Metamath Proof Explorer


Table of Contents - 6.1.4. Even and odd numbers

The set of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4. Instead of defining new class variables and to represent these sets, we use the idiom to say that " is even" (which implies , see evenelz) and to say that " is odd" (under the assumption that ). The previously proven theorems about even and odd numbers, like zneo, zeo, zeo2, etc. use different representations, which are equivalent to the representations using the divides relation, see evend2 and oddp1d2. The corresponding theorems are zeneo, zeo3 and zeo4.

  1. evenelz
  2. zeo3
  3. zeo4
  4. zeneo
  5. odd2np1lem
  6. odd2np1
  7. even2n
  8. oddm1even
  9. oddp1even
  10. oexpneg
  11. mod2eq0even
  12. mod2eq1n2dvds
  13. oddnn02np1
  14. oddge22np1
  15. evennn02n
  16. evennn2n
  17. 2tp1odd
  18. mulsucdiv2z
  19. sqoddm1div8z
  20. 2teven
  21. zeo5
  22. evend2
  23. oddp1d2
  24. zob
  25. oddm1d2
  26. ltoddhalfle
  27. halfleoddlt
  28. opoe
  29. omoe
  30. opeo
  31. omeo
  32. z0even
  33. n2dvds1
  34. n2dvdsm1
  35. z2even
  36. n2dvds3
  37. z4even
  38. 4dvdseven
  39. m1expe
  40. m1expo
  41. m1exp1
  42. nn0enne
  43. nn0ehalf
  44. nnehalf
  45. nn0onn
  46. nn0o1gt2
  47. nno
  48. nn0o
  49. nn0ob
  50. nn0oddm1d2
  51. nnoddm1d2
  52. sumeven
  53. sumodd
  54. evensumodd
  55. oddsumodd
  56. pwp1fsum
  57. oddpwp1fsum