lemma A.506 Locality

open in the book · appendices/A-long-proofs.tex:24749 · p. 3040

Rests on

depends_on definition A.503 The substitution property ¶
- 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 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 5.15 Basis ¶
  - 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 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 ¶
      - 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 7.125 Multiple integral ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
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 ¶ ↺
depends_on lemma A.505 A continuous partition of unity ¶
- depends_on definition 6.9 Compact set ¶ ↺
- depends_on definition 6.6 Continuous map ¶
  - depends_on definition 3.53 Preimage ¶
  - depends_on definition 6.2 Open set ¶ ↺
  - depends_on definition 6.1 Topological space ¶ ↺
- depends_on definition 6.24 Metric ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- proves proof app:A-long-proofs@proof-300 ¶
proves proof app:A-long-proofs@proof-301 ¶

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`	→	The substitution property	declared	appendices/A-long-proofs.tex:24756
`depends_on`	→	Compact set	declared	appendices/A-long-proofs.tex:24756
`depends_on`	→	A continuous partition of unity	declared	appendices/A-long-proofs.tex:24756
`depends_on`	←	The substitution property is universal	declared	appendices/A-long-proofs.tex:24974
`proves`	←	app:A-long-proofs@proof-301	declared	appendices/A-long-proofs.tex:24760