lemma A.510 Every diffeomorphism factorises locally

open in the book · appendices/A-long-proofs.tex:24902 · p. 3041

Rests on

depends_on corollary 7.113 Inverse function theorem ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - 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 ¶
- depends_on proposition 7.104 Chain rule in several variables ¶
  - depends_on definition 7.13 Composite function ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on definition 7.99 Differentiability at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on equation 6.12 eq:top-euclidean-metric ¶
  - depends_on definition 7.102 Differential of a function ¶
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
  - proves proof ch:05-real-analysis@proof-63 ¶
- depends_on theorem 7.112 Implicit function theorem ¶
  - depends_on definition 7.99 Differentiability at a point ¶ ↺
  - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
  - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-62 ¶
  - proves proof ch:05-real-analysis@prooflink-1 ¶
- proves proof ch:05-real-analysis@prooflink-2 ¶
depends_on definition A.508 Primitive map ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
- depends_on equation 5.19 eq:lin-leibniz-det ¶
depends_on proposition 7.104 Chain rule in several variables ¶ ↺
proves proof app:A-long-proofs@proof-304 ¶

Supports

depends_on proposition A.511 The substitution property is universal ¶
- depends_on remark A.516 The hypotheses of the global form ¶
- depends_on theorem A.513 Change of variables ¶
  - depends_on corollary A.514 Degeneracy on a negligible set ¶
  - depends_on remark A.516 The hypotheses of the global form ¶ ↺

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`	→	Inverse function theorem	declared	appendices/A-long-proofs.tex:24909
`depends_on`	→	Primitive map	declared	appendices/A-long-proofs.tex:24909
`depends_on`	→	Chain rule in several variables	declared	appendices/A-long-proofs.tex:24909
`depends_on`	←	The substitution property is universal	declared	appendices/A-long-proofs.tex:24974
`proves`	←	app:A-long-proofs@proof-304	declared	appendices/A-long-proofs.tex:24913