theorem A.550 Quoted: discrete subgroups of a real vector space

open in the book · appendices/A-long-proofs.tex:26676 · p. 3059

Rests on

depends_on definition 4.21 Group ¶
- depends_on definition 4.11 Inverse element ¶
  - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
- depends_on definition 4.20 Monoid ¶
  - depends_on definition 4.9 Neutral element ¶ ↺
  - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
      - depends_on definition 4.2 Binary operation ¶ ↺
depends_on definition 6.9 Compact set ¶
- depends_on definition 6.5 Open cover ¶
  - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on equation 3.51 eq:set-indexed ¶ ↺

Supports

depends_on proposition A.552 The component is a torus ¶
- depends_on corollary A.553 Quasi-periodic motion ¶
- depends_on definition A.554 Actions on the torus ¶
  - depends_on lemma A.555 The actions are well defined ¶
  - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
    - depends_on example A.557 The check that fixes the convention ¶
- depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶ ↺

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`	→	Group	declared	appendices/A-long-proofs.tex:26689
`depends_on`	→	Compact set	declared	appendices/A-long-proofs.tex:26689
`depends_on`	←	The component is a torus	declared	appendices/A-long-proofs.tex:26724