Home | Metamath
Proof Explorer Theorem List (p. 148 of 424) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-27159) |
Hilbert Space Explorer
(27160-28684) |
Users' Mathboxes
(28685-42360) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tancld 14701 | 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 14702 | 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 14703 | 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 14704 | 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 14705 | 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 14706* | Separate out the first four terms of the infinite series expansion of the exponential function of an imaginary number. (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 14707* | 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 14708* | 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 14709 | The sine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) | ||
Theorem | recoscl 14710 | The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ) | ||
Theorem | retancl 14711 | 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 14712 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (sin‘𝐴) ∈ ℝ) | ||
Theorem | recoscld 14713 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ) | ||
Theorem | retancld 14714 | 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 14715 | 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 14716 | 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 14717 | 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 14718 | Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.) |
⊢ (sin‘0) = 0 | ||
Theorem | cos0 14719 | Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.) |
⊢ (cos‘0) = 1 | ||
Theorem | tan0 14720 | The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.) |
⊢ (tan‘0) = 0 | ||
Theorem | efival 14721 | 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 14722 | 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 14723 | 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 14724 | 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 14725 | 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 14726 | 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 14727 | The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ) | ||
Theorem | retanhcl 14728 | 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 14729 | 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 14730 | 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 14731 | 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 14732 | 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 14733 | 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 14734 | 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 14735 | A useful intermediate step in tanadd 14736 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 14736 | 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 14737 | 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 14738 | 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 14739 | 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 14740 | 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 14741 | Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 14734 and cossub 14738. (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 14742 | Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 14734 and cossub 14738. (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 14743 | 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 14744 | 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 14745 | 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 14746 | Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴)))) | ||
Theorem | cos2t 14747 | Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1)) | ||
Theorem | cos2tsin 14748 | 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 14749 | 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 14750 | 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 14751 | 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 14752 | 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 14753* | Lemma for sin01bnd 14754 and cos01bnd 14755. (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 14754 | 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 14755 | 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 14756 | Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3)) | ||
Theorem | cos2bnd 14757 | Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9)) | ||
Theorem | sinltx 14758 | 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 14759 | The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) Replace OLD theorem. (Revised by Wolf Lammen, 25-Sep-2020.) |
⊢ (𝐴 ∈ (0(,]1) → 0 < (sin‘𝐴)) | ||
Theorem | cos01gt0 14760 | 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 14761 | 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 14762 | The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (0 < (sin‘1) ∧ 0 < (cos‘1)) | ||
Theorem | sincos2sgn 14763 | The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | ||
Theorem | sin4lt0 14764 | The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (sin‘4) < 0 | ||
Theorem | absefi 14765 | The absolute value of the exponential function 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 14766 | The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.) |
⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴))) | ||
Theorem | absefib 14767 | A number is real iff its imaginary exponential has absolute value one. (Contributed by NM, 21-Aug-2008.) |
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (abs‘(exp‘(i · 𝐴))) = 1)) | ||
Theorem | efieq1re 14768 | 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 14769 | 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 14770 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 14770 | Alternate proof of demoivre 14769. 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))))) | ||
Theorem | eirrlem 14771* | Lemma for eirr 14772. (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 14772 | e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.) |
⊢ e ∉ ℚ | ||
Theorem | egt2lt3 14773 | Euler's constant e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Mario Carneiro, 29-Apr-2014.) |
⊢ (2 < e ∧ e < 3) | ||
Theorem | epos 14774 | Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.) |
⊢ 0 < e | ||
Theorem | epr 14775 | Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.) |
⊢ e ∈ ℝ+ | ||
Theorem | ene0 14776 | e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.) |
⊢ e ≠ 0 | ||
Theorem | ene1 14777 | e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.) |
⊢ e ≠ 1 | ||
Theorem | xpnnen 14778 | 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 | znnenlem 14779 | Lemma for znnen 14780. (Contributed by NM, 31-Jul-2004.) |
⊢ (((0 ≤ 𝑥 ∧ ¬ 0 ≤ 𝑦) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 = 𝑦 ↔ (2 · 𝑥) = ((-2 · 𝑦) + 1))) | ||
Theorem | znnen 14780 | 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 14781 | 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 | ||
Theorem | rpnnen2lem1 14782* | Lemma for rpnnen2 14794. (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 14783* | Lemma for rpnnen2 14794. (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 14784* | Lemma for rpnnen2 14794. (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 14785* | Lemma for rpnnen2 14794. (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 14786* | Lemma for rpnnen2 14794. (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 14787* | Lemma for rpnnen2 14794. (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 14788* | Lemma for rpnnen2 14794. (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 14789* | Lemma for rpnnen2 14794. (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 14790* | Lemma for rpnnen2 14794. (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 14791* | Lemma for rpnnen2 14794. (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 14792* | Lemma for rpnnen2 14794. (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 14793* | Lemma for rpnnen2 14794. (Contributed by Mario Carneiro, 13-May-2013.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) ⇒ ⊢ 𝒫 ℕ ≼ (0[,]1) | ||
Theorem | rpnnen2 14794 |
The other half of rpnnen 14795, 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... 14451). 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 Σ𝑘 ∈ 𝐴(3↑-𝑘) = Σ𝑘 ∈ ℕ((𝐹‘𝐴)‘𝑘), where ((𝐹‘𝐴)‘𝑘) = if(𝑘 ∈ 𝐴, (3↑-𝑘), 0)) (rpnnen2lem1 14782). This is an infinite sum of real numbers (rpnnen2lem2 14783), and since 𝐴 ⊆ 𝐵 implies (𝐹‘𝐴) ≤ (𝐹‘𝐵) (rpnnen2lem4 14785) and (𝐹‘ℕ) converges to 1 / 2 (rpnnen2lem3 14784) by geoisum1 14449, the sum is convergent to some real (rpnnen2lem5 14786 and rpnnen2lem6 14787) by the comparison test for convergence cvgcmp 14389. The comparison test also tells us that 𝐴 ⊆ 𝐵 implies Σ(𝐹‘𝐴) ≤ Σ(𝐹‘𝐵) (rpnnen2lem7 14788). Putting it all together, if we have two sets 𝑥 ≠ 𝑦, there must differ somewhere, and so there must be an 𝑚 such that ∀𝑛 < 𝑚(𝑛 ∈ 𝑥 ↔ 𝑛 ∈ 𝑦) but 𝑚 ∈ (𝑥 ∖ 𝑦) or vice versa. In this case, we split off the first 𝑚 − 1 terms (rpnnen2lem8 14789) and cancel them (rpnnen2lem10 14791), since these are the same for both sets. For the remaining terms, we use the subset property to establish that Σ(𝐹‘𝑦) ≤ Σ(𝐹‘(ℕ ∖ {𝑚})) and Σ(𝐹‘{𝑚}) ≤ Σ(𝐹‘𝑥) (where these sums are only over (ℤ≥‘𝑚)), and since Σ(𝐹‘(ℕ ∖ {𝑚})) = (3↑-𝑚) / 2 (rpnnen2lem9 14790) and Σ(𝐹‘{𝑚}) = (3↑-𝑚), we establish that Σ(𝐹‘𝑦) < Σ(𝐹‘𝑥) (rpnnen2lem11 14792) so that they must be different. By contraposition (rpnnen2lem12 14793), 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 ℕ ≼ (0[,]1) | ||
Theorem | rpnnen 14795 | The cardinality of the continuum is the same as the powerset of ω. This is a stronger statement than ruc 14811, which only asserts that ℝ is uncountable, i.e. has a cardinality larger than ω. The main proof is in two parts, rpnnen1 11696 and rpnnen2 14794, each showing an injection in one direction, and this last part uses sbth 7965 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 ≈ 𝒫 ℕ | ||
Theorem | rexpen 14796 | 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 8723 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) |
⊢ (ℝ × ℝ) ≈ ℝ | ||
Theorem | cpnnen 14797 | The complex numbers are equinumerous to the powerset of the positive integers. (Contributed by Mario Carneiro, 16-Jun-2013.) |
⊢ ℂ ≈ 𝒫 ℕ | ||
Theorem | rucALT 14798 | Alternate proof of ruc 14811. This proof is a simple corollary of rpnnen 14795, which determines the exact cardinality of the reals. For an alternate proof discussed at mmcomplex.html#uncountable, see ruc 14811. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℕ ≺ ℝ | ||
Theorem | ruclem1 14799* | Lemma for ruc 14811 (the reals are uncountable). Substitutions for the function 𝐷. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.) |
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑋 = (1st ‘(〈𝐴, 𝐵〉𝐷𝑀)) & ⊢ 𝑌 = (2nd ‘(〈𝐴, 𝐵〉𝐷𝑀)) ⇒ ⊢ (𝜑 → ((〈𝐴, 𝐵〉𝐷𝑀) ∈ (ℝ × ℝ) ∧ 𝑋 = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)) ∧ 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))) | ||
Theorem | ruclem2 14800* | Lemma for ruc 14811. Ordering property for the input to 𝐷. (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑋 = (1st ‘(〈𝐴, 𝐵〉𝐷𝑀)) & ⊢ 𝑌 = (2nd ‘(〈𝐴, 𝐵〉𝐷𝑀)) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝑋 ∧ 𝑋 < 𝑌 ∧ 𝑌 ≤ 𝐵)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |