PF000X · theorem body

pythagorean_primitive_hypotenuse_coprime_second_leg

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

In every primitive Pythagorean triple, the second leg and hypotenuse have no nonunit common divisor.

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

∀ x. ∀ y. ∀ z. PrimitivePythagorean(x,y,z) → Coprime(y,z)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

CF0013 PrimitivePythagorean PD0005 Coprime

In local proof propositions

none
Exact expanded first-order statement
forall x y z. ((((x) * (x) + (y) * (y) = (z) * (z)) /\ (forall pff_divisor_pairwise_source. (exists pff_left_pairwise_source. (x) = pff_divisor_pairwise_source * pff_left_pairwise_source) -> (exists pff_right_pairwise_source. (y) = pff_divisor_pairwise_source * pff_right_pairwise_source) -> pff_divisor_pairwise_source = 1))) -> (forall pff_divisor_second_hypotenuse_result. (exists pff_left_second_hypotenuse_result. (y) = pff_divisor_second_hypotenuse_result * pff_left_second_hypotenuse_result) -> (exists pff_right_second_hypotenuse_result. (z) = pff_divisor_second_hypotenuse_result * pff_right_second_hypotenuse_result) -> pff_divisor_second_hypotenuse_result = 1)

Proof neighborhood

Direct theorem prerequisites

PF000C pythagorean_primitive_leg_swap PF000W pythagorean_primitive_hypotenuse_coprime_first_leg

Direct theorem dependents

PF000Y pythagorean_primitive_pairwise_coprime

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

13 script commands · 2 reading checkpoints · 0 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)
pythagorean primitive leg swappythagorean primitive hypotenuse coprime first leg
01Fix variables and assumptionsL1–4

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro z
  4. L4
    intro hprimitive
02Use earlier factsL5–13

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

  1. L5
    specialize pythagorean_primitive_hypotenuse_coprime_first_leg y
  2. L6
    specialize pythagorean_primitive_hypotenuse_coprime_first_leg x
  3. L7
    specialize pythagorean_primitive_hypotenuse_coprime_first_leg z
  4. L8
    apply pythagorean_primitive_hypotenuse_coprime_first_leg
  5. L9
    specialize pythagorean_primitive_leg_swap x
  6. L10
    specialize pythagorean_primitive_leg_swap y
  7. L11
    specialize pythagorean_primitive_leg_swap z
  8. L12
    apply pythagorean_primitive_leg_swap
  9. L13
    exact hprimitive

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004intro hprimitive
  5. 0005specialize pythagorean_primitive_hypotenuse_coprime_first_leg y
  6. 0006specialize pythagorean_primitive_hypotenuse_coprime_first_leg x
  7. 0007specialize pythagorean_primitive_hypotenuse_coprime_first_leg z
  8. 0008apply pythagorean_primitive_hypotenuse_coprime_first_leg
  9. 0009specialize pythagorean_primitive_leg_swap x
  10. 0010specialize pythagorean_primitive_leg_swap y
  11. 0011specialize pythagorean_primitive_leg_swap z
  12. 0012apply pythagorean_primitive_leg_swap
  13. 0013exact hprimitive