Home Metamath Proof ExplorerTheorem List (p. 154 of 437) < 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 (1-28347) Hilbert Space Explorer (28348-29872) Users' Mathboxes (29873-43661)

Theorem List for Metamath Proof Explorer - 15301-15400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcossub 15301 Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴𝐵)) = (((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))))

Theoremaddsin 15302 Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) + (sin‘𝐵)) = (2 · ((sin‘((𝐴 + 𝐵) / 2)) · (cos‘((𝐴𝐵) / 2)))))

Theoremsubsin 15303 Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) − (sin‘𝐵)) = (2 · ((cos‘((𝐴 + 𝐵) / 2)) · (sin‘((𝐴𝐵) / 2)))))

Theoremsinmul 15304 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 15297 and cossub 15301. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴𝐵)) − (cos‘(𝐴 + 𝐵))) / 2))

Theoremcosmul 15305 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 15297 and cossub 15301. (Contributed by David A. Wheeler, 26-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) · (cos‘𝐵)) = (((cos‘(𝐴𝐵)) + (cos‘(𝐴 + 𝐵))) / 2))

Theoremaddcos 15306 Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) + (cos‘𝐵)) = (2 · ((cos‘((𝐴 + 𝐵) / 2)) · (cos‘((𝐴𝐵) / 2)))))

Theoremsubcos 15307 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‘((𝐴𝐵) / 2)))))

Theoremsincossq 15308 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)

Theoremsin2t 15309 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘(2 · 𝐴)) = (2 · ((sin‘𝐴) · (cos‘𝐴))))

Theoremcos2t 15310 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1))

Theoremcos2tsin 15311 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))))

Theoremsinbnd 15312 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))

Theoremcosbnd 15313 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))

Theoremsinbnd2 15314 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))

Theoremcosbnd2 15315 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))

Theoremef01bndlem 15316* Lemma for sin01bnd 15317 and cos01bnd 15318. (Contributed by Paul Chapman, 19-Jan-2008.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))       (𝐴 ∈ (0(,]1) → (abs‘Σ𝑘 ∈ (ℤ‘4)(𝐹𝑘)) < ((𝐴↑4) / 6))

Theoremsin01bnd 15317 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))

Theoremcos01bnd 15318 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))))

Theoremcos1bnd 15319 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3))

Theoremcos2bnd 15320 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9))

Theoremsinltx 15321 The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴)

Theoremsin01gt0 15322 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))

Theoremcos01gt0 15323 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))

Theoremsin02gt0 15324 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))

Theoremsincos1sgn 15325 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sin‘1) ∧ 0 < (cos‘1))

Theoremsincos2sgn 15326 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
(0 < (sin‘2) ∧ (cos‘2) < 0)

Theoremsin4lt0 15327 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
(sin‘4) < 0

Theoremabsefi 15328 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)

Theoremabsef 15329 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)))

Theoremabsefib 15330 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))

Theoremefieq1re 15331 A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
((𝐴 ∈ ℂ ∧ (exp‘(i · 𝐴)) = 1) → 𝐴 ∈ ℝ)

Theoremdemoivre 15332 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 15333 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)))))

TheoremdemoivreALT 15333 Alternate proof of demoivre 15332. 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)

Syntaxctau 15334 Extend class notation to include the constant tau, τ = 6.28318....
class τ

Definitiondf-tau 15335 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 . 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

Theoremeirrlem 15336* Lemma for eirr 15337. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))    &   (𝜑𝑃 ∈ ℤ)    &   (𝜑𝑄 ∈ ℕ)    &   (𝜑 → e = (𝑃 / 𝑄))        ¬ 𝜑

Theoremeirr 15337 e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
e ∉ ℚ

Theoremegt2lt3 15338 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)

Theoremepos 15339 Euler's constant e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
0 < e

Theoremepr 15340 Euler's constant e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
e ∈ ℝ+

Theoremene0 15341 e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
e ≠ 0

Theoremene1 15342 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

Theoremxpnnen 15343 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

TheoremznnenlemOLD 15344 Obsolete as of 6-Sep-2022. Used to be a lemma for znnen 15345. (Contributed by NM, 31-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(((0 ≤ 𝑥 ∧ ¬ 0 ≤ 𝑦) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 = 𝑦 ↔ (2 · 𝑥) = ((-2 · 𝑦) + 1)))

Theoremznnen 15345 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

Theoremqnnen 15346 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

Theoremrpnnen2lem1 15347* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       ((𝐴 ⊆ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐹𝐴)‘𝑁) = if(𝑁𝐴, ((1 / 3)↑𝑁), 0))

Theoremrpnnen2lem2 15348* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       (𝐴 ⊆ ℕ → (𝐹𝐴):ℕ⟶ℝ)

Theoremrpnnen2lem3 15349* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2)

Theoremrpnnen2lem4 15350* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       ((𝐴𝐵𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹𝐴)‘𝑘) ∧ ((𝐹𝐴)‘𝑘) ≤ ((𝐹𝐵)‘𝑘)))

Theoremrpnnen2lem5 15351* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → seq𝑀( + , (𝐹𝐴)) ∈ dom ⇝ )

Theoremrpnnen2lem6 15352* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → Σ𝑘 ∈ (ℤ𝑀)((𝐹𝐴)‘𝑘) ∈ ℝ)

Theoremrpnnen2lem7 15353* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       ((𝐴𝐵𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → Σ𝑘 ∈ (ℤ𝑀)((𝐹𝐴)‘𝑘) ≤ Σ𝑘 ∈ (ℤ𝑀)((𝐹𝐵)‘𝑘))

Theoremrpnnen2lem8 15354* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → Σ𝑘 ∈ ℕ ((𝐹𝐴)‘𝑘) = (Σ𝑘 ∈ (1...(𝑀 − 1))((𝐹𝐴)‘𝑘) + Σ𝑘 ∈ (ℤ𝑀)((𝐹𝐴)‘𝑘)))

Theoremrpnnen2lem9 15355* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       (𝑀 ∈ ℕ → Σ𝑘 ∈ (ℤ𝑀)((𝐹‘(ℕ ∖ {𝑀}))‘𝑘) = (0 + (((1 / 3)↑(𝑀 + 1)) / (1 − (1 / 3)))))

Theoremrpnnen2lem10 15356* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑𝐵 ⊆ ℕ)    &   (𝜑𝑚 ∈ (𝐴𝐵))    &   (𝜑 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛𝐴𝑛𝐵)))    &   (𝜓 ↔ Σ𝑘 ∈ ℕ ((𝐹𝐴)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹𝐵)‘𝑘))       ((𝜑𝜓) → Σ𝑘 ∈ (ℤ𝑚)((𝐹𝐴)‘𝑘) = Σ𝑘 ∈ (ℤ𝑚)((𝐹𝐵)‘𝑘))

Theoremrpnnen2lem11 15357* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑𝐵 ⊆ ℕ)    &   (𝜑𝑚 ∈ (𝐴𝐵))    &   (𝜑 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛𝐴𝑛𝐵)))    &   (𝜓 ↔ Σ𝑘 ∈ ℕ ((𝐹𝐴)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹𝐵)‘𝑘))       (𝜑 → ¬ 𝜓)

Theoremrpnnen2lem12 15358* Lemma for rpnnen2 15359. (Contributed by Mario Carneiro, 13-May-2013.)
𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛𝑥, ((1 / 3)↑𝑛), 0)))       𝒫 ℕ ≼ (0[,]1)

Theoremrpnnen2 15359 The other half of rpnnen 15360, 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... 15016). 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 Σ𝑘𝐴(3↑-𝑘) = Σ𝑘 ∈ ℕ((𝐹𝐴)‘𝑘), where ((𝐹𝐴)‘𝑘) = if(𝑘𝐴, (3↑-𝑘), 0)) (rpnnen2lem1 15347). This is an infinite sum of real numbers (rpnnen2lem2 15348), and since 𝐴𝐵 implies (𝐹𝐴) ≤ (𝐹𝐵) (rpnnen2lem4 15350) and (𝐹‘ℕ) converges to 1 / 2 (rpnnen2lem3 15349) by geoisum1 15014, the sum is convergent to some real (rpnnen2lem5 15351 and rpnnen2lem6 15352) by the comparison test for convergence cvgcmp 14952. The comparison test also tells us that 𝐴𝐵 implies Σ(𝐹𝐴) ≤ Σ(𝐹𝐵) (rpnnen2lem7 15353).

Putting it all together, if we have two sets 𝑥𝑦, there must differ somewhere, and so there must be an 𝑚 such that 𝑛 < 𝑚(𝑛𝑥𝑛𝑦) but 𝑚 ∈ (𝑥𝑦) or vice versa. In this case, we split off the first 𝑚 − 1 terms (rpnnen2lem8 15354) and cancel them (rpnnen2lem10 15356), since these are the same for both sets. For the remaining terms, we use the subset property to establish that Σ(𝐹𝑦) ≤ Σ(𝐹‘(ℕ ∖ {𝑚})) and Σ(𝐹‘{𝑚}) ≤ Σ(𝐹𝑥) (where these sums are only over (ℤ𝑚)), and since Σ(𝐹‘(ℕ ∖ {𝑚})) = (3↑-𝑚) / 2 (rpnnen2lem9 15355) and Σ(𝐹‘{𝑚}) = (3↑-𝑚), we establish that Σ(𝐹𝑦) < Σ(𝐹𝑥) (rpnnen2lem11 15357) so that they must be different. By contraposition (rpnnen2lem12 15358), 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)

Theoremrpnnen 15360 The cardinality of the continuum is the same as the powerset of ω. This is a stronger statement than ruc 15376, which only asserts that is uncountable, i.e. has a cardinality larger than ω. The main proof is in two parts, rpnnen1 12130 and rpnnen2 15359, each showing an injection in one direction, and this last part uses sbth 8368 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

Theoremrexpen 15361 The real numbers are equinumerous to their own Cartesian product, even though it is not necessarily true that is well-orderable (so we cannot use infxpidm2 9173 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)
(ℝ × ℝ) ≈ ℝ

Theoremcpnnen 15362 The complex numbers are equinumerous to the powerset of the positive integers. (Contributed by Mario Carneiro, 16-Jun-2013.)
ℂ ≈ 𝒫 ℕ

TheoremrucALT 15363 Alternate proof of ruc 15376. This proof is a simple corollary of rpnnen 15360, which determines the exact cardinality of the reals. For an alternate proof discussed at mmcomplex.html#uncountable, see ruc 15376. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℕ ≺ ℝ

Theoremruclem1 15364* Lemma for ruc 15376 (the reals are uncountable). Substitutions for the function 𝐷. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   𝑋 = (1st ‘(⟨𝐴, 𝐵𝐷𝑀))    &   𝑌 = (2nd ‘(⟨𝐴, 𝐵𝐷𝑀))       (𝜑 → ((⟨𝐴, 𝐵𝐷𝑀) ∈ (ℝ × ℝ) ∧ 𝑋 = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)) ∧ 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵)))

Theoremruclem2 15365* Lemma for ruc 15376. Ordering property for the input to 𝐷. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   𝑋 = (1st ‘(⟨𝐴, 𝐵𝐷𝑀))    &   𝑌 = (2nd ‘(⟨𝐴, 𝐵𝐷𝑀))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴𝑋𝑋 < 𝑌𝑌𝐵))

Theoremruclem3 15366* Lemma for ruc 15376. The constructed interval [𝑋, 𝑌] always excludes 𝑀. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   𝑋 = (1st ‘(⟨𝐴, 𝐵𝐷𝑀))    &   𝑌 = (2nd ‘(⟨𝐴, 𝐵𝐷𝑀))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝑀 < 𝑋𝑌 < 𝑀))

Theoremruclem4 15367* Lemma for ruc 15376. Initial value of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       (𝜑 → (𝐺‘0) = ⟨0, 1⟩)

Theoremruclem6 15368* Lemma for ruc 15376. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       (𝜑𝐺:ℕ0⟶(ℝ × ℝ))

Theoremruclem7 15369* Lemma for ruc 15376. Successor value for the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       ((𝜑𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺𝑁)𝐷(𝐹‘(𝑁 + 1))))

Theoremruclem8 15370* Lemma for ruc 15376. The intervals of the 𝐺 sequence are all nonempty. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       ((𝜑𝑁 ∈ ℕ0) → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))

Theoremruclem9 15371* Lemma for ruc 15376. The first components of the 𝐺 sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))

Theoremruclem10 15372* Lemma for ruc 15376. 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.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (1st ‘(𝐺𝑀)) < (2nd ‘(𝐺𝑁)))

Theoremruclem11 15373* Lemma for ruc 15376. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))

Theoremruclem12 15374* Lemma for ruc 15376. The supremum of the increasing sequence 1st𝐺 is a real number that is not in the range of 𝐹. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)    &   𝑆 = sup(ran (1st𝐺), ℝ, < )       (𝜑𝑆 ∈ (ℝ ∖ ran 𝐹))

Theoremruclem13 15375 Lemma for ruc 15376. There is no function that maps onto . (Use nex 1844 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

Theoremruc 15376 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 15364 through ruclem13 15375 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 15375 for the function existence version of this theorem. For an informal discussion of this proof, see mmcomplex.html#uncountable. For an alternate proof see rucALT 15363. This is Metamath 100 proof #22. (Contributed by NM, 13-Oct-2004.)
ℕ ≺ ℝ

Theoremresdomq 15377 The set of rationals is strictly less equinumerous than the set of reals ( strictly dominates ). (Contributed by NM, 18-Dec-2004.)
ℚ ≺ ℝ

Theoremaleph1re 15378 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)
(ℵ‘1o) ≼ ℝ

Theoremaleph1irr 15379 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)
(ℵ‘1o) ≼ (ℝ ∖ ℚ)

Theoremcnso 15380 The complex numbers can be linearly ordered. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑥 𝑥 Or ℂ

PART 6  ELEMENTARY NUMBER THEORY

Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

6.1  Elementary properties of divisibility

6.1.1  Irrationality of square root of 2

Theoremsqrt2irrlem 15381 Lemma for sqrt2irr 15382. 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.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → (√‘2) = (𝐴 / 𝐵))       (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ))

Theoremsqrt2irr 15382 The square root of 2 is irrational. See zsqrtelqelz 15870 for a generalization to all non-square integers. The proof's core is proven in sqrt2irrlem 15381, 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/. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
(√‘2) ∉ ℚ

Theoremsqrt2re 15383 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
(√‘2) ∈ ℝ

Theoremsqrt2irr0 15384 The square root of 2 is an irrational number. (Contributed by AV, 23-Dec-2022.)
(√‘2) ∈ (ℝ ∖ ℚ)

6.1.2  Some Number sets are chains of proper subsets

Theoremnthruc 15385 The sequence , , , , and 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 but not , one-half belongs to but not , the square root of 2 belongs to but not , and finally that the imaginary number i belongs to but not . See nthruz 15386 for a further refinement. (Contributed by NM, 12-Jan-2002.)
((ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) ∧ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ))

Theoremnthruz 15386 The sequence , 0, and 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 0 but not and minus one belongs to but not 0. This theorem refines the chain of proper subsets nthruc 15385. (Contributed by NM, 9-May-2004.)
(ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ)

6.1.3  The divides relation

Syntaxcdvds 15387 Extend the definition of a class to include the divides relation. See df-dvds 15388.
class

Definitiondf-dvds 15388* 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)}

Theoremdivides 15389* Define the divides relation. 𝑀𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 27888). As proven in dvdsval3 15391, 𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 15389 and dvdsval2 15390 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))

Theoremdvdsval2 15390 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) → (𝑀𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))

Theoremdvdsval3 15391 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) → (𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0))

Theoremdvdszrcl 15392 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
(𝑋𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))

Theoremdvdsmod0 15393 If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.)
((𝑀 ∈ ℕ ∧ 𝑀𝑁) → (𝑁 mod 𝑀) = 0)

Theoremp1modz1 15394 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.)
((𝑀𝐴 ∧ 1 < 𝑀) → ((𝐴 + 1) mod 𝑀) = 1)

Theoremdvdsmodexp 15395 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 15893). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.)
((𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁𝐴) → ((𝐴𝐵) mod 𝑁) = (𝐴 mod 𝑁))

Theoremnndivdvds 15396 Strong form of dvdsval2 15390 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ))

Theoremnndivides 15397* Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℕ (𝑛 · 𝑀) = 𝑁))

Theoremmoddvds 15398 Two ways to say 𝐴𝐵 (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 ∥ (𝐴𝐵)))

Theoremdvds0lem 15399 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) → 𝑀𝑁)

Theoremdvds1lem 15400* 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))    &   ((𝜑𝑥 ∈ ℤ) → 𝑍 ∈ ℤ)    &   ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁))       (𝜑 → (𝐽𝐾𝑀𝑁))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43661
 Copyright terms: Public domain < Previous  Next >