PA003N

prime_not_divides_coprime

Stable checked-use theorem · independently closed

A prime not dividing a natural is coprime to that natural.

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.

Exact expanded PA statement

forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1

Structural proof guide

Generated structural guide

A prime not dividing a natural is coprime to that natural.

Use the direct prerequisites prime_coprime_or_divides as previously established PA formulas.

The proof proceeds by case analysis (1), intermediate claims (1).

Referenced ingredients

PA003M prime_coprime_or_divides

Proof neighborhood

Direct dependencies

PA003M prime_coprime_or_divides

Direct dependents

PA003S prime_mod_cancel PA0062 prime_mod_inverse PA0082 prime_positive_bounded_product_coprime PA008K prime_range_product_coprime

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

14 script commands · 7 reading checkpoints · 1 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 (1)
prime coprime or divides
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro hp
  4. L4
    intro hnot
02Use earlier factsL5–6

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

  1. L5
    specialize prime_coprime_or_divides p
  2. L6
    specialize prime_coprime_or_divides a
03Establish hsplitL7–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime coprime or divides.

  1. L7
    have hsplit : (forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1) \/ exists k. a = p * k
  2. L8
    apply prime_coprime_or_divides
  3. L9
    exact hp
04Separate the logical casesL10–10

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

  1. L10
    cases hsplit
05Use earlier factsL11–11

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

  1. L11
    exact hsplit_left
06Separate the logical casesL12–12

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

  1. L12
    exfalso
07Use earlier factsL13–14

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

  1. L13
    apply hnot
  2. L14
    exact hsplit_right

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro hnot
  5. 0005specialize prime_coprime_or_divides p
  6. 0006specialize prime_coprime_or_divides a
  7. 0007have hsplit : (forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1) \/ exists k. a = p * k
  8. 0008apply prime_coprime_or_divides
  9. 0009exact hp
  10. 0010cases hsplit
  11. 0011exact hsplit_left
  12. 0012exfalso
  13. 0013apply hnot
  14. 0014exact hsplit_right