BT008R

prime_not_divides_coprime

Stable checked-use theorem · independently kernel verified

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

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

Direct prerequisites: prime_coprime_or_divides. The authored body proceeds by case analysis (1), intermediate claims (1).

Proof neighborhood

Direct dependencies

BT008Q prime_coprime_or_divides

Direct dependents

BT008S distinct_primes_coprime

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

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