PA000O · theorem

divisor_one

Stable checked-use theorem · independently closed

Every natural divisor of one equals one.

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

∀ d. Dvd(d,1) → d = 1

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0003 Dvd

1 occurrences

In local proof propositions

none

0 occurrences

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

Proof neighborhood

Direct theorem prerequisites

PA000N mul_eq_one_components

Direct theorem dependents

PA000P coprime_one_left PA0015 beta_modulus_coprime_base PA0037 is_gcd_one_to_coprime

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

11 script commands · 6 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)
mul eq one components
01Fix variables and assumptionsL1–2

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

  1. L1
    intro d
  2. L2
    intro h
02Separate the logical casesL3–3

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L3
    cases h
03Use earlier factsL4–5

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

  1. L4
    specialize mul_eq_one_components d
  2. L5
    specialize mul_eq_one_components x
04Establish partsL6–9

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

  1. L6
    have parts : d = 1 /\ x = 1
  2. L7
    apply mul_eq_one_components
  3. L8
    symm
  4. L9
    exact h_witness
05Separate the logical casesL10–10

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L10
    cases parts
06Use earlier factsL11–11

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

  1. L11
    exact parts_left

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro d
  2. 0002intro h
  3. 0003cases h
  4. 0004specialize mul_eq_one_components d
  5. 0005specialize mul_eq_one_components x
  6. 0006have parts : d = 1 /\ x = 1
  7. 0007apply mul_eq_one_components
  8. 0008symm
  9. 0009exact h_witness
  10. 0010cases parts
  11. 0011exact parts_left