If 𝑎 and 𝑏 are integers for which 𝑎|𝑏 and 𝑏|𝑎, then 𝑎=±𝑏.

Question

If 𝑎 and 𝑏 are integers for which 𝑎|𝑏 and 𝑏|𝑎, then 𝑎=±𝑏. 

Answer 1

PROOF REASONS

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1. 𝑎 and 𝑏 are integers for which 𝑎|𝑏 and 𝑏|𝑎 (premise)

2. a is an integer (1,simp rule)

3. b is an integer (1,simp rule)

4. a | b (1,simp rule)

5. b | a (1,simp rule)

6. b = ak for some k in Z (4, definition)

7. a = bk for some k in Z (5, definition)

8. b/k = a for some k in Z (6, algebra)

9. a/k = b for some k in Z (7, algebra)

10. b/i = a for a particular i in Z (8, EI rule)

11. a/j = b for a particular j in Z (9, EI rule)

12. (a/j)/i = a (10,11 subst.rule)

13. a/(ji) = a (12, algebra)

14. a = a(ji) (13, algebra)

15. a/a = ji (14, algebra)

16. 1 = ji (15, algebra)

17. j is an integer (10, definition)

18. i is an integer (11, definition)

19. (1 = -1 x -1) and (1 = 1 x 1) (THEOREM)

20. (1 = -1 x -1) (19, simp rule)

21. (1 = 1 x 1) (19, simp rule)

22. 1 x 1 = ji (16,21,transitive)

23. (1 x 1)/i = j (22, algebra)

24. 1/i = j (23, algebra)

25. ------| not (i = 1 or i = -1) (subpremise)

26. ------| i != 1 and i != -1 (25, demorgans law)

27. ------|(i = 0 (26, equivalence)

------| or i is postive but not 1

------| or i is negative but not 1)

28. ------|-----| i = 0 (sub sub premise)

29. ------|-----| 1 x 1 = j x 0 (22,28, subst.rule)

30. ------|-----| 1 x 1 = 0 (29, algebra)

31. ------|-----| 1 = 0 (30, algebra)

32. ------|-----| FALSE (31, CONTRADICTON)

33. ------| not(i = 0) (28 - 32, IP rule)

34. ------|-----| i is positive but not 1 (sub sub premise)

35. ------|-----| 1/i is not an integer (34, definition of Z)

36. ------|-----| 1/i = j states that (24,35 reanalysis)

------|-----| j is not an integer

37. ------|-----| FALSE (17,36 CONTRADICTION)

38. ------| not(i is positive but not 1) (34-37, IP rule)

39. ------| (i is postive but not 1 (27,33, DS rule)

------| or i is negative but not 1)

40. ------| i is negative but not 1 (38,39, DS rule)

41. ------| 1/i is not an integer (34, definition of Z)

42. ------|(1/i = j states that (24,41 reanalysis)

------| j is not an integer)

43. ------| FALSE (17,42, CONTRADICTION)

44. not(not (i = 1 or i = -1) ) (25-27,33,38-43 IP rule)

45. i = 1 or i = -1 (44, double negation)

46. ------| i = 1 (subpremise)

47. ------| b/1 = a (10,46 subst. rule)

48. ------| b = a (47, algebra)

49. ------| a = b (48, symmetric )

50. if i = 1 then a = b (46-49 CP rule)

51. ------| i = -1 (subpremise)

52. ------| b/-1 = a (10,51 subst. rule)

53. ------| -b = a (52, algebra)

54. ------| a = -b (53, symmetric)

55. if i = -1 then a = -b (51-54 CP rule)

56. (a = b) or (a = -b) (45,50,55, CD rule)

57. a = ±b (56, definition of ±)

QED