EU0019

euler_nonunit_factor_unchanged_congruence

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

A nonunit index and its image both contribute one, so neither adds to the exponent.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic statement

forall a m i r u v. (forall eut_divisor_eu_no_scale_multiplier. (exists eut_left_eu_no_scale_multiplier. (a) = eut_divisor_eu_no_scale_multiplier * eut_left_eu_no_scale_multiplier) -> (exists eut_right_eu_no_scale_multiplier. (m) = eut_divisor_eu_no_scale_multiplier * eut_right_eu_no_scale_multiplier) -> eut_divisor_eu_no_scale_multiplier = 1) -> (exists eu_mod_left_no_scale_residue eu_mod_right_no_scale_residue. (a*i) + (m) * eu_mod_left_no_scale_residue = (r) + (m) * eu_mod_right_no_scale_residue) -> ((((forall eut_divisor_eu_no_scale_source_coprime. (exists eut_left_eu_no_scale_source_coprime. (i) = eut_divisor_eu_no_scale_source_coprime * eut_left_eu_no_scale_source_coprime) -> (exists eut_right_eu_no_scale_source_coprime. (m) = eut_divisor_eu_no_scale_source_coprime * eut_right_eu_no_scale_source_coprime) -> eut_divisor_eu_no_scale_source_coprime = 1) /\ (u)=(i)) \/ (~(forall eut_divisor_eu_no_scale_source_coprime. (exists eut_left_eu_no_scale_source_coprime. (i) = eut_divisor_eu_no_scale_source_coprime * eut_left_eu_no_scale_source_coprime) -> (exists eut_right_eu_no_scale_source_coprime. (m) = eut_divisor_eu_no_scale_source_coprime * eut_right_eu_no_scale_source_coprime) -> eut_divisor_eu_no_scale_source_coprime = 1) /\ (u)=1))) -> ((((forall eut_divisor_eu_no_scale_target_coprime. (exists eut_left_eu_no_scale_target_coprime. (r) = eut_divisor_eu_no_scale_target_coprime * eut_left_eu_no_scale_target_coprime) -> (exists eut_right_eu_no_scale_target_coprime. (m) = eut_divisor_eu_no_scale_target_coprime * eut_right_eu_no_scale_target_coprime) -> eut_divisor_eu_no_scale_target_coprime = 1) /\ (v)=(r)) \/ (~(forall eut_divisor_eu_no_scale_target_coprime. (exists eut_left_eu_no_scale_target_coprime. (r) = eut_divisor_eu_no_scale_target_coprime * eut_left_eu_no_scale_target_coprime) -> (exists eut_right_eu_no_scale_target_coprime. (m) = eut_divisor_eu_no_scale_target_coprime * eut_right_eu_no_scale_target_coprime) -> eut_divisor_eu_no_scale_target_coprime = 1) /\ (v)=1))) -> ~(forall eut_divisor_eu_no_scale_index. (exists eut_left_eu_no_scale_index. (i) = eut_divisor_eu_no_scale_index * eut_left_eu_no_scale_index) -> (exists eut_right_eu_no_scale_index. (m) = eut_divisor_eu_no_scale_index * eut_right_eu_no_scale_index) -> eut_divisor_eu_no_scale_index = 1) -> (exists eu_mod_left_no_scale_result eu_mod_right_no_scale_result. (u) + (m) * eu_mod_left_no_scale_result = (v) + (m) * eu_mod_right_no_scale_result)

Constructive proof overview

Generated structural guide

A nonunit index and its image both contribute one, so neither adds to the exponent.

The unchanged tactic script uses 3 declared prerequisites and contains 42 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

EU0002 euler_multiplier_coprime_iff EU0010 euler_unit_product_factor_nonunit_value mod_eq_refl Stable theorem; checked-use authorized

Direct dependents

EU001B euler_unit_product_reindex_scale

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

42 script commands · 8 reading checkpoints · 3 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Named ingredients (2)
euler multiplier coprime iffeuler unit product factor nonunit value

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–10

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro i
  4. L4
    intro r
  5. L5
    intro u
  6. L6
    intro v
  7. L7
    intro ha
  8. L8
    intro hmod
  9. L9
    intro hu
  10. L10
    intro hv
02Fix variables and assumptionsL11–11

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro hi
03Establish hequivL12–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply euler multiplier coprime iff.

  1. L12
    have hequiv : (Coprime(i,m) → Coprime(r,m)) ∧ (Coprime(r,m) → Coprime(i,m))Definitions: Coprime
  2. L13
    specialize euler_multiplier_coprime_iff (a)
  3. L14
    specialize euler_multiplier_coprime_iff (m)
  4. L15
    specialize euler_multiplier_coprime_iff (i)
  5. L16
    specialize euler_multiplier_coprime_iff (r)
  6. L17
    apply euler_multiplier_coprime_iff
  7. L18
    exact ha
  8. L19
    exact hmod
04Separate the logical casesL20–20

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L20
    cases hequiv
05Establish heL21–27

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply euler unit product factor nonunit value.

  1. L21
    have he : u=1
  2. L22
    specialize euler_unit_product_factor_nonunit_value (m)
  3. L23
    specialize euler_unit_product_factor_nonunit_value (i)
  4. L24
    specialize euler_unit_product_factor_nonunit_value (u)
  5. L25
    apply euler_unit_product_factor_nonunit_value
  6. L26
    exact hi
  7. L27
    exact hu
06Establish hfL28–37

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply euler unit product factor nonunit value.

  1. L28
    have hf : v=1
  2. L29
    specialize euler_unit_product_factor_nonunit_value (m)
  3. L30
    specialize euler_unit_product_factor_nonunit_value (r)
  4. L31
    specialize euler_unit_product_factor_nonunit_value (v)
  5. L32
    apply euler_unit_product_factor_nonunit_value
  6. L33
    intro hr
  7. L34
    apply hi
  8. L35
    apply hequiv_right
  9. L36
    exact hr
  10. L37
    exact hv
07Calculate and transport equalitiesL38–39

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L38
    rewrite he
  2. L39
    rewrite hf
08Use earlier factsL40–42

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L40
    specialize mod_eq_refl (m)
  2. L41
    specialize mod_eq_refl (1)
  3. L42
    apply mod_eq_refl

Library-wide reading audit

Original exact command ledger · 42 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro i
  4. 0004intro r
  5. 0005intro u
  6. 0006intro v
  7. 0007intro ha
  8. 0008intro hmod
  9. 0009intro hu
  10. 0010intro hv
  11. 0011intro hi
  12. 0012have hequiv : ((forall eut_divisor_eu_no_scale_left. (exists eut_left_eu_no_scale_left. (i) = eut_divisor_eu_no_scale_left * eut_left_eu_no_scale_left) -> (exists eut_right_eu_no_scale_left. (m) = eut_divisor_eu_no_scale_left * eut_right_eu_no_scale_left) -> eut_divisor_eu_no_scale_left = 1) -> (forall eut_divisor_eu_no_scale_right. (exists eut_left_eu_no_scale_right. (r) = eut_divisor_eu_no_scale_right * eut_left_eu_no_scale_right) -> (exists eut_right_eu_no_scale_right. (m) = eut_divisor_eu_no_scale_right * eut_right_eu_no_scale_right) -> eut_divisor_eu_no_scale_right = 1)) /\ ((forall eut_divisor_eu_no_scale_right_back. (exists eut_left_eu_no_scale_right_back. (r) = eut_divisor_eu_no_scale_right_back * eut_left_eu_no_scale_right_back) -> (exists eut_right_eu_no_scale_right_back. (m) = eut_divisor_eu_no_scale_right_back * eut_right_eu_no_scale_right_back) -> eut_divisor_eu_no_scale_right_back = 1) -> (forall eut_divisor_eu_no_scale_left_back. (exists eut_left_eu_no_scale_left_back. (i) = eut_divisor_eu_no_scale_left_back * eut_left_eu_no_scale_left_back) -> (exists eut_right_eu_no_scale_left_back. (m) = eut_divisor_eu_no_scale_left_back * eut_right_eu_no_scale_left_back) -> eut_divisor_eu_no_scale_left_back = 1))
  13. 0013specialize euler_multiplier_coprime_iff (a)
  14. 0014specialize euler_multiplier_coprime_iff (m)
  15. 0015specialize euler_multiplier_coprime_iff (i)
  16. 0016specialize euler_multiplier_coprime_iff (r)
  17. 0017apply euler_multiplier_coprime_iff
  18. 0018exact ha
  19. 0019exact hmod
  20. 0020cases hequiv
  21. 0021have he : u=1
  22. 0022specialize euler_unit_product_factor_nonunit_value (m)
  23. 0023specialize euler_unit_product_factor_nonunit_value (i)
  24. 0024specialize euler_unit_product_factor_nonunit_value (u)
  25. 0025apply euler_unit_product_factor_nonunit_value
  26. 0026exact hi
  27. 0027exact hu
  28. 0028have hf : v=1
  29. 0029specialize euler_unit_product_factor_nonunit_value (m)
  30. 0030specialize euler_unit_product_factor_nonunit_value (r)
  31. 0031specialize euler_unit_product_factor_nonunit_value (v)
  32. 0032apply euler_unit_product_factor_nonunit_value
  33. 0033intro hr
  34. 0034apply hi
  35. 0035apply hequiv_right
  36. 0036exact hr
  37. 0037exact hv
  38. 0038rewrite he
  39. 0039rewrite hf
  40. 0040specialize mod_eq_refl (m)
  41. 0041specialize mod_eq_refl (1)
  42. 0042apply mod_eq_refl