lemma A.564 Dimension and double orthogonal

open in the book · appendices/A-long-proofs.tex:27141 · p. 3064

Rests on

depends_on definition 24.2 Symplectic vector space ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  - depends_on definition 5.110 Bilinear map ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
  - depends_on definition 5.113 Non-degenerate form ¶
    - depends_on definition 5.110 Bilinear map ¶ ↺
  - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
    - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
  - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
  - proves proof ch:03-linear-algebra-representations@proof-54 ¶
depends_on theorem 5.40 Rank–nullity ¶
- depends_on definition 5.15 Basis ¶
  - depends_on definition 5.12 Subspace generated by a set of vectors ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-3 ¶
  - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.5 Linear combination ¶ ↺
- 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 lemma 5.38 Exchange and completion ¶
  - depends_on definition 5.15 Basis ¶ ↺
  - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
  - depends_on definition 5.14 Linear independence ¶ ↺
  - proves proof ch:03-linear-algebra-representations@proof-11 ¶
- proves proof ch:03-linear-algebra-representations@proof-13 ¶
proves proof app:A-long-proofs@proof-339 ¶

Supports

depends_on proposition A.565 Freeness makes every value regular ¶
- depends_on lemma A.567 The differential of the momentum map along the orbit ¶
  - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
- depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
  - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶ ↺
depends_on theorem A.568 The radical is the isotropy orbit ¶ ↺

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`	→	Symplectic vector space	declared	appendices/A-long-proofs.tex:27153
`depends_on`	→	Rank–nullity	declared	appendices/A-long-proofs.tex:27153
`depends_on`	←	Freeness makes every value regular	declared	appendices/A-long-proofs.tex:27181
`depends_on`	←	The radical is the isotropy orbit	declared	appendices/A-long-proofs.tex:27291
`proves`	←	app:A-long-proofs@proof-339	declared	appendices/A-long-proofs.tex:27156