Home Metamath Proof ExplorerTheorem List (p. 115 of 425) < 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 (1-26947) Hilbert Space Explorer (26948-28472) Users' Mathboxes (28473-42426)

Theorem List for Metamath Proof Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem8t7e56 11401 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 7) = 56

Theorem8t8e64 11402 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 8) = 64

Theorem9t2e18 11403 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 2) = 18

Theorem9t3e27 11404 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 3) = 27

Theorem9t4e36 11405 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 4) = 36

Theorem9t5e45 11406 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 5) = 45

Theorem9t6e54 11407 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 6) = 54

Theorem9t7e63 11408 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 7) = 63

Theorem9t8e72 11409 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 8) = 72

Theorem9t9e81 11410 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 9) = 81

Theorem9t11e99 11411 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
(9 · 11) = 99

Theorem9t11e99OLD 11412 Obsolete proof of 9t11e99 11411 as of 6-Sep-2021. (Contributed by AV, 14-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
(9 · 11) = 99

Theorem9lt10 11413 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
9 < 10

Theorem8lt10 11414 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
8 < 10

Theorem7lt10 11415 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
7 < 10

Theorem6lt10 11416 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
6 < 10

Theorem5lt10 11417 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
5 < 10

Theorem4lt10 11418 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
4 < 10

Theorem3lt10 11419 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
3 < 10

Theorem2lt10 11420 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
2 < 10

Theorem1lt10 11421 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
1 < 10

Theoremdecbin0 11422 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       (4 · 𝐴) = (2 · (2 · 𝐴))

Theoremdecbin2 11423 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))

Theoremdecbin3 11424 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1)

Theoremhalfthird 11425 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
((1 / 2) − (1 / 3)) = (1 / 6)

Theorem5recm6rec 11426 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
((1 / 5) − (1 / 6)) = (1 / 30)

5.4.10  Upper sets of integers

Syntaxcuz 11427 Extend class notation with the upper integer function. Read "𝑀 " as "the set of integers greater than or equal to 𝑀."
class

Definitiondf-uz 11428* Define a function whose value at 𝑗 is the semi-infinite set of contiguous integers starting at 𝑗, which we will also call the upper integers starting at 𝑗. Read "𝑀 " as "the set of integers greater than or equal to 𝑀." See uzval 11429 for its value, uzssz 11447 for its relationship to , nnuz 11463 and nn0uz 11462 for its relationships to and 0, and eluz1 11431 and eluz2 11433 for its membership relations. (Contributed by NM, 5-Sep-2005.)
= (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})

Theoremuzval 11429* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ ℤ → (ℤ𝑁) = {𝑘 ∈ ℤ ∣ 𝑁𝑘})

Theoremuzf 11430 The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
:ℤ⟶𝒫 ℤ

Theoremeluz1 11431 Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
(𝑀 ∈ ℤ → (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁)))

Theoremeluzel2 11432 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)

Theoremeluz2 11433 Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show 𝑀 ∈ ℤ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))

Theoremeluzmn 11434 Membership in an earlier upper set of integers. (Contributed by Thierry Arnoux, 8-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑀 ∈ (ℤ‘(𝑀𝑁)))

Theoremeluz1i 11435 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
𝑀 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁))

Theoremeluzuzle 11436 An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
((𝐵 ∈ ℤ ∧ 𝐵𝐴) → (𝐶 ∈ (ℤ𝐴) → 𝐶 ∈ (ℤ𝐵)))

Theoremeluzelz 11437 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)

Theoremeluzelre 11438 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℝ)

Theoremeluzelcn 11439 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℂ)

Theoremeluzle 11440 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → 𝑀𝑁)

Theoremeluz 11441 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))

Theoremuzid 11442 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
(𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))

Theoremuzn0 11443 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
(𝑀 ∈ ran ℤ𝑀 ≠ ∅)

Theoremuztrn 11444 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
((𝑀 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝑀 ∈ (ℤ𝑁))

Theoremuztrn2 11445 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝐾)       ((𝑁𝑍𝑀 ∈ (ℤ𝑁)) → 𝑀𝑍)

Theoremuzneg 11446 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
(𝑁 ∈ (ℤ𝑀) → -𝑀 ∈ (ℤ‘-𝑁))

Theoremuzssz 11447 An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(ℤ𝑀) ⊆ ℤ

Theoremuzss 11448 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))

Theoremuztric 11449 Totality of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ∨ 𝑀 ∈ (ℤ𝑁)))

Theoremuz11 11450 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
(𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝑁) ↔ 𝑀 = 𝑁))

Theoremeluzp1m1 11451 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ𝑀))

Theoremeluzp1l 11452 Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑁)

Theoremeluzp1p1 11453 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ‘(𝑀 + 1)))

Theoremeluzaddi 11454 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) → (𝑁 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))

Theoremeluzsubi 11455 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ‘(𝑀 + 𝐾)) → (𝑁𝐾) ∈ (ℤ𝑀))

Theoremeluzadd 11456 Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))

Theoremeluzsub 11457 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝑁𝐾) ∈ (ℤ𝑀))

Theoremuzm1 11458 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ𝑀)))

Theoremuznn0sub 11459 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁𝑀) ∈ ℕ0)

Theoremuzin 11460 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((ℤ𝑀) ∩ (ℤ𝑁)) = (ℤ‘if(𝑀𝑁, 𝑁, 𝑀)))

Theoremuzp1 11461 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 = 𝑀𝑁 ∈ (ℤ‘(𝑀 + 1))))

Theoremnn0uz 11462 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
0 = (ℤ‘0)

Theoremnnuz 11463 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
ℕ = (ℤ‘1)

Theoremelnnuz 11464 A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))

Theoremelnn0uz 11465 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))

Theoremeluz2nn 11466 An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
(𝐴 ∈ (ℤ‘2) → 𝐴 ∈ ℕ)

Theoremeluzge2nn0 11467 If an integer is greater than or equal to 2, then it is a nonnegative integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV, 3-Nov-2018.)
(𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ0)

Theoremeluz2n0 11468 An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
(𝑁 ∈ (ℤ‘2) → 𝑁 ≠ 0)

Theoremuzuzle23 11469 An integer in the upper set of integers starting at 3 is element of the upper set of integers starting at 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝐴 ∈ (ℤ‘3) → 𝐴 ∈ (ℤ‘2))

Theoremeluzge3nn 11470 If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 ∈ (ℤ‘3) → 𝑁 ∈ ℕ)

Theoremuz3m2nn 11471 An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 11503. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 ∈ (ℤ‘3) → (𝑁 − 2) ∈ ℕ)

Theorem1eluzge0 11472 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 ∈ (ℤ‘0)

Theorem2eluzge0 11473 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
2 ∈ (ℤ‘0)

Theorem2eluzge1 11474 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 ∈ (ℤ‘1)

Theoremuznnssnn 11475 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
(𝑁 ∈ ℕ → (ℤ𝑁) ⊆ ℕ)

Theoremraluz 11476* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremraluz2 11477* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremrexuz 11478* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremrexuz2 11479* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theorem2rexuz 11480* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(∃𝑚𝑛 ∈ (ℤ𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚𝑛𝜑))

Theorempeano2uz 11481 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ𝑀))

Theorempeano2uzs 11482 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (𝑁 + 1) ∈ 𝑍)

Theorempeano2uzr 11483 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑁 ∈ (ℤ𝑀))

Theoremuzaddcl 11484 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ𝑀))

Theoremnn0pzuz 11485 The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
((𝑁 ∈ ℕ0𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ𝑍))

Theoremuzind4 11486* Induction on the upper set of integers that starts at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.)
(𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   (𝑀 ∈ ℤ → 𝜓)    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))       (𝑁 ∈ (ℤ𝑀) → 𝜏)

Theoremuzind4ALT 11487* Induction on the upper set of integers that starts at an integer 𝑀. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 11486 or uzind4ALT 11487 may be used; see comment for nnind 10793. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑀 ∈ ℤ → 𝜓)    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))    &   (𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))       (𝑁 ∈ (ℤ𝑀) → 𝜏)

Theoremuzind4s 11488* Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)
(𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑)    &   (𝑘 ∈ (ℤ𝑀) → (𝜑[(𝑘 + 1) / 𝑘]𝜑))       (𝑁 ∈ (ℤ𝑀) → [𝑁 / 𝑘]𝜑)

Theoremuzind4s2 11489* Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 11488 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.)
(𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)    &   (𝑘 ∈ (ℤ𝑀) → ([𝑘 / 𝑗]𝜑[(𝑘 + 1) / 𝑗]𝜑))       (𝑁 ∈ (ℤ𝑀) → [𝑁 / 𝑗]𝜑)

Theoremuzind4i 11490* Induction on the upper integers that start at 𝑀. The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 4-Sep-2005.)
𝑀 ∈ ℤ    &   (𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   𝜓    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))       (𝑁 ∈ (ℤ𝑀) → 𝜏)

Theoremuzwo 11491* Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by NM, 8-Oct-2005.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗𝑆𝑘𝑆 𝑗𝑘)

Theoremuzwo2 11492* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. (Contributed by NM, 8-Oct-2005.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝑆 ≠ ∅) → ∃!𝑗𝑆𝑘𝑆 𝑗𝑘)

Theoremnnwo 11493* Well-ordering principle: any nonempty set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.)
((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremnnwof 11494* Well-ordering principle: any nonempty set of positive integers has a least element. This version allows 𝑥 and 𝑦 to be present in 𝐴 as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑦𝐴       ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremnnwos 11495* Well-ordering principle: any nonempty set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥 ∈ ℕ 𝜑 → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓𝑥𝑦)))

Theoremindstr 11496* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑))       (𝑥 ∈ ℕ → 𝜑)

Theoremeluznn0 11497 Membership in a nonnegative upper set of integers implies membership in 0. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ0)

Theoremeluznn 11498 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ)

Theoremeluz2b1 11499 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
(𝑁 ∈ (ℤ‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))

Theoremeluz2gt1 11500 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
(𝑁 ∈ (ℤ‘2) → 1 < 𝑁)

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 42401-42426
 Copyright terms: Public domain < Previous  Next >