PA0020 · theorem

mod_eq_mul_left

Stable checked-use theorem · independently closed

Balanced congruence is preserved by multiplication on the left.

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

∀ m. ∀ a. ∀ b. ∀ c. ModEq(m,a,b) → ModEq(m,c · a,c · b)

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

PD0008 ModEq

2 occurrences

In local proof propositions

PD0008 ModEq

1 occurrences

Exact expanded native-PA statement
forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (c * a) + m * r = (c * b) + m * s

Proof neighborhood

Direct theorem prerequisites

PA001Y mod_eq_mul_right PA000H mul_comm

Direct theorem dependents

PA0026 binary_crt PA003R mod_eq_cancel_coprime PA004E mod_eq_mul PA0063 prime_bounded_nonzero_mod_inverse PA00AK bounded_mod_inverse_unique PA008N scaled_inverse_from_unit_inverse PA008O scaled_inverse_transport_right

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

25 script commands · 12 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 (2)
mod eq mul rightmul comm
01Fix variables and assumptionsL1–5

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

  1. L1
    intro m
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro h
02Use earlier factsL6–9

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

  1. L6
    specialize mod_eq_mul_right m
  2. L7
    specialize mod_eq_mul_right a
  3. L8
    specialize mod_eq_mul_right b
  4. L9
    specialize mod_eq_mul_right c
03Establish hrL10–12

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

  1. L10
    have hr : ModEq(m,a · c,b · c)Definitions: ModEq(m,a · c,b · c)Original native command in the exact edition
  2. L11
    apply mod_eq_mul_right
  3. L12
    exact h
04Separate the logical casesL13–14

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

  1. L13
    cases hr
  2. L14
    cases hr_witness
05Construct an explicit witnessL15–16

Supply the displayed value, then prove that it has the required property.

  1. L15
    exists x
  2. L16
    exists x1
06Calculate and transport equalitiesL17–18

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

  1. L17
    trans a * c + m * x
  2. L18
    congr
07Use earlier factsL19–19

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

  1. L19
    apply mul_comm
08Calculate and transport equalitiesL20–21

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

  1. L20
    refl
  2. L21
    trans b * c + m * x1
09Use earlier factsL22–22

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

  1. L22
    exact hr_witness_witness
10Calculate and transport equalitiesL23–23

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

  1. L23
    congr
11Use earlier factsL24–24

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

  1. L24
    apply mul_comm
12Calculate and transport equalitiesL25–25

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

  1. L25
    refl

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro m
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro h
  6. 0006specialize mod_eq_mul_right m
  7. 0007specialize mod_eq_mul_right a
  8. 0008specialize mod_eq_mul_right b
  9. 0009specialize mod_eq_mul_right c
  10. 0010have hr : ModEq(m,a · c,b · c)
    Exact native replay linehave hr : exists r s. (a * c) + m * r = (b * c) + m * s
  11. 0011apply mod_eq_mul_right
  12. 0012exact h
  13. 0013cases hr
  14. 0014cases hr_witness
  15. 0015exists x
  16. 0016exists x1
  17. 0017trans a * c + m * x
  18. 0018congr
  19. 0019apply mul_comm
  20. 0020refl
  21. 0021trans b * c + m * x1
  22. 0022exact hr_witness_witness
  23. 0023congr
  24. 0024apply mul_comm
  25. 0025refl