𝐴)) / i) < 1)
 
Theoremtanhbnd 16129 The hyperbolic tangent of a real number is bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.)
(𝐴 ∈ ℝ β†’ ((tanβ€˜(i Β· 𝐴)) / i) ∈ (-1(,)1))
 
Theoremefeul 16130 Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)
(𝐴 ∈ β„‚ β†’ (expβ€˜π΄) = ((expβ€˜(β„œβ€˜π΄)) Β· ((cosβ€˜(β„‘β€˜π΄)) + (i Β· (sinβ€˜(β„‘β€˜π΄))))))
 
Theoremefieq 16131 The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((expβ€˜(i Β· 𝐴)) = (expβ€˜(i Β· 𝐡)) ↔ ((cosβ€˜π΄) = (cosβ€˜π΅) ∧ (sinβ€˜π΄) = (sinβ€˜π΅))))
 
Theoremsinadd 16132 Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (sinβ€˜(𝐴 + 𝐡)) = (((sinβ€˜π΄) Β· (cosβ€˜π΅)) + ((cosβ€˜π΄) Β· (sinβ€˜π΅))))
 
Theoremcosadd 16133 Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (cosβ€˜(𝐴 + 𝐡)) = (((cosβ€˜π΄) Β· (cosβ€˜π΅)) βˆ’ ((sinβ€˜π΄) Β· (sinβ€˜π΅))))
 
Theoremtanaddlem 16134 A useful intermediate step in tanadd 16135 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ ((cosβ€˜π΄) β‰  0 ∧ (cosβ€˜π΅) β‰  0)) β†’ ((cosβ€˜(𝐴 + 𝐡)) β‰  0 ↔ ((tanβ€˜π΄) Β· (tanβ€˜π΅)) β‰  1))
 
Theoremtanadd 16135 Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ ((cosβ€˜π΄) β‰  0 ∧ (cosβ€˜π΅) β‰  0 ∧ (cosβ€˜(𝐴 + 𝐡)) β‰  0)) β†’ (tanβ€˜(𝐴 + 𝐡)) = (((tanβ€˜π΄) + (tanβ€˜π΅)) / (1 βˆ’ ((tanβ€˜π΄) Β· (tanβ€˜π΅)))))
 
Theoremsinsub 16136 Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (sinβ€˜(𝐴 βˆ’ 𝐡)) = (((sinβ€˜π΄) Β· (cosβ€˜π΅)) βˆ’ ((cosβ€˜π΄) Β· (sinβ€˜π΅))))
 
Theoremcossub 16137 Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (cosβ€˜(𝐴 βˆ’ 𝐡)) = (((cosβ€˜π΄) Β· (cosβ€˜π΅)) + ((sinβ€˜π΄) Β· (sinβ€˜π΅))))
 
Theoremaddsin 16138 Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((sinβ€˜π΄) + (sinβ€˜π΅)) = (2 Β· ((sinβ€˜((𝐴 + 𝐡) / 2)) Β· (cosβ€˜((𝐴 βˆ’ 𝐡) / 2)))))
 
Theoremsubsin 16139 Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((sinβ€˜π΄) βˆ’ (sinβ€˜π΅)) = (2 Β· ((cosβ€˜((𝐴 + 𝐡) / 2)) Β· (sinβ€˜((𝐴 βˆ’ 𝐡) / 2)))))
 
Theoremsinmul 16140 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 16133 and cossub 16137. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((sinβ€˜π΄) Β· (sinβ€˜π΅)) = (((cosβ€˜(𝐴 βˆ’ 𝐡)) βˆ’ (cosβ€˜(𝐴 + 𝐡))) / 2))
 
Theoremcosmul 16141 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 16133 and cossub 16137. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((cosβ€˜π΄) Β· (cosβ€˜π΅)) = (((cosβ€˜(𝐴 βˆ’ 𝐡)) + (cosβ€˜(𝐴 + 𝐡))) / 2))
 
Theoremaddcos 16142 Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((cosβ€˜π΄) + (cosβ€˜π΅)) = (2 Β· ((cosβ€˜((𝐴 + 𝐡) / 2)) Β· (cosβ€˜((𝐴 βˆ’ 𝐡) / 2)))))
 
Theoremsubcos 16143 Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((cosβ€˜π΅) βˆ’ (cosβ€˜π΄)) = (2 Β· ((sinβ€˜((𝐴 + 𝐡) / 2)) Β· (sinβ€˜((𝐴 βˆ’ 𝐡) / 2)))))
 
Theoremsincossq 16144 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.)
(𝐴 ∈ β„‚ β†’ (((sinβ€˜π΄)↑2) + ((cosβ€˜π΄)↑2)) = 1)
 
Theoremsin2t 16145 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
(𝐴 ∈ β„‚ β†’ (sinβ€˜(2 Β· 𝐴)) = (2 Β· ((sinβ€˜π΄) Β· (cosβ€˜π΄))))
 
Theoremcos2t 16146 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ β„‚ β†’ (cosβ€˜(2 Β· 𝐴)) = ((2 Β· ((cosβ€˜π΄)↑2)) βˆ’ 1))
 
Theoremcos2tsin 16147 Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)
(𝐴 ∈ β„‚ β†’ (cosβ€˜(2 Β· 𝐴)) = (1 βˆ’ (2 Β· ((sinβ€˜π΄)↑2))))
 
Theoremsinbnd 16148 The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
(𝐴 ∈ ℝ β†’ (-1 ≀ (sinβ€˜π΄) ∧ (sinβ€˜π΄) ≀ 1))
 
Theoremcosbnd 16149 The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
(𝐴 ∈ ℝ β†’ (-1 ≀ (cosβ€˜π΄) ∧ (cosβ€˜π΄) ≀ 1))
 
Theoremsinbnd2 16150 The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
(𝐴 ∈ ℝ β†’ (sinβ€˜π΄) ∈ (-1[,]1))
 
Theoremcosbnd2 16151 The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
(𝐴 ∈ ℝ β†’ (cosβ€˜π΄) ∈ (-1[,]1))
 
Theoremef01bndlem 16152* Lemma for sin01bnd 16153 and cos01bnd 16154. (Contributed by Paul Chapman, 19-Jan-2008.)
𝐹 = (𝑛 ∈ β„•0 ↦ (((i Β· 𝐴)↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ (0(,]1) β†’ (absβ€˜Ξ£π‘˜ ∈ (β„€β‰₯β€˜4)(πΉβ€˜π‘˜)) < ((𝐴↑4) / 6))
 
Theoremsin01bnd 16153 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.)
(𝐴 ∈ (0(,]1) β†’ ((𝐴 βˆ’ ((𝐴↑3) / 3)) < (sinβ€˜π΄) ∧ (sinβ€˜π΄) < 𝐴))
 
Theoremcos01bnd 16154 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.)
(𝐴 ∈ (0(,]1) β†’ ((1 βˆ’ (2 Β· ((𝐴↑2) / 3))) < (cosβ€˜π΄) ∧ (cosβ€˜π΄) < (1 βˆ’ ((𝐴↑2) / 3))))
 
Theoremcos1bnd 16155 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
((1 / 3) < (cosβ€˜1) ∧ (cosβ€˜1) < (2 / 3))
 
Theoremcos2bnd 16156 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(-(7 / 9) < (cosβ€˜2) ∧ (cosβ€˜2) < -(1 / 9))
 
Theoremsinltx 16157 The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ ℝ+ β†’ (sinβ€˜π΄) < 𝐴)
 
Theoremsin01gt0 16158 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.)
(𝐴 ∈ (0(,]1) β†’ 0 < (sinβ€˜π΄))
 
Theoremcos01gt0 16159 The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
(𝐴 ∈ (0(,]1) β†’ 0 < (cosβ€˜π΄))
 
Theoremsin02gt0 16160 The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
(𝐴 ∈ (0(,]2) β†’ 0 < (sinβ€˜π΄))
 
Theoremsincos1sgn 16161 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sinβ€˜1) ∧ 0 < (cosβ€˜1))
 
Theoremsincos2sgn 16162 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sinβ€˜2) ∧ (cosβ€˜2) < 0)
 
Theoremsin4lt0 16163 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
(sinβ€˜4) < 0
 
Theoremabsefi 16164 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.)
(𝐴 ∈ ℝ β†’ (absβ€˜(expβ€˜(i Β· 𝐴))) = 1)
 
Theoremabsef 16165 The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
(𝐴 ∈ β„‚ β†’ (absβ€˜(expβ€˜π΄)) = (expβ€˜(β„œβ€˜π΄)))
 
Theoremabsefib 16166 A complex number is real iff the exponential of its product with i has absolute value one. (Contributed by NM, 21-Aug-2008.)
(𝐴 ∈ β„‚ β†’ (𝐴 ∈ ℝ ↔ (absβ€˜(expβ€˜(i Β· 𝐴))) = 1))
 
Theoremefieq1re 16167 A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
((𝐴 ∈ β„‚ ∧ (expβ€˜(i Β· 𝐴)) = 1) β†’ 𝐴 ∈ ℝ)
 
Theoremdemoivre 16168 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 16169 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„€) β†’ (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄)))↑𝑁) = ((cosβ€˜(𝑁 Β· 𝐴)) + (i Β· (sinβ€˜(𝑁 Β· 𝐴)))))
 
TheoremdemoivreALT 16169 Alternate proof of demoivre 16168. 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.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄)))↑𝑁) = ((cosβ€˜(𝑁 Β· 𝐴)) + (i Β· (sinβ€˜(𝑁 Β· 𝐴)))))
 
5.11.1.1  The circle constant (tau = 2 pi)
 
Syntaxctau 16170 Extend class notation to include the constant tau, Ο„ = 6.28318....
class Ο„
 
Definitiondf-tau 16171 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 𝜏. 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.)
Ο„ = inf((ℝ+ ∩ (β—‘cos β€œ {1})), ℝ, < )
 
5.11.2  _e is irrational
 
Theoremeirrlem 16172* Lemma for eirr 16173. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
𝐹 = (𝑛 ∈ β„•0 ↦ (1 / (!β€˜π‘›)))    &   (πœ‘ β†’ 𝑃 ∈ β„€)    &   (πœ‘ β†’ 𝑄 ∈ β„•)    &   (πœ‘ β†’ e = (𝑃 / 𝑄))    β‡’    Β¬ πœ‘
 
Theoremeirr 16173 e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
e βˆ‰ β„š
 
Theoremegt2lt3 16174 Euler's constant e = 2.71828... is strictly bounded below by 2 and above by 3. (Contributed by NM, 28-Nov-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
(2 < e ∧ e < 3)
 
Theoremepos 16175 Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
0 < e
 
Theoremepr 16176 Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
e ∈ ℝ+
 
Theoremene0 16177 e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
e β‰  0
 
Theoremene1 16178 e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
e β‰  1
 
5.12  Cardinality of real and complex number subsets
 
5.12.1  Countability of integers and rationals
 
Theoremxpnnen 16179 The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
(β„• Γ— β„•) β‰ˆ β„•
 
Theoremznnen 16180 The set of integers and the set of positive integers are equinumerous. Exercise 1 of [Gleason] p. 140. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.)
β„€ β‰ˆ β„•
 
Theoremqnnen 16181 The rational numbers are countable. This proof does not use the Axiom of Choice, even though it uses an onto function, because the base set (β„€ Γ— β„•) is numerable. Exercise 2 of [Enderton] p. 133. For purposes of the Metamath 100 list, we are considering Mario Carneiro's revision as the date this proof was completed. This is Metamath 100 proof #3. (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 3-Mar-2013.)
β„š β‰ˆ β„•
 
5.12.2  The reals are uncountable
 
Theoremrpnnen2lem1 16182* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   ((𝐴 βŠ† β„• ∧ 𝑁 ∈ β„•) β†’ ((πΉβ€˜π΄)β€˜π‘) = if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0))
 
Theoremrpnnen2lem2 16183* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   (𝐴 βŠ† β„• β†’ (πΉβ€˜π΄):β„•βŸΆβ„)
 
Theoremrpnnen2lem3 16184* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   seq1( + , (πΉβ€˜β„•)) ⇝ (1 / 2)
 
Theoremrpnnen2lem4 16185* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   ((𝐴 βŠ† 𝐡 ∧ 𝐡 βŠ† β„• ∧ π‘˜ ∈ β„•) β†’ (0 ≀ ((πΉβ€˜π΄)β€˜π‘˜) ∧ ((πΉβ€˜π΄)β€˜π‘˜) ≀ ((πΉβ€˜π΅)β€˜π‘˜)))
 
Theoremrpnnen2lem5 16186* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   ((𝐴 βŠ† β„• ∧ 𝑀 ∈ β„•) β†’ seq𝑀( + , (πΉβ€˜π΄)) ∈ dom ⇝ )
 
Theoremrpnnen2lem6 16187* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   ((𝐴 βŠ† β„• ∧ 𝑀 ∈ β„•) β†’ Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘€)((πΉβ€˜π΄)β€˜π‘˜) ∈ ℝ)
 
Theoremrpnnen2lem7 16188* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   ((𝐴 βŠ† 𝐡 ∧ 𝐡 βŠ† β„• ∧ 𝑀 ∈ β„•) β†’ Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘€)((πΉβ€˜π΄)β€˜π‘˜) ≀ Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘€)((πΉβ€˜π΅)β€˜π‘˜))
 
Theoremrpnnen2lem8 16189* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   ((𝐴 βŠ† β„• ∧ 𝑀 ∈ β„•) β†’ Ξ£π‘˜ ∈ β„• ((πΉβ€˜π΄)β€˜π‘˜) = (Ξ£π‘˜ ∈ (1...(𝑀 βˆ’ 1))((πΉβ€˜π΄)β€˜π‘˜) + Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘€)((πΉβ€˜π΄)β€˜π‘˜)))
 
Theoremrpnnen2lem9 16190* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   (𝑀 ∈ β„• β†’ Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘€)((πΉβ€˜(β„• βˆ– {𝑀}))β€˜π‘˜) = (0 + (((1 / 3)↑(𝑀 + 1)) / (1 βˆ’ (1 / 3)))))
 
Theoremrpnnen2lem10 16191* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    &   (πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ 𝐡 βŠ† β„•)    &   (πœ‘ β†’ π‘š ∈ (𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• (𝑛 < π‘š β†’ (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐡)))    &   (πœ“ ↔ Ξ£π‘˜ ∈ β„• ((πΉβ€˜π΄)β€˜π‘˜) = Ξ£π‘˜ ∈ β„• ((πΉβ€˜π΅)β€˜π‘˜))    β‡’   ((πœ‘ ∧ πœ“) β†’ Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘š)((πΉβ€˜π΄)β€˜π‘˜) = Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘š)((πΉβ€˜π΅)β€˜π‘˜))
 
Theoremrpnnen2lem11 16192* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    &   (πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ 𝐡 βŠ† β„•)    &   (πœ‘ β†’ π‘š ∈ (𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• (𝑛 < π‘š β†’ (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐡)))    &   (πœ“ ↔ Ξ£π‘˜ ∈ β„• ((πΉβ€˜π΄)β€˜π‘˜) = Ξ£π‘˜ ∈ β„• ((πΉβ€˜π΅)β€˜π‘˜))    β‡’   (πœ‘ β†’ Β¬ πœ“)
 
Theoremrpnnen2lem12 16193* Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.)
𝐹 = (π‘₯ ∈ 𝒫 β„• ↦ (𝑛 ∈ β„• ↦ if(𝑛 ∈ π‘₯, ((1 / 3)↑𝑛), 0)))    β‡’   π’« β„• β‰Ό (0[,]1)
 
Theoremrpnnen2 16194 The other half of rpnnen 16195, where we show an injection from sets of positive integers to real numbers. The obvious choice for this is binary expansion, but it has the unfortunate property that it does not produce an injection on numbers which end with all 0's or all 1's (the more well-known decimal version of this is 0.999... 15851). Instead, we opt for a ternary expansion, which produces (a scaled version of) the Cantor set. Since the Cantor set is riddled with gaps, we can show that any two sequences that are not equal must differ somewhere, and when they do, they are placed a finite distance apart, thus ensuring that the map is injective.

Our map assigns to each subset 𝐴 of the positive integers the number Ξ£π‘˜ ∈ 𝐴(3↑-π‘˜) = Ξ£π‘˜ ∈ β„•((πΉβ€˜π΄)β€˜π‘˜), where ((πΉβ€˜π΄)β€˜π‘˜) = if(π‘˜ ∈ 𝐴, (3↑-π‘˜), 0)) (rpnnen2lem1 16182). This is an infinite sum of real numbers (rpnnen2lem2 16183), and since 𝐴 βŠ† 𝐡 implies (πΉβ€˜π΄) ≀ (πΉβ€˜π΅) (rpnnen2lem4 16185) and (πΉβ€˜β„•) converges to 1 / 2 (rpnnen2lem3 16184) by geoisum1 15849, the sum is convergent to some real (rpnnen2lem5 16186 and rpnnen2lem6 16187) by the comparison test for convergence cvgcmp 15786. The comparison test also tells us that 𝐴 βŠ† 𝐡 implies Ξ£(πΉβ€˜π΄) ≀ Ξ£(πΉβ€˜π΅) (rpnnen2lem7 16188).

Putting it all together, if we have two sets π‘₯ β‰  𝑦, there must differ somewhere, and so there must be an π‘š such that βˆ€π‘› < π‘š(𝑛 ∈ π‘₯ ↔ 𝑛 ∈ 𝑦) but π‘š ∈ (π‘₯ βˆ– 𝑦) or vice versa. In this case, we split off the first π‘š βˆ’ 1 terms (rpnnen2lem8 16189) and cancel them (rpnnen2lem10 16191), since these are the same for both sets. For the remaining terms, we use the subset property to establish that Ξ£(πΉβ€˜π‘¦) ≀ Ξ£(πΉβ€˜(β„• βˆ– {π‘š})) and Ξ£(πΉβ€˜{π‘š}) ≀ Ξ£(πΉβ€˜π‘₯) (where these sums are only over (β„€β‰₯β€˜π‘š)), and since Ξ£(πΉβ€˜(β„• βˆ– {π‘š})) = (3↑-π‘š) / 2 (rpnnen2lem9 16190) and Ξ£(πΉβ€˜{π‘š}) = (3↑-π‘š), we establish that Ξ£(πΉβ€˜π‘¦) < Ξ£(πΉβ€˜π‘₯) (rpnnen2lem11 16192) so that they must be different. By contraposition (rpnnen2lem12 16193), we find that this map is an injection. (Contributed by Mario Carneiro, 13-May-2013.) (Proof shortened by Mario Carneiro, 30-Apr-2014.) (Revised by NM, 17-Aug-2021.)

𝒫 β„• β‰Ό (0[,]1)
 
Theoremrpnnen 16195 The cardinality of the continuum is the same as the powerset of Ο‰. This is a stronger statement than ruc 16211, which only asserts that ℝ is uncountable, i.e. has a cardinality larger than Ο‰. The main proof is in two parts, rpnnen1 12989 and rpnnen2 16194, each showing an injection in one direction, and this last part uses sbth 9109 to prove that the sets are equinumerous. By constructing explicit injections, we avoid the use of AC. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
ℝ β‰ˆ 𝒫 β„•
 
Theoremrexpen 16196 The real numbers are equinumerous to their own Cartesian product, even though it is not necessarily true that ℝ is well-orderable (so we cannot use infxpidm2 10032 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)
(ℝ Γ— ℝ) β‰ˆ ℝ
 
Theoremcpnnen 16197 The complex numbers are equinumerous to the powerset of the positive integers. (Contributed by Mario Carneiro, 16-Jun-2013.)
β„‚ β‰ˆ 𝒫 β„•
 
TheoremrucALT 16198 Alternate proof of ruc 16211. This proof is a simple corollary of rpnnen 16195, which determines the exact cardinality of the reals. For an alternate proof discussed at mmcomplex.html#uncountable 16195, see ruc 16211. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
β„• β‰Ί ℝ
 
Theoremruclem1 16199* Lemma for ruc 16211 (the reals are uncountable). Substitutions for the function 𝐷. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)
(πœ‘ β†’ 𝐹:β„•βŸΆβ„)    &   (πœ‘ β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ ℝ)    &   π‘‹ = (1st β€˜(⟨𝐴, π΅βŸ©π·π‘€))    &   π‘Œ = (2nd β€˜(⟨𝐴, π΅βŸ©π·π‘€))    β‡’   (πœ‘ β†’ ((⟨𝐴, π΅βŸ©π·π‘€) ∈ (ℝ Γ— ℝ) ∧ 𝑋 = if(((𝐴 + 𝐡) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐡) / 2) + 𝐡) / 2)) ∧ π‘Œ = if(((𝐴 + 𝐡) / 2) < 𝑀, ((𝐴 + 𝐡) / 2), 𝐡)))
 
Theoremruclem2 16200* Lemma for ruc 16211. Ordering property for the input to 𝐷. (Contributed by Mario Carneiro, 28-May-2014.)
(πœ‘ β†’ 𝐹:β„•βŸΆβ„)    &   (πœ‘ β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ ℝ)    &   π‘‹ = (1st β€˜(⟨𝐴, π΅βŸ©π·π‘€))    &   π‘Œ = (2nd β€˜(⟨𝐴, π΅βŸ©π·π‘€))    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ (𝐴 ≀ 𝑋 ∧ 𝑋 < π‘Œ ∧ π‘Œ ≀ 𝐡))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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