PA008N · theorem

scaled_inverse_from_unit_inverse

Alpha v34 checked-use theorem · independently closed; not Stable

Multiplying an ordinary inverse by the target gives a scaled inverse.

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 original first-admission records.

Statement with defined notation

∀ p. ∀ a. ∀ x. ∀ z. BalancedInverse(p,x,z) → ModEq(p,x · (a · z),a)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0008 ModEq PD0031 BalancedInverse

2 occurrences

In local proof propositions

PD0008 ModEq

1 occurrences

Exact expanded native-PA statement
forall p a x z. (exists esi_mod_left_from_unit_source esi_mod_right_from_unit_source. (x * z) + p * esi_mod_left_from_unit_source = (1) + p * esi_mod_right_from_unit_source) -> (exists esi_mod_left_from_unit_result esi_mod_right_from_unit_result. (x * (a * z)) + p * esi_mod_left_from_unit_result = (a) + p * esi_mod_right_from_unit_result)

Proof neighborhood

Direct theorem prerequisites

PA0020 mod_eq_mul_left PA000B mul_assoc PA000H mul_comm PA0002 mul_one

Direct theorem dependents

PA008Q prime_scaled_inverse_exists

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

33 script commands · 6 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (4)
mod eq mul leftmul assocmul commmul one
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro x
  4. L4
    intro z
  5. L5
    intro hxz
02Establish hscaledL6–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq mul left.

  1. L6
    have hscaled : ModEq(p,a · (x · z),a · 1)Definitions: ModEq(p,a · (x · z),a · 1)Original native command in the exact edition
  2. L7
    specialize mod_eq_mul_left p
  3. L8
    specialize mod_eq_mul_left (x * z)
  4. L9
    specialize mod_eq_mul_left 1
  5. L10
    specialize mod_eq_mul_left a
  6. L11
    apply mod_eq_mul_left
  7. L12
    exact hxz
03Establish hleftL13–22

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul assoc.

  1. L13
    have hleft : a * (x * z) = x * (a * z)
  2. L14
    trans (a * x) * z
  3. L15
    symm
  4. L16
    specialize mul_assoc a
  5. L17
    specialize mul_assoc x
  6. L18
    specialize mul_assoc z
  7. L19
    apply mul_assoc
  8. L20
    trans (x * a) * z
  9. L21
    congr
  10. L22
    apply mul_comm
04Calculate and transport equalitiesL23–23

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

  1. L23
    refl
05Use earlier factsL24–27

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

  1. L24
    specialize mul_assoc x
  2. L25
    specialize mul_assoc a
  3. L26
    specialize mul_assoc z
  4. L27
    apply mul_assoc
06Establish hrightL28–33

Establish this local claim before using it. It is not an additional assumption.

  1. L28
    have hright : a * 1 = a
  2. L29
    specialize mul_one a
  3. L30
    exact mul_one
  4. L31
    rewrite hleft at hscaled
  5. L32
    rewrite hright at hscaled
  6. L33
    exact hscaled

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro x
  4. 0004intro z
  5. 0005intro hxz
  6. 0006have hscaled : ModEq(p,a · (x · z),a · 1)
    Exact native replay linehave hscaled : exists esi_mod_left_from_unit_scaled esi_mod_right_from_unit_scaled. (a * (x * z)) + p * esi_mod_left_from_unit_scaled = (a * 1) + p * esi_mod_right_from_unit_scaled
  7. 0007specialize mod_eq_mul_left p
  8. 0008specialize mod_eq_mul_left (x * z)
  9. 0009specialize mod_eq_mul_left 1
  10. 0010specialize mod_eq_mul_left a
  11. 0011apply mod_eq_mul_left
  12. 0012exact hxz
  13. 0013have hleft : a * (x * z) = x * (a * z)
  14. 0014trans (a * x) * z
  15. 0015symm
  16. 0016specialize mul_assoc a
  17. 0017specialize mul_assoc x
  18. 0018specialize mul_assoc z
  19. 0019apply mul_assoc
  20. 0020trans (x * a) * z
  21. 0021congr
  22. 0022apply mul_comm
  23. 0023refl
  24. 0024specialize mul_assoc x
  25. 0025specialize mul_assoc a
  26. 0026specialize mul_assoc z
  27. 0027apply mul_assoc
  28. 0028have hright : a * 1 = a
  29. 0029specialize mul_one a
  30. 0030exact mul_one
  31. 0031rewrite hleft at hscaled
  32. 0032rewrite hright at hscaled
  33. 0033exact hscaled