FS005N · theorem body

four_square_signed_cross_negative

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

Two negative signed coordinate orientations also have congruent crossed bilinear products.

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

∀ k. ∀ a. ∀ b. ∀ e. ∀ f. ModEq(k,a + e,0) → ModEq(k,b + f,0) → ModEq(k,a · f,b · e)

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

PD0008 ModEq

In local proof propositions

PD0008 ModEq
Exact expanded first-order statement
forall k a b e f. (exists ftcn_left_fssq_cross_negative_a ftcn_right_fssq_cross_negative_a. (a + e) + (k) * ftcn_left_fssq_cross_negative_a = (0) + (k) * ftcn_right_fssq_cross_negative_a) -> (exists ftcn_left_fssq_cross_negative_b ftcn_right_fssq_cross_negative_b. (b + f) + (k) * ftcn_left_fssq_cross_negative_b = (0) + (k) * ftcn_right_fssq_cross_negative_b) -> (exists ftcn_left_fssq_cross_negative_result ftcn_right_fssq_cross_negative_result. (a * f) + (k) * ftcn_left_fssq_cross_negative_result = (b * e) + (k) * ftcn_right_fssq_cross_negative_result)

Proof neighborhood

Direct theorem prerequisites

FS005K four_square_signed_negative_scale_zero mul_comm · Stable closed FS005L four_square_signed_common_zero_cancel

Direct theorem dependents

FS004N four_square_signed_conjugate_negative_blocks FS004O four_square_signed_natural_positive_first_blocks FS005Y four_square_signed_conjugate_positive_blocks FS005Z four_square_signed_conjugate_mixed_blocks FS0060 four_square_signed_natural_negative_first_blocks

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

31 script commands · 4 reading checkpoints · 3 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)
four square signed negative scale zerofour square signed common zero cancel
01Fix variables and assumptionsL1–7

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

  1. L1
    intro k
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro e
  5. L5
    intro f
  6. L6
    intro ha
  7. L7
    intro hb
02Establish hfirstL8–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square signed negative scale zero.

  1. L8
    have hfirst : ModEq(k,a · f + e · f,0)Definitions: ModEq(k,a · f + e · f,0)Original native command in the exact edition
  2. L9
    specialize four_square_signed_negative_scale_zero k
  3. L10
    specialize four_square_signed_negative_scale_zero a
  4. L11
    specialize four_square_signed_negative_scale_zero e
  5. L12
    specialize four_square_signed_negative_scale_zero f
  6. L13
    apply four_square_signed_negative_scale_zero
  7. L14
    exact ha
03Establish hsecondL15–21

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square signed negative scale zero.

  1. L15
    have hsecond : ModEq(k,b · e + f · e,0)Definitions: ModEq(k,b · e + f · e,0)Original native command in the exact edition
  2. L16
    specialize four_square_signed_negative_scale_zero k
  3. L17
    specialize four_square_signed_negative_scale_zero b
  4. L18
    specialize four_square_signed_negative_scale_zero f
  5. L19
    specialize four_square_signed_negative_scale_zero e
  6. L20
    apply four_square_signed_negative_scale_zero
  7. L21
    exact hb
04Establish hswapL22–31

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

  1. L22
    have hswap : f * e = e * f
  2. L23
    apply mul_comm
  3. L24
    rewrite hswap at hsecond
  4. L25
    specialize four_square_signed_common_zero_cancel k
  5. L26
    specialize four_square_signed_common_zero_cancel (a * f)
  6. L27
    specialize four_square_signed_common_zero_cancel (b * e)
  7. L28
    specialize four_square_signed_common_zero_cancel (e * f)
  8. L29
    apply four_square_signed_common_zero_cancel
  9. L30
    exact hfirst
  10. L31
    exact hsecond

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro e
  5. 0005intro f
  6. 0006intro ha
  7. 0007intro hb
  8. 0008have hfirst : ModEq(k,a · f + e · f,0)
    Exact native replay linehave hfirst : exists ftcn_left_fssq_cross_negative_first ftcn_right_fssq_cross_negative_first. (a * f + e * f) + (k) * ftcn_left_fssq_cross_negative_first = (0) + (k) * ftcn_right_fssq_cross_negative_first
  9. 0009specialize four_square_signed_negative_scale_zero k
  10. 0010specialize four_square_signed_negative_scale_zero a
  11. 0011specialize four_square_signed_negative_scale_zero e
  12. 0012specialize four_square_signed_negative_scale_zero f
  13. 0013apply four_square_signed_negative_scale_zero
  14. 0014exact ha
  15. 0015have hsecond : ModEq(k,b · e + f · e,0)
    Exact native replay linehave hsecond : exists ftcn_left_fssq_cross_negative_second ftcn_right_fssq_cross_negative_second. (b * e + f * e) + (k) * ftcn_left_fssq_cross_negative_second = (0) + (k) * ftcn_right_fssq_cross_negative_second
  16. 0016specialize four_square_signed_negative_scale_zero k
  17. 0017specialize four_square_signed_negative_scale_zero b
  18. 0018specialize four_square_signed_negative_scale_zero f
  19. 0019specialize four_square_signed_negative_scale_zero e
  20. 0020apply four_square_signed_negative_scale_zero
  21. 0021exact hb
  22. 0022have hswap : f * e = e * f
  23. 0023apply mul_comm
  24. 0024rewrite hswap at hsecond
  25. 0025specialize four_square_signed_common_zero_cancel k
  26. 0026specialize four_square_signed_common_zero_cancel (a * f)
  27. 0027specialize four_square_signed_common_zero_cancel (b * e)
  28. 0028specialize four_square_signed_common_zero_cancel (e * f)
  29. 0029apply four_square_signed_common_zero_cancel
  30. 0030exact hfirst
  31. 0031exact hsecond