BT008R · Bertrand theorem

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.

Statement with defined notation

∀ p. ∀ a. Prime(p) → ¬Dvd(p,a) → Coprime(p,a)

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0003 Dvd PD0004 Prime PD0005 Coprime

3 occurrences

In local proof propositions

PD0003 Dvd PD0005 Coprime

2 occurrences

Exact expanded native-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

Proof neighborhood

Direct theorem prerequisites

BT008Q prime_coprime_or_divides

Direct theorem dependents

BT008S distinct_primes_coprime

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

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.

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 (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 : Coprime(p,a) ∨ Dvd(p,a)Definitions: Coprime(p,a)Dvd(p,a)Original native command in the exact edition
  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 defined 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 : Coprime(p,a) ∨ Dvd(p,a)
    Exact native replay linehave 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