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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | recoshcl 15501 | The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ) | ||
Theorem | retanhcl 15502 | 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 15503 | 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 15504 | 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 15505 | 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 15506 | 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 15507 | 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 15508 | 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 15509 | A useful intermediate step in tanadd 15510 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 15510 | 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 15511 | 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 15512 | 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 15513 | 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 15514 | 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 15515 | Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 15508 and cossub 15512. (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 15516 | Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 15508 and cossub 15512. (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 15517 | 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 15518 | 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 15519 | 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 15520 | Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴)))) | ||
Theorem | cos2t 15521 | Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1)) | ||
Theorem | cos2tsin 15522 | 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 15523 | 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 15524 | 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 15525 | 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 15526 | 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 15527* | Lemma for sin01bnd 15528 and cos01bnd 15529. (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 15528 | 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 15529 | 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 15530 | Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3)) | ||
Theorem | cos2bnd 15531 | Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9)) | ||
Theorem | sinltx 15532 | 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 15533 | 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 15534 | 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 15535 | 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 15536 | The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (0 < (sin‘1) ∧ 0 < (cos‘1)) | ||
Theorem | sincos2sgn 15537 | The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | ||
Theorem | sin4lt0 15538 | The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.) |
⊢ (sin‘4) < 0 | ||
Theorem | absefi 15539 | 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 15540 | 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 15541 | 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 15542 | 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 15543 | 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 15544 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 15544 | Alternate proof of demoivre 15543. 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))))) | ||
Syntax | ctau 15545 | Extend class notation to include the constant tau, τ = 6.28318.... |
class τ | ||
Definition | df-tau 15546 | 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, < ) | ||
Theorem | eirrlem 15547* | Lemma for eirr 15548. (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 15548 | e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.) |
⊢ e ∉ ℚ | ||
Theorem | egt2lt3 15549 | 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 15550 | Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.) |
⊢ 0 < e | ||
Theorem | epr 15551 | Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.) |
⊢ e ∈ ℝ+ | ||
Theorem | ene0 15552 | e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.) |
⊢ e ≠ 0 | ||
Theorem | ene1 15553 | e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.) |
⊢ e ≠ 1 | ||
Theorem | xpnnen 15554 | 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 15555 | 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 15556 | 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 15557* | Lemma for rpnnen2 15569. (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 15558* | Lemma for rpnnen2 15569. (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 15559* | Lemma for rpnnen2 15569. (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 15560* | Lemma for rpnnen2 15569. (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 15561* | Lemma for rpnnen2 15569. (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 15562* | Lemma for rpnnen2 15569. (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 15563* | Lemma for rpnnen2 15569. (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 15564* | Lemma for rpnnen2 15569. (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 15565* | Lemma for rpnnen2 15569. (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 15566* | Lemma for rpnnen2 15569. (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 15567* | Lemma for rpnnen2 15569. (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 15568* | Lemma for rpnnen2 15569. (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 15569 |
The other half of rpnnen 15570, 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... 15227). 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 15557). This is an infinite sum of real numbers (rpnnen2lem2 15558), and since 𝐴 ⊆ 𝐵 implies (𝐹‘𝐴) ≤ (𝐹‘𝐵) (rpnnen2lem4 15560) and (𝐹‘ℕ) converges to 1 / 2 (rpnnen2lem3 15559) by geoisum1 15225, the sum is convergent to some real (rpnnen2lem5 15561 and rpnnen2lem6 15562) by the comparison test for convergence cvgcmp 15161. The comparison test also tells us that 𝐴 ⊆ 𝐵 implies Σ(𝐹‘𝐴) ≤ Σ(𝐹‘𝐵) (rpnnen2lem7 15563). 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 15564) and cancel them (rpnnen2lem10 15566), 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 15565) and Σ(𝐹‘{𝑚}) = (3↑-𝑚), we establish that Σ(𝐹‘𝑦) < Σ(𝐹‘𝑥) (rpnnen2lem11 15567) so that they must be different. By contraposition (rpnnen2lem12 15568), 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 15570 | The cardinality of the continuum is the same as the powerset of ω. This is a stronger statement than ruc 15586, which only asserts that ℝ is uncountable, i.e. has a cardinality larger than ω. The main proof is in two parts, rpnnen1 12372 and rpnnen2 15569, each showing an injection in one direction, and this last part uses sbth 8626 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 15571 | 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 9432 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) |
⊢ (ℝ × ℝ) ≈ ℝ | ||
Theorem | cpnnen 15572 | The complex numbers are equinumerous to the powerset of the positive integers. (Contributed by Mario Carneiro, 16-Jun-2013.) |
⊢ ℂ ≈ 𝒫 ℕ | ||
Theorem | rucALT 15573 | Alternate proof of ruc 15586. This proof is a simple corollary of rpnnen 15570, which determines the exact cardinality of the reals. For an alternate proof discussed at mmcomplex.html#uncountable 15570, see ruc 15586. (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 15574* | Lemma for ruc 15586 (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 15575* | Lemma for ruc 15586. 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)) | ||
Theorem | ruclem3 15576* | Lemma for ruc 15586. The constructed interval [𝑋, 𝑌] always excludes 𝑀. (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 → (𝑀 < 𝑋 ∨ 𝑌 < 𝑀)) | ||
Theorem | ruclem4 15577* | Lemma for ruc 15586. Initial value of the interval sequence. (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)〉))) & ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) & ⊢ 𝐺 = seq0(𝐷, 𝐶) ⇒ ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) | ||
Theorem | ruclem6 15578* | Lemma for ruc 15586. Domain and range of the interval sequence. (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)〉))) & ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) & ⊢ 𝐺 = seq0(𝐷, 𝐶) ⇒ ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ)) | ||
Theorem | ruclem7 15579* | Lemma for ruc 15586. Successor value for the interval sequence. (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)〉))) & ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) & ⊢ 𝐺 = seq0(𝐷, 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) | ||
Theorem | ruclem8 15580* | Lemma for ruc 15586. The intervals of the 𝐺 sequence are all nonempty. (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)〉))) & ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) & ⊢ 𝐺 = seq0(𝐷, 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (1st ‘(𝐺‘𝑁)) < (2nd ‘(𝐺‘𝑁))) | ||
Theorem | ruclem9 15581* | Lemma for ruc 15586. The first components of the 𝐺 sequence are increasing, and the second components are decreasing. (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)〉))) & ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) & ⊢ 𝐺 = seq0(𝐷, 𝐶) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑁)) ∧ (2nd ‘(𝐺‘𝑁)) ≤ (2nd ‘(𝐺‘𝑀)))) | ||
Theorem | ruclem10 15582* | Lemma for ruc 15586. Every first component of the 𝐺 sequence is less than every second component. That is, the sequences form a chain a1 < a2 <... < b2 < b1, where ai are the first components and bi are the second components. (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)〉))) & ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) & ⊢ 𝐺 = seq0(𝐷, 𝐶) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) < (2nd ‘(𝐺‘𝑁))) | ||
Theorem | ruclem11 15583* | Lemma for ruc 15586. Closure lemmas for supremum. (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)〉))) & ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) & ⊢ 𝐺 = seq0(𝐷, 𝐶) ⇒ ⊢ (𝜑 → (ran (1st ∘ 𝐺) ⊆ ℝ ∧ ran (1st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1)) | ||
Theorem | ruclem12 15584* | Lemma for ruc 15586. The supremum of the increasing sequence 1st ∘ 𝐺 is a real number that is not in the range of 𝐹. (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)〉))) & ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) & ⊢ 𝐺 = seq0(𝐷, 𝐶) & ⊢ 𝑆 = sup(ran (1st ∘ 𝐺), ℝ, < ) ⇒ ⊢ (𝜑 → 𝑆 ∈ (ℝ ∖ ran 𝐹)) | ||
Theorem | ruclem13 15585 | Lemma for ruc 15586. There is no function that maps ℕ onto ℝ. (Use nex 1792 if you want this in the form ¬ ∃𝑓𝑓:ℕ–onto→ℝ.) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |
⊢ ¬ 𝐹:ℕ–onto→ℝ | ||
Theorem | ruc 15586 | The set of positive integers is strictly dominated by the set of real numbers, i.e. the real numbers are uncountable. The proof consists of lemmas ruclem1 15574 through ruclem13 15585 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 15585 for the function existence version of this theorem. For an informal discussion of this proof, see mmcomplex.html#uncountable 15585. For an alternate proof see rucALT 15573. This is Metamath 100 proof #22. (Contributed by NM, 13-Oct-2004.) |
⊢ ℕ ≺ ℝ | ||
Theorem | resdomq 15587 | The set of rationals is strictly less equinumerous than the set of reals (ℝ strictly dominates ℚ). (Contributed by NM, 18-Dec-2004.) |
⊢ ℚ ≺ ℝ | ||
Theorem | aleph1re 15588 | There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.) |
⊢ (ℵ‘1o) ≼ ℝ | ||
Theorem | aleph1irr 15589 | There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.) |
⊢ (ℵ‘1o) ≼ (ℝ ∖ ℚ) | ||
Theorem | cnso 15590 | The complex numbers can be linearly ordered. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ ∃𝑥 𝑥 Or ℂ | ||
Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory. | ||
Theorem | sqrt2irrlem 15591 | Lemma for sqrt2irr 15592. This is the core of the proof: if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). This is Metamath 100 proof #1. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) (Proof shortened by JV, 4-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) ⇒ ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) | ||
Theorem | sqrt2irr 15592 | The square root of 2 is irrational. See zsqrtelqelz 16088 for a generalization to all non-square integers. The proof's core is proven in sqrt2irrlem 15591, which shows that if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd. An older version of this proof was included in The Seventeen Provers of the World compiled by Freek Wiedijk. It is also the first of the "top 100" mathematical theorems whose formalization is tracked by Freek Wiedijk on his Formalizing 100 Theorems page at http://www.cs.ru.nl/~freek/100/ 15591. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ (√‘2) ∉ ℚ | ||
Theorem | sqrt2re 15593 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
⊢ (√‘2) ∈ ℝ | ||
Theorem | sqrt2irr0 15594 | The square root of 2 is an irrational number. (Contributed by AV, 23-Dec-2022.) |
⊢ (√‘2) ∈ (ℝ ∖ ℚ) | ||
Theorem | nthruc 15595 | The sequence ℕ, ℤ, ℚ, ℝ, and ℂ forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to ℤ but not ℕ, one-half belongs to ℚ but not ℤ, the square root of 2 belongs to ℝ but not ℚ, and finally that the imaginary number i belongs to ℂ but not ℝ. See nthruz 15596 for a further refinement. (Contributed by NM, 12-Jan-2002.) |
⊢ ((ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) ∧ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ)) | ||
Theorem | nthruz 15596 | The sequence ℕ, ℕ0, and ℤ forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to ℕ0 but not ℕ and minus one belongs to ℤ but not ℕ0. This theorem refines the chain of proper subsets nthruc 15595. (Contributed by NM, 9-May-2004.) |
⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) | ||
Syntax | cdvds 15597 | Extend the definition of a class to include the divides relation. See df-dvds 15598. |
class ∥ | ||
Definition | df-dvds 15598* | 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 15599* | Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 28163). As proven in dvdsval3 15601, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 15599 and dvdsval2 15600 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) | ||
Theorem | dvdsval2 15600 | 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)) |
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