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Theorem List for Intuitionistic Logic Explorer - 11701-11800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeft0val 11701 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.)
(𝐴 ∈ β„‚ β†’ ((𝐴↑0) / (!β€˜0)) = 1)
 
Theoremef4p 11702* 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.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ β„‚ β†’ (expβ€˜π΄) = ((((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) + Ξ£π‘˜ ∈ (β„€β‰₯β€˜4)(πΉβ€˜π‘˜)))
 
Theoremefgt1p2 11703 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.)
(𝐴 ∈ ℝ+ β†’ ((1 + 𝐴) + ((𝐴↑2) / 2)) < (expβ€˜π΄))
 
Theoremefgt1p 11704 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.)
(𝐴 ∈ ℝ+ β†’ (1 + 𝐴) < (expβ€˜π΄))
 
Theoremefgt1 11705 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.)
(𝐴 ∈ ℝ+ β†’ 1 < (expβ€˜π΄))
 
Theoremefltim 11706 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 < 𝐡 β†’ (expβ€˜π΄) < (expβ€˜π΅)))
 
Theoremreef11 11707 The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((expβ€˜π΄) = (expβ€˜π΅) ↔ 𝐴 = 𝐡))
 
Theoremreeff1 11708 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.)
(exp β†Ύ ℝ):ℝ–1-1→ℝ+
 
Theoremeflegeo 11709 The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 0 ≀ 𝐴)    &   (πœ‘ β†’ 𝐴 < 1)    β‡’   (πœ‘ β†’ (expβ€˜π΄) ≀ (1 / (1 βˆ’ 𝐴)))
 
Theoremsinval 11710 Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (sinβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)))
 
Theoremcosval 11711 Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (cosβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2))
 
Theoremsinf 11712 Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
sin:β„‚βŸΆβ„‚
 
Theoremcosf 11713 Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
cos:β„‚βŸΆβ„‚
 
Theoremsincl 11714 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝐴 ∈ β„‚ β†’ (sinβ€˜π΄) ∈ β„‚)
 
Theoremcoscl 11715 Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝐴 ∈ β„‚ β†’ (cosβ€˜π΄) ∈ β„‚)
 
Theoremtanvalap 11716 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) = ((sinβ€˜π΄) / (cosβ€˜π΄)))
 
Theoremtanclap 11717 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.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) ∈ β„‚)
 
Theoremsincld 11718 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (sinβ€˜π΄) ∈ β„‚)
 
Theoremcoscld 11719 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (cosβ€˜π΄) ∈ β„‚)
 
Theoremtanclapd 11720 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ (cosβ€˜π΄) # 0)    β‡’   (πœ‘ β†’ (tanβ€˜π΄) ∈ β„‚)
 
Theoremtanval2ap 11721 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (i Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
 
Theoremtanval3ap 11722 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
((𝐴 ∈ β„‚ ∧ ((expβ€˜(2 Β· (i Β· 𝐴))) + 1) # 0) β†’ (tanβ€˜π΄) = (((expβ€˜(2 Β· (i Β· 𝐴))) βˆ’ 1) / (i Β· ((expβ€˜(2 Β· (i Β· 𝐴))) + 1))))
 
Theoremresinval 11723 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 11724 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 11725* 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 11726* 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 11727* 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 11728 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ β†’ (sinβ€˜π΄) ∈ ℝ)
 
Theoremrecoscl 11729 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ β†’ (cosβ€˜π΄) ∈ ℝ)
 
Theoremretanclap 11730 The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
((𝐴 ∈ ℝ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) ∈ ℝ)
 
Theoremresincld 11731 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 11732 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (cosβ€˜π΄) ∈ ℝ)
 
Theoremretanclapd 11733 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (cosβ€˜π΄) # 0)    β‡’   (πœ‘ β†’ (tanβ€˜π΄) ∈ ℝ)
 
Theoremsinneg 11734 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 11735 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ β„‚ β†’ (cosβ€˜-𝐴) = (cosβ€˜π΄))
 
Theoremtannegap 11736 The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
 
Theoremsin0 11737 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
(sinβ€˜0) = 0
 
Theoremcos0 11738 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
(cosβ€˜0) = 1
 
Theoremtan0 11739 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
(tanβ€˜0) = 0
 
Theoremefival 11740 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 11741 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΄))))
 
Theoremefeul 11742 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 11743 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 11744 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 11745 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΅))))
 
Theoremtanaddaplem 11746 A useful intermediate step in tanaddap 11747 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ ((cosβ€˜π΄) # 0 ∧ (cosβ€˜π΅) # 0)) β†’ ((cosβ€˜(𝐴 + 𝐡)) # 0 ↔ ((tanβ€˜π΄) Β· (tanβ€˜π΅)) # 1))
 
Theoremtanaddap 11747 Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ ((cosβ€˜π΄) # 0 ∧ (cosβ€˜π΅) # 0 ∧ (cosβ€˜(𝐴 + 𝐡)) # 0)) β†’ (tanβ€˜(𝐴 + 𝐡)) = (((tanβ€˜π΄) + (tanβ€˜π΅)) / (1 βˆ’ ((tanβ€˜π΄) Β· (tanβ€˜π΅)))))
 
Theoremsinsub 11748 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 11749 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 11750 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 11751 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 11752 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11745 and cossub 11749. (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 11753 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11745 and cossub 11749. (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 11754 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 11755 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 11756 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 11757 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 11758 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 11759 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 11760 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 11761 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 11762 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 11763 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 11764* Lemma for sin01bnd 11765 and cos01bnd 11766. (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 11765 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 11766 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 11767 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
((1 / 3) < (cosβ€˜1) ∧ (cosβ€˜1) < (2 / 3))
 
Theoremcos2bnd 11768 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(-(7 / 9) < (cosβ€˜2) ∧ (cosβ€˜2) < -(1 / 9))
 
Theoremsin01gt0 11769 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 11770 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 11771 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 11772 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sinβ€˜1) ∧ 0 < (cosβ€˜1))
 
Theoremsincos2sgn 11773 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sinβ€˜2) ∧ (cosβ€˜2) < 0)
 
Theoremsin4lt0 11774 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
(sinβ€˜4) < 0
 
Theoremcos12dec 11775 Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
((𝐴 ∈ (1[,]2) ∧ 𝐡 ∈ (1[,]2) ∧ 𝐴 < 𝐡) β†’ (cosβ€˜π΅) < (cosβ€˜π΄))
 
Theoremabsefi 11776 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 11777 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 11778 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 11779 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 11780 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 11781 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 11781 Alternate proof of demoivre 11780. 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)))))
 
4.9.1.1  The circle constant (tau = 2 pi)
 
Syntaxctau 11782 Extend class notation to include the constant tau, Ο„ = 6.28318....
class Ο„
 
Definitiondf-tau 11783 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, < )
 
4.9.2  _e is irrational
 
Theoremeirraplem 11784* Lemma for eirrap 11785. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)
𝐹 = (𝑛 ∈ β„•0 ↦ (1 / (!β€˜π‘›)))    &   (πœ‘ β†’ 𝑃 ∈ β„€)    &   (πœ‘ β†’ 𝑄 ∈ β„•)    β‡’   (πœ‘ β†’ e # (𝑃 / 𝑄))
 
Theoremeirrap 11785 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 11786. (Contributed by Jim Kingdon, 6-Jan-2023.)
(𝑄 ∈ β„š β†’ e # 𝑄)
 
Theoremeirr 11786 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 11785. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
e βˆ‰ β„š
 
Theoremegt2lt3 11787 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.)
(2 < e ∧ e < 3)
 
Theoremepos 11788 Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
0 < e
 
Theoremepr 11789 Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
e ∈ ℝ+
 
Theoremene0 11790 e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
e β‰  0
 
Theoremeap0 11791 e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 0
 
Theoremene1 11792 e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
e β‰  1
 
Theoremeap1 11793 e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 1
 
PART 5  ELEMENTARY NUMBER THEORY

This part introduces elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

 
5.1  Elementary properties of divisibility
 
5.1.1  The divides relation
 
Syntaxcdvds 11794 Extend the definition of a class to include the divides relation. See df-dvds 11795.
class βˆ₯
 
Definitiondf-dvds 11795* Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
βˆ₯ = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) ∧ βˆƒπ‘› ∈ β„€ (𝑛 Β· π‘₯) = 𝑦)}
 
Theoremdivides 11796* Define the divides relation. 𝑀 βˆ₯ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 βˆ₯ 6 (ex-dvds 14485). As proven in dvdsval3 11798, 𝑀 βˆ₯ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 11796 and dvdsval2 11797 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀 βˆ₯ 𝑁 ↔ βˆƒπ‘› ∈ β„€ (𝑛 Β· 𝑀) = 𝑁))
 
Theoremdvdsval2 11797 One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
((𝑀 ∈ β„€ ∧ 𝑀 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝑀 βˆ₯ 𝑁 ↔ (𝑁 / 𝑀) ∈ β„€))
 
Theoremdvdsval3 11798 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.)
((𝑀 ∈ β„• ∧ 𝑁 ∈ β„€) β†’ (𝑀 βˆ₯ 𝑁 ↔ (𝑁 mod 𝑀) = 0))
 
Theoremdvdszrcl 11799 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
(𝑋 βˆ₯ π‘Œ β†’ (𝑋 ∈ β„€ ∧ π‘Œ ∈ β„€))
 
Theoremdvdsmod0 11800 If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.)
((𝑀 ∈ β„• ∧ 𝑀 βˆ₯ 𝑁) β†’ (𝑁 mod 𝑀) = 0)
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