BT004S · Bertrand theorem

coprime_mul_right

Stable checked-use theorem · independently kernel verified

Coprimality with a fixed left operand is closed under multiplication on the right.

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

∀ n. ∀ a. ∀ b. Coprime(n,a) → Coprime(n,b) → Coprime(n,a · b)

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

PD0005 Coprime

3 occurrences

In local proof propositions

PD0005 Coprime

3 occurrences

Exact expanded native-PA statement
forall n a b. (forall d. (exists x. n = d * x) -> (exists y. a = d * y) -> d = 1) -> (forall d. (exists x. n = d * x) -> (exists y. b = d * y) -> d = 1) -> forall d. (exists x. n = d * x) -> (exists y. a * b = d * y) -> d = 1

Proof neighborhood

Direct theorem prerequisites

BT004R coprime_mul_left BT002W coprime_symm

Direct theorem dependents

BT00YU coprime_power_right

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

26 script commands · 5 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 (2)
coprime mul leftcoprime symm
01Fix variables and assumptionsL1–5

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

  1. L1
    intro n
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro hna
  5. L5
    intro hnb
02Establish hanL6–10

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

  1. L6
  2. L7
    specialize coprime_symm n
  3. L8
    specialize coprime_symm a
  4. L9
    apply coprime_symm
  5. L10
    exact hna
03Establish hbnL11–15

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

  1. L11
  2. L12
    specialize coprime_symm n
  3. L13
    specialize coprime_symm b
  4. L14
    apply coprime_symm
  5. L15
    exact hnb
04Establish habnL16–25

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

  1. L16
    have habn : Coprime(a · b,n)Definitions: Coprime(a · b,n)Original native command in the exact edition
  2. L17
    specialize coprime_mul_left a
  3. L18
    specialize coprime_mul_left b
  4. L19
    specialize coprime_mul_left n
  5. L20
    apply coprime_mul_left
  6. L21
    exact han
  7. L22
    exact hbn
  8. L23
    specialize coprime_symm (a * b)
  9. L24
    specialize coprime_symm n
  10. L25
    apply coprime_symm
05Use earlier factsL26–26

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

  1. L26
    exact habn

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro n
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hna
  5. 0005intro hnb
  6. 0006have han : Coprime(a,n)
    Exact native replay linehave han : forall d. (exists x. a = d * x) -> (exists y. n = d * y) -> d = 1
  7. 0007specialize coprime_symm n
  8. 0008specialize coprime_symm a
  9. 0009apply coprime_symm
  10. 0010exact hna
  11. 0011have hbn : Coprime(b,n)
    Exact native replay linehave hbn : forall d. (exists x. b = d * x) -> (exists y. n = d * y) -> d = 1
  12. 0012specialize coprime_symm n
  13. 0013specialize coprime_symm b
  14. 0014apply coprime_symm
  15. 0015exact hnb
  16. 0016have habn : Coprime(a · b,n)
    Exact native replay linehave habn : forall d. (exists x. a * b = d * x) -> (exists y. n = d * y) -> d = 1
  17. 0017specialize coprime_mul_left a
  18. 0018specialize coprime_mul_left b
  19. 0019specialize coprime_mul_left n
  20. 0020apply coprime_mul_left
  21. 0021exact han
  22. 0022exact hbn
  23. 0023specialize coprime_symm (a * b)
  24. 0024specialize coprime_symm n
  25. 0025apply coprime_symm
  26. 0026exact habn