PA000J

add_eq_zero_left

Stable checked-use theorem · independently closed

A sum equal to zero has zero as its left addend.

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 a b. a + b = 0 -> a = 0

Structural proof guide

Generated structural guide

A sum equal to zero has zero as its left addend.

Use the direct prerequisites add_comm, add_eq_zero_right as previously established PA formulas.

The proof proceeds by direct introduction and elimination.

Referenced ingredients

PA000F add_comm PA0004 add_eq_zero_right

Proof neighborhood

Direct dependencies

PA000F add_comm PA0004 add_eq_zero_right

Direct dependents

PA000K bounded_common_multiple_exists PA007C gauss_signed_half_magnitude_injective

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

9 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.

Named ingredients (2)
add commadd eq zero right
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
02Use earlier factsL4–6

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

  1. L4
    specialize add_eq_zero_right b
  2. L5
    specialize add_eq_zero_right a
  3. L6
    apply add_eq_zero_right
03Calculate and transport equalitiesL7–7

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

  1. L7
    trans a + b
04Use earlier factsL8–9

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

  1. L8
    apply add_comm
  2. L9
    exact h

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004specialize add_eq_zero_right b
  5. 0005specialize add_eq_zero_right a
  6. 0006apply add_eq_zero_right
  7. 0007trans a + b
  8. 0008apply add_comm
  9. 0009exact h