P. 126 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tanvalap 12501	Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
Theorem	tanclap 12502	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 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) ∈ ℂ)
Theorem	sincld 12503	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ)
Theorem	coscld 12504	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)
Theorem	tanclapd 12505	Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (cos‘𝐴) # 0)    ⇒   ⊢ (𝜑 → (tan‘𝐴) ∈ ℂ)
Theorem	tanval2ap 12506	Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘( -i · 𝐴))) / (i · ((exp‘(i · 𝐴)) + (exp‘( -i · 𝐴))))))
Theorem	tanval3ap 12507	Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
⊢ ((𝐴 ∈ ℂ ∧ ((exp‘(2 · (i · 𝐴))) + 1) # 0) → (tan‘𝐴) = (((exp‘(2 · (i · 𝐴))) − 1) / (i · ((exp‘(2 · (i · 𝐴))) + 1))))
Theorem	resinval 12508	The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴))))
Theorem	recosval 12509	The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = (ℜ‘(exp‘(i · 𝐴))))
Theorem	efi4p 12510*	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 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))
Theorem	resin4p 12511*	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 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = ((𝐴 − ((𝐴↑3) / 6)) + (ℑ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))))
Theorem	recos4p 12512*	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 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))))
Theorem	resincl 12513	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ)
Theorem	recoscl 12514	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ)
Theorem	retanclap 12515	The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
⊢ ((𝐴 ∈ ℝ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) ∈ ℝ)
Theorem	resincld 12516	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℝ)
Theorem	recoscld 12517	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ)
Theorem	retanclapd 12518	Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → (cos‘𝐴) # 0)    ⇒   ⊢ (𝜑 → (tan‘𝐴) ∈ ℝ)
Theorem	sinneg 12519	The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (sin‘ -𝐴) = -(sin‘𝐴))
Theorem	cosneg 12520	The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (cos‘ -𝐴) = (cos‘𝐴))
Theorem	tannegap 12521	The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘ -𝐴) = -(tan‘𝐴))
Theorem	sin0 12522	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
⊢ (sin‘0) = 0
Theorem	cos0 12523	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
⊢ (cos‘0) = 1
Theorem	tan0 12524	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
⊢ (tan‘0) = 0
Theorem	efival 12525	The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = ((cos‘𝐴) + (i · (sin‘𝐴))))
Theorem	efmival 12526	The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (exp‘( -i · 𝐴)) = ((cos‘𝐴) − (i · (sin‘𝐴))))
Theorem	efeul 12527	Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((exp‘(ℜ‘𝐴)) · ((cos‘(ℑ‘𝐴)) + (i · (sin‘(ℑ‘𝐴))))))
Theorem	efieq 12528	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 ∧ 𝐵 ∈ ℝ) → ((exp‘(i · 𝐴)) = (exp‘(i · 𝐵)) ↔ ((cos‘𝐴) = (cos‘𝐵) ∧ (sin‘𝐴) = (sin‘𝐵))))
Theorem	sinadd 12529	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 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (sin‘(𝐴 + 𝐵)) = (((sin‘𝐴) · (cos‘𝐵)) + ((cos‘𝐴) · (sin‘𝐵))))
Theorem	cosadd 12530	Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴 + 𝐵)) = (((cos‘𝐴) · (cos‘𝐵)) − ((sin‘𝐴) · (sin‘𝐵))))
Theorem	tanaddaplem 12531	A useful intermediate step in tanaddap 12532 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 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((cos‘𝐴) # 0 ∧ (cos‘𝐵) # 0)) → ((cos‘(𝐴 + 𝐵)) # 0 ↔ ((tan‘𝐴) · (tan‘𝐵)) # 1))
Theorem	tanaddap 12532	Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((cos‘𝐴) # 0 ∧ (cos‘𝐵) # 0 ∧ (cos‘(𝐴 + 𝐵)) # 0)) → (tan‘(𝐴 + 𝐵)) = (((tan‘𝐴) + (tan‘𝐵)) / (1 − ((tan‘𝐴) · (tan‘𝐵)))))
Theorem	sinsub 12533	Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (sin‘(𝐴 − 𝐵)) = (((sin‘𝐴) · (cos‘𝐵)) − ((cos‘𝐴) · (sin‘𝐵))))
Theorem	cossub 12534	Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴 − 𝐵)) = (((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))))
Theorem	addsin 12535	Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) + (sin‘𝐵)) = (2 · ((sin‘((𝐴 + 𝐵) / 2)) · (cos‘((𝐴 − 𝐵) / 2)))))
Theorem	subsin 12536	Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) − (sin‘𝐵)) = (2 · ((cos‘((𝐴 + 𝐵) / 2)) · (sin‘((𝐴 − 𝐵) / 2)))))
Theorem	sinmul 12537	Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12530 and cossub 12534. (Contributed by David A. Wheeler, 26-May-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴 − 𝐵)) − (cos‘(𝐴 + 𝐵))) / 2))
Theorem	cosmul 12538	Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12530 and cossub 12534. (Contributed by David A. Wheeler, 26-May-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) · (cos‘𝐵)) = (((cos‘(𝐴 − 𝐵)) + (cos‘(𝐴 + 𝐵))) / 2))
Theorem	addcos 12539	Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) + (cos‘𝐵)) = (2 · ((cos‘((𝐴 + 𝐵) / 2)) · (cos‘((𝐴 − 𝐵) / 2)))))
Theorem	subcos 12540	Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐵) − (cos‘𝐴)) = (2 · ((sin‘((𝐴 + 𝐵) / 2)) · (sin‘((𝐴 − 𝐵) / 2)))))
Theorem	sincossq 12541	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 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1)
Theorem	sin2t 12542	Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴))))
Theorem	cos2t 12543	Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1))
Theorem	cos2tsin 12544	Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = (1 − (2 · ((sin‘𝐴)↑2))))
Theorem	sinbnd 12545	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 → ( -1 ≤ (sin‘𝐴) ∧ (sin‘𝐴) ≤ 1))
Theorem	cosbnd 12546	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 → ( -1 ≤ (cos‘𝐴) ∧ (cos‘𝐴) ≤ 1))
Theorem	sinbnd2 12547	The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ( -1[,]1))
Theorem	cosbnd2 12548	The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ( -1[,]1))
Theorem	ef01bndlem 12549*	Lemma for sin01bnd 12550 and cos01bnd 12551. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ (0(,]1) → (abs‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)) < ((𝐴↑4) / 6))
Theorem	sin01bnd 12550	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) → ((𝐴 − ((𝐴↑3) / 3)) < (sin‘𝐴) ∧ (sin‘𝐴) < 𝐴))
Theorem	cos01bnd 12551	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) → ((1 − (2 · ((𝐴↑2) / 3))) < (cos‘𝐴) ∧ (cos‘𝐴) < (1 − ((𝐴↑2) / 3))))
Theorem	cos1bnd 12552	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3))
Theorem	cos2bnd 12553	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ ( -(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9))
Theorem	sinltxirr 12554*	The sine of a positive irrational number is less than its argument. Here irrational means apart from any rational number. (Contributed by Mario Carneiro, 29-Jul-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ ∀𝑞 ∈ ℚ 𝐴 # 𝑞) → (sin‘𝐴) < 𝐴)
Theorem	sin01gt0 12555	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) → 0 < (sin‘𝐴))
Theorem	cos01gt0 12556	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) → 0 < (cos‘𝐴))
Theorem	sin02gt0 12557	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) → 0 < (sin‘𝐴))
Theorem	sincos1sgn 12558	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (0 < (sin‘1) ∧ 0 < (cos‘1))
Theorem	sincos2sgn 12559	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (0 < (sin‘2) ∧ (cos‘2) < 0)
Theorem	sin4lt0 12560	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (sin‘4) < 0
Theorem	cos12dec 12561	Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
⊢ ((𝐴 ∈ (1[,]2) ∧ 𝐵 ∈ (1[,]2) ∧ 𝐴 < 𝐵) → (cos‘𝐵) < (cos‘𝐴))
Theorem	absefi 12562	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 → (abs‘(exp‘(i · 𝐴))) = 1)
Theorem	absef 12563	The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴)))
Theorem	absefib 12564	A complex number is real iff the exponential of its product with i has absolute value one. (Contributed by NM, 21-Aug-2008.)
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (abs‘(exp‘(i · 𝐴))) = 1))
Theorem	efieq1re 12565	A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
⊢ ((𝐴 ∈ ℂ ∧ (exp‘(i · 𝐴)) = 1) → 𝐴 ∈ ℝ)
Theorem	demoivre 12566	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 12567 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
Theorem	demoivreALT 12567	Alternate proof of demoivre 12566. 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 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
4.10.1.1  The circle constant (tau = 2 pi)
Syntax	ctau 12568	Extend class notation to include the constant tau, τ = 6.28318....
class τ
Definition	df-tau 12569	Define the circle constant tau, τ = 6.28318..., which is the smallest positive real number whose cosine is one. Various notations have been used or proposed for this number including τ, a three-legged variant of π, or 2π. Note the difference between this constant τ 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.)
⊢ τ = inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < )
4.10.2  _e is irrational
Theorem	eirraplem 12570*	Lemma for eirrap 12571. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))    &   ⊢ (𝜑 → 𝑃 ∈ ℤ)    &   ⊢ (𝜑 → 𝑄 ∈ ℕ)    ⇒   ⊢ (𝜑 → e # (𝑃 / 𝑄))
Theorem	eirrap 12571	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 12572. (Contributed by Jim Kingdon, 6-Jan-2023.)
⊢ (𝑄 ∈ ℚ → e # 𝑄)
Theorem	eirr 12572	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 12571. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
⊢ e ∉ ℚ
Theorem	egt2lt3 12573	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)
Theorem	epos 12574	Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
⊢ 0 < e
Theorem	epr 12575	Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
⊢ e ∈ ℝ+
Theorem	ene0 12576	e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
⊢ e ≠ 0
Theorem	eap0 12577	e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
⊢ e # 0
Theorem	ene1 12578	e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
⊢ e ≠ 1
Theorem	eap1 12579	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
Syntax	cdvds 12580	Extend the definition of a class to include the divides relation. See df-dvds 12581.
class ∥
Definition	df-dvds 12581*	Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
Theorem	divides 12582*	Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 16961). As proven in dvdsval3 12584, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 12582 and dvdsval2 12583 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
Theorem	dvdsval2 12583	One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))
Theorem	dvdsval3 12584	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 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0))
Theorem	dvdszrcl 12585	Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))
Theorem	dvdsmod0 12586	If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑀 ∥ 𝑁) → (𝑁 mod 𝑀) = 0)
Theorem	p1modz1 12587	If a number greater than 1 divides another number, the second number increased by 1 is 1 modulo the first number. (Contributed by AV, 19-Mar-2022.)
⊢ ((𝑀 ∥ 𝐴 ∧ 1 < 𝑀) → ((𝐴 + 1) mod 𝑀) = 1)
Theorem	dvdsmodexp 12588	If a positive integer divides another integer, this other integer is equal to its positive powers modulo the positive integer. (Formerly part of the proof for fermltl 13042). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∥ 𝐴) → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁))
Theorem	nndivdvds 12589	Strong form of dvdsval2 12583 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵 ∥ 𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ))
Theorem	nndivides 12590*	Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (𝑛 · 𝑀) = 𝑁))
Theorem	dvdsdc 12591	Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ∥ 𝑁)
Theorem	moddvds 12592	Two ways to say 𝐴≡𝐵 (mod 𝑁), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) ↔ 𝑁 ∥ (𝐴 − 𝐵)))
Theorem	modm1div 12593	An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℤ) → ((𝐴 mod 𝑁) = 1 ↔ 𝑁 ∥ (𝐴 − 1)))
Theorem	dvds0lem 12594	A lemma to assist theorems of ∥ with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 · 𝑀) = 𝑁) → 𝑀 ∥ 𝑁)
Theorem	dvds1lem 12595*	A lemma to assist theorems of ∥ with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))    &   ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑍 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁))    ⇒   ⊢ (𝜑 → (𝐽 ∥ 𝐾 → 𝑀 ∥ 𝑁))
Theorem	dvds2lem 12596*	A lemma to assist theorems of ∥ with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝜑 → (𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ))    &   ⊢ (𝜑 → (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ))    &   ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑍 ∈ ℤ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → (𝑍 · 𝑀) = 𝑁))    ⇒   ⊢ (𝜑 → ((𝐼 ∥ 𝐽 ∧ 𝐾 ∥ 𝐿) → 𝑀 ∥ 𝑁))
Theorem	iddvds 12597	An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 𝑁)
Theorem	1dvds 12598	1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
Theorem	dvds0 12599	Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 0)
Theorem	negdvdsb 12600	An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ -𝑀 ∥ 𝑁))