PF002R · theorem body

fermat_four_solutions_have_zero_coordinate

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

Fermat exponent four has no nontrivial natural solution, without any separate positivity assumptions.

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

forall a b h. a * a * a * a + b * b * b * b = h * h * h * h -> (a = 0 \/ b = 0)

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

none

In local proof propositions

none
Exact expanded first-order statement
forall a b h. a * a * a * a + b * b * b * b = h * h * h * h -> (a = 0 \/ b = 0)

Proof neighborhood

Direct theorem prerequisites

PF002Q fermat_four_square_solutions_have_zero_coordinate fourth_power_regroup · Stable closed

Direct theorem dependents

PF002S fermat_four_complete_classification

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

11 script commands · 4 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 (1)
fermat four square solutions have zero coordinate
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
  4. L4
    intro hequation
02Use earlier factsL5–8

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

  1. L5
    specialize fermat_four_square_solutions_have_zero_coordinate (a)
  2. L6
    specialize fermat_four_square_solutions_have_zero_coordinate (b)
  3. L7
    specialize fermat_four_square_solutions_have_zero_coordinate (h * h)
  4. L8
    apply fermat_four_square_solutions_have_zero_coordinate
03Calculate and transport equalitiesL9–9

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    trans h * h * h * h
04Use earlier factsL10–11

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

  1. L10
    exact hequation
  2. L11
    apply fourth_power_regroup

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hequation
  5. 0005specialize fermat_four_square_solutions_have_zero_coordinate (a)
  6. 0006specialize fermat_four_square_solutions_have_zero_coordinate (b)
  7. 0007specialize fermat_four_square_solutions_have_zero_coordinate (h * h)
  8. 0008apply fermat_four_square_solutions_have_zero_coordinate
  9. 0009trans h * h * h * h
  10. 0010exact hequation
  11. 0011apply fourth_power_regroup