P. 162 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sincl 16101	Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ)
Theorem	coscl 16102	Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ)
Theorem	tanval 16103	Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
Theorem	tancl 16104	The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ ℂ)
Theorem	sincld 16105	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ)
Theorem	coscld 16106	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)
Theorem	tancld 16107	Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (cos‘𝐴) ≠ 0)    ⇒   ⊢ (𝜑 → (tan‘𝐴) ∈ ℂ)
Theorem	tanval2 16108	Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (i · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))
Theorem	tanval3 16109	Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
⊢ ((𝐴 ∈ ℂ ∧ ((exp‘(2 · (i · 𝐴))) + 1) ≠ 0) → (tan‘𝐴) = (((exp‘(2 · (i · 𝐴))) − 1) / (i · ((exp‘(2 · (i · 𝐴))) + 1))))
Theorem	resinval 16110	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 16111	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 16112*	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 16113*	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 16114*	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 16115	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ)
Theorem	recoscl 16116	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ)
Theorem	retancl 16117	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 16118	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (sin‘𝐴) ∈ ℝ)
Theorem	recoscld 16119	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ)
Theorem	retancld 16120	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 16121	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 16122	The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴))
Theorem	tanneg 16123	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 16124	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
⊢ (sin‘0) = 0
Theorem	cos0 16125	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
⊢ (cos‘0) = 1
Theorem	tan0 16126	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
⊢ (tan‘0) = 0
Theorem	efival 16127	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 16128	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	sinhval 16129	Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (𝐴 ∈ ℂ → ((sin‘(i · 𝐴)) / i) = (((exp‘𝐴) − (exp‘-𝐴)) / 2))
Theorem	coshval 16130	Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘𝐴) + (exp‘-𝐴)) / 2))
Theorem	resinhcl 16131	The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (𝐴 ∈ ℝ → ((sin‘(i · 𝐴)) / i) ∈ ℝ)
Theorem	rpcoshcl 16132	The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+)
Theorem	recoshcl 16133	The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ)
Theorem	retanhcl 16134	The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ)
Theorem	tanhlt1 16135	The hyperbolic tangent of a real number is upper bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) < 1)
Theorem	tanhbnd 16136	The hyperbolic tangent of a real number is bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ (-1(,)1))
Theorem	efeul 16137	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 16138	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 16139	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 16140	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	tanaddlem 16141	A useful intermediate step in tanadd 16142 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((cos‘𝐴) ≠ 0 ∧ (cos‘𝐵) ≠ 0)) → ((cos‘(𝐴 + 𝐵)) ≠ 0 ↔ ((tan‘𝐴) · (tan‘𝐵)) ≠ 1))
Theorem	tanadd 16142	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 16143	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 16144	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 16145	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 16146	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 16147	Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 16140 and cossub 16144. (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 16148	Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 16140 and cossub 16144. (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 16149	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 16150	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 16151	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 16152	Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴))))
Theorem	cos2t 16153	Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1))
Theorem	cos2tsin 16154	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 16155	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 16156	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 16157	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 16158	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 16159*	Lemma for sin01bnd 16160 and cos01bnd 16161. (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 16160	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 16161	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 16162	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3))
Theorem	cos2bnd 16163	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9))
Theorem	sinltx 16164	The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)
⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴)
Theorem	sin01gt0 16165	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 16166	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 16167	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 16168	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (0 < (sin‘1) ∧ 0 < (cos‘1))
Theorem	sincos2sgn 16169	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (0 < (sin‘2) ∧ (cos‘2) < 0)
Theorem	sin4lt0 16170	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
⊢ (sin‘4) < 0
Theorem	absefi 16171	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 16172	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 16173	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 16174	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 16175	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 16176 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 16176	Alternate proof of demoivre 16175. 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)))))
5.11.1.1  The circle constant (tau = 2 pi)
Syntax	ctau 16177	Extend class notation to include the constant tau, τ = 6.28318....
class τ
Definition	df-tau 16178	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, < )
5.11.2  _e is irrational
Theorem	eirrlem 16179*	Lemma for eirr 16180. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))    &   ⊢ (𝜑 → 𝑃 ∈ ℤ)    &   ⊢ (𝜑 → 𝑄 ∈ ℕ)    &   ⊢ (𝜑 → e = (𝑃 / 𝑄))    ⇒   ⊢ ¬ 𝜑
Theorem	eirr 16180	e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
⊢ e ∉ ℚ
Theorem	egt2lt3 16181	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)
Theorem	epos 16182	Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
⊢ 0 < e
Theorem	epr 16183	Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
⊢ e ∈ ℝ+
Theorem	ene0 16184	e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
⊢ e ≠ 0
Theorem	ene1 16185	e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
⊢ e ≠ 1
5.12  Cardinality of real and complex number subsets
5.12.1  Countability of integers and rationals
Theorem	xpnnen 16186	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.)
⊢ (ℕ × ℕ) ≈ ℕ
Theorem	znnen 16187	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.)
⊢ ℤ ≈ ℕ
Theorem	qnnen 16188	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 (ℤ × ℕ) 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.)
⊢ ℚ ≈ ℕ
5.12.2  The reals are uncountable
Theorem	rpnnen2lem1 16189*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ ((𝐴 ⊆ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝐴)‘𝑁) = if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0))
Theorem	rpnnen2lem2 16190*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ (𝐴 ⊆ ℕ → (𝐹‘𝐴):ℕ⟶ℝ)
Theorem	rpnnen2lem3 16191*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2)
Theorem	rpnnen2lem4 16192*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝐴)‘𝑘) ∧ ((𝐹‘𝐴)‘𝑘) ≤ ((𝐹‘𝐵)‘𝑘)))
Theorem	rpnnen2lem5 16193*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → seq𝑀( + , (𝐹‘𝐴)) ∈ dom ⇝ )
Theorem	rpnnen2lem6 16194*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘𝑀)((𝐹‘𝐴)‘𝑘) ∈ ℝ)
Theorem	rpnnen2lem7 16195*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘𝑀)((𝐹‘𝐴)‘𝑘) ≤ Σ𝑘 ∈ (ℤ≥‘𝑀)((𝐹‘𝐵)‘𝑘))
Theorem	rpnnen2lem8 16196*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → Σ𝑘 ∈ ℕ ((𝐹‘𝐴)‘𝑘) = (Σ𝑘 ∈ (1...(𝑀 − 1))((𝐹‘𝐴)‘𝑘) + Σ𝑘 ∈ (ℤ≥‘𝑀)((𝐹‘𝐴)‘𝑘)))
Theorem	rpnnen2lem9 16197*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ (𝑀 ∈ ℕ → Σ𝑘 ∈ (ℤ≥‘𝑀)((𝐹‘(ℕ ∖ {𝑀}))‘𝑘) = (0 + (((1 / 3)↑(𝑀 + 1)) / (1 − (1 / 3)))))
Theorem	rpnnen2lem10 16198*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    &   ⊢ (𝜑 → 𝐴 ⊆ ℕ)    &   ⊢ (𝜑 → 𝐵 ⊆ ℕ)    &   ⊢ (𝜑 → 𝑚 ∈ (𝐴 ∖ 𝐵))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵)))    &   ⊢ (𝜓 ↔ Σ𝑘 ∈ ℕ ((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝐵)‘𝑘))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → Σ𝑘 ∈ (ℤ≥‘𝑚)((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ (ℤ≥‘𝑚)((𝐹‘𝐵)‘𝑘))
Theorem	rpnnen2lem11 16199*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    &   ⊢ (𝜑 → 𝐴 ⊆ ℕ)    &   ⊢ (𝜑 → 𝐵 ⊆ ℕ)    &   ⊢ (𝜑 → 𝑚 ∈ (𝐴 ∖ 𝐵))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵)))    &   ⊢ (𝜓 ↔ Σ𝑘 ∈ ℕ ((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝐵)‘𝑘))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	rpnnen2lem12 16200*	Lemma for rpnnen2 16201. (Contributed by Mario Carneiro, 13-May-2013.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))    ⇒   ⊢ 𝒫 ℕ ≼ (0[,]1)