remark 24.6 Connectedness, and what it rests on

Rests on

• depends_on proposition 24.5 Properties of the symplectic group ¶
  • depends_on definition 24.4 Symplectic group ¶
    • depends_on definition 5.135 The symplectic group ¶
      • depends_on definition 5.113 Non-degenerate form ¶
        • depends_on definition 5.110 Bilinear map ¶
      • depends_on equation 5.138 eq:lin-bilinear-expansion ¶
      • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
        • depends_on definition 5.110 Bilinear map ¶ ↺
        • depends_on definition 5.113 Non-degenerate form ¶ ↺
        • depends_on definition 5.111 Symmetric and antisymmetric forms ¶
        • depends_on equation 5.139 eq:lin-bilinear-congruence ¶
        • proves proof ch:03-linear-algebra-representations@proof-54 ¶
    • depends_on proposition 24.3 Even dimension and the canonical basis ¶
      • depends_on definition 24.2 Symplectic vector space ¶
        • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
      • depends_on equation 22.56 eq:ham-symplectic-matrix ¶
      • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
      • proves proof ch:07-symplectic-geometry@proof-1 ¶
  • depends_on proposition 5.139 Dimension of the symplectic group ¶
    • depends_on definition 5.127 Lie algebra ¶
      • depends_on definition 5.121 Algebra ¶
        • depends_on definition 4.4 Internal binary operation; magma ¶
        • depends_on definition 4.33 Vector space ¶
    • depends_on definition 5.135 The symplectic group ¶ ↺
    • depends_on proposition 5.125 prop:lin-matrix-algebra ¶
      • depends_on definition 5.121 Algebra ¶ ↺
      • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶ ↺
      • depends_on equation 5.150 eq:lin-ctesesc2 ¶
      • depends_on equation 5.74 eq:lin-matrix-rep ¶
      • depends_on proposition 5.45 prop:lin-matrix-unique ¶
        • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
        • depends_on equation 5.29 eq:lin-indepli2 ¶
        • depends_on equation 5.74 eq:lin-matrix-rep ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-16 ¶
      • proves proof ch:03-linear-algebra-representations@proof-55 ¶
    • proves proof ch:03-linear-algebra-representations@proof-62 ¶
  • depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
    • depends_on definition 5.135 The symplectic group ¶ ↺
    • depends_on lemma 5.137 The alternating top form is unique up to scale ¶
      • depends_on definition 5.15 Basis ¶ ↺
      • depends_on definition 5.110 Bilinear map ¶ ↺
      • depends_on equation 5.19 eq:lin-leibniz-det ¶
      • proves proof ch:03-linear-algebra-representations@proof-60 ¶
    • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-61 ¶
  • proves proof ch:07-symplectic-geometry@proof-2 ¶

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