lemma A.671 The three products are independent

open in the book · appendices/A-long-proofs.tex:32669 · p. 3119

Rests on

depends_on equation 5.168 eq:rep-isotropic-4 ¶
depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
- depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
  - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
    - depends_on definition 4.48 Symmetric group ¶
      - depends_on definition 3.47 Bijective map ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.27 Orthonormal basis ¶
      - depends_on definition 5.26 Orthogonal basis ¶
      - depends_on definition 5.25 Unit vector ¶
  - depends_on equation 5.19 eq:lin-leibniz-det ¶
  - proves proof ch:03-linear-algebra-representations@proof-8 ¶
- depends_on lemma 5.132 Parity constraint ¶
  - depends_on definition 5.131 Isotropic Cartesian tensor ¶
    - depends_on definition 5.32 Levi–Civita symbol; cross product ¶ ↺
    - depends_on equation 5.162 eq:rep-special-orthogonal ¶
  - depends_on equation 5.162 eq:rep-special-orthogonal ¶ ↺
  - proves proof ch:03-linear-algebra-representations@proof-57 ¶
- depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
  - depends_on definition 5.18 Inner product ¶ ↺
  - depends_on equation 5.44 eq:lin-norm-assoc ¶
  - depends_on proposition 5.49 Injective, surjective, invertible ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-17 ¶
    - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on lemma 5.38 Exchange and completion ¶
      - proves proof ch:03-linear-algebra-representations@proof-13 ¶
    - proves proof ch:03-linear-algebra-representations@proof-18 ¶
  - proves proof ch:03-linear-algebra-representations@proof-56 ¶
- proves proof ch:03-linear-algebra-representations@proof-58 ¶
proves proof app:A-long-proofs@proof-401 ¶

Supports

depends_on proposition A.673 Cubic tensors of rank four ¶
- depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
  - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:rep-isotropic-4	declared	appendices/A-long-proofs.tex:32676
`depends_on`	→	Isotropic Cartesian tensors of rank at most four	declared	appendices/A-long-proofs.tex:32676
`depends_on`	←	Cubic tensors of rank four	declared	appendices/A-long-proofs.tex:32744
`proves`	←	app:A-long-proofs@proof-401	declared	appendices/A-long-proofs.tex:32679