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Theorem List for Intuitionistic Logic Explorer - 11701-11800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremefsep 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.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    &   π‘ = (𝑀 + 1)    &   π‘€ ∈ β„•0    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ (expβ€˜π΄) = (𝐡 + Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘€)(πΉβ€˜π‘˜)))    &   (πœ‘ β†’ (𝐡 + ((𝐴↑𝑀) / (!β€˜π‘€))) = 𝐷)    β‡’   (πœ‘ β†’ (expβ€˜π΄) = (𝐷 + Ξ£π‘˜ ∈ (β„€β‰₯β€˜π‘)(πΉβ€˜π‘˜)))
 
Theoremeffsumlt 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.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (seq0( + , 𝐹)β€˜π‘) < (expβ€˜π΄))
 
Theoremeft0val 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.)
(𝐴 ∈ β„‚ β†’ ((𝐴↑0) / (!β€˜0)) = 1)
 
Theoremef4p 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.)
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 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.)
(𝐴 ∈ ℝ+ β†’ ((1 + 𝐴) + ((𝐴↑2) / 2)) < (expβ€˜π΄))
 
Theoremefgt1p 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.)
(𝐴 ∈ ℝ+ β†’ (1 + 𝐴) < (expβ€˜π΄))
 
Theoremefgt1 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.)
(𝐴 ∈ ℝ+ β†’ 1 < (expβ€˜π΄))
 
Theoremefltim 11708 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 11709 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 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.)
(exp β†Ύ ℝ):ℝ–1-1→ℝ+
 
Theoremeflegeo 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.)
(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 11712 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 11713 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 11714 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 11715 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 11716 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 11717 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 11718 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 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.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) ∈ β„‚)
 
Theoremsincld 11720 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 11721 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 11722 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 11723 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 11724 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 11725 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 11726 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 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.)
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 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.)
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 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.)
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 11730 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ β†’ (sinβ€˜π΄) ∈ ℝ)
 
Theoremrecoscl 11731 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℝ β†’ (cosβ€˜π΄) ∈ ℝ)
 
Theoremretanclap 11732 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 11733 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 11734 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 11735 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 11736 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 11737 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 11738 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 11739 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
(sinβ€˜0) = 0
 
Theoremcos0 11740 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
(cosβ€˜0) = 1
 
Theoremtan0 11741 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
(tanβ€˜0) = 0
 
Theoremefival 11742 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 11743 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 11744 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 11745 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 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (sinβ€˜(𝐴 + 𝐡)) = (((sinβ€˜π΄) Β· (cosβ€˜π΅)) + ((cosβ€˜π΄) Β· (sinβ€˜π΅))))
 
Theoremcosadd 11747 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 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.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ ((cosβ€˜π΄) # 0 ∧ (cosβ€˜π΅) # 0)) β†’ ((cosβ€˜(𝐴 + 𝐡)) # 0 ↔ ((tanβ€˜π΄) Β· (tanβ€˜π΅)) # 1))
 
Theoremtanaddap 11749 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 11750 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 11751 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 11752 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 11753 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 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((sinβ€˜π΄) Β· (sinβ€˜π΅)) = (((cosβ€˜(𝐴 βˆ’ 𝐡)) βˆ’ (cosβ€˜(𝐴 + 𝐡))) / 2))
 
Theoremcosmul 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((cosβ€˜π΄) Β· (cosβ€˜π΅)) = (((cosβ€˜(𝐴 βˆ’ 𝐡)) + (cosβ€˜(𝐴 + 𝐡))) / 2))
 
Theoremaddcos 11756 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 11757 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 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.)
(𝐴 ∈ β„‚ β†’ (((sinβ€˜π΄)↑2) + ((cosβ€˜π΄)↑2)) = 1)
 
Theoremsin2t 11759 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 11760 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 11761 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 11762 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 11763 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 11764 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 11765 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 11766* Lemma for sin01bnd 11767 and cos01bnd 11768. (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 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.)
(𝐴 ∈ (0(,]1) β†’ ((𝐴 βˆ’ ((𝐴↑3) / 3)) < (sinβ€˜π΄) ∧ (sinβ€˜π΄) < 𝐴))
 
Theoremcos01bnd 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.)
(𝐴 ∈ (0(,]1) β†’ ((1 βˆ’ (2 Β· ((𝐴↑2) / 3))) < (cosβ€˜π΄) ∧ (cosβ€˜π΄) < (1 βˆ’ ((𝐴↑2) / 3))))
 
Theoremcos1bnd 11769 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
((1 / 3) < (cosβ€˜1) ∧ (cosβ€˜1) < (2 / 3))
 
Theoremcos2bnd 11770 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(-(7 / 9) < (cosβ€˜2) ∧ (cosβ€˜2) < -(1 / 9))
 
Theoremsin01gt0 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.)
(𝐴 ∈ (0(,]1) β†’ 0 < (sinβ€˜π΄))
 
Theoremcos01gt0 11772 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 11773 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 11774 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sinβ€˜1) ∧ 0 < (cosβ€˜1))
 
Theoremsincos2sgn 11775 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sinβ€˜2) ∧ (cosβ€˜2) < 0)
 
Theoremsin4lt0 11776 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
(sinβ€˜4) < 0
 
Theoremcos12dec 11777 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 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.)
(𝐴 ∈ ℝ β†’ (absβ€˜(expβ€˜(i Β· 𝐴))) = 1)
 
Theoremabsef 11779 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 11780 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 11781 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 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.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„€) β†’ (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄)))↑𝑁) = ((cosβ€˜(𝑁 Β· 𝐴)) + (i Β· (sinβ€˜(𝑁 Β· 𝐴)))))
 
TheoremdemoivreALT 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.)
((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 11784 Extend class notation to include the constant tau, Ο„ = 6.28318....
class Ο„
 
Definitiondf-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 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 11786* Lemma for eirrap 11787. (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 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.)
(𝑄 ∈ β„š β†’ e # 𝑄)
 
Theoremeirr 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.)
e βˆ‰ β„š
 
Theoremegt2lt3 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.)
(2 < e ∧ e < 3)
 
Theoremepos 11790 Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
0 < e
 
Theoremepr 11791 Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
e ∈ ℝ+
 
Theoremene0 11792 e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
e β‰  0
 
Theoremeap0 11793 e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 0
 
Theoremene1 11794 e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
e β‰  1
 
Theoremeap1 11795 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 11796 Extend the definition of a class to include the divides relation. See df-dvds 11797.
class βˆ₯
 
Definitiondf-dvds 11797* 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 11798* Define the divides relation. 𝑀 βˆ₯ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 βˆ₯ 6 (ex-dvds 14567). As proven in dvdsval3 11800, 𝑀 βˆ₯ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 11798 and dvdsval2 11799 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 11799 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 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.)
((𝑀 ∈ β„• ∧ 𝑁 ∈ β„€) β†’ (𝑀 βˆ₯ 𝑁 ↔ (𝑁 mod 𝑀) = 0))
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