BT004S

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.

Exact expanded 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

Structural proof guide

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

Direct prerequisites: coprime_mul_left, coprime_symm. The authored body proceeds by intermediate claims (3).

Proof neighborhood

Direct dependencies

BT004R coprime_mul_left BT002W coprime_symm

Direct dependents

BT00YU coprime_power_right

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

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.

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
    have han : forall d. (exists x. a = d * x) -> (exists y. n = d * y) -> d = 1
  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
    have hbn : forall d. (exists x. b = d * x) -> (exists y. n = d * y) -> d = 1
  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 : forall d. (exists x. a * b = d * x) -> (exists y. n = d * y) -> d = 1
  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 exact command ledger · 26 lines
  1. 0001intro n
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hna
  5. 0005intro hnb
  6. 0006have 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 : 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 : 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