definition A.503 The substitution property

open in the book · appendices/A-long-proofs.tex:24661 · p. 3039

ground object -- no derivation owed

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 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 $\R$ ¶ ↺

Supports

depends_on lemma A.506 Locality ¶
- 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 ¶ ↺
depends_on lemma A.504 Transitivity ¶

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:24675
`depends_on`	→	Multiple integral	declared	appendices/A-long-proofs.tex:24675
`depends_on`	←	Locality	declared	appendices/A-long-proofs.tex:24756
`depends_on`	←	Transitivity	declared	appendices/A-long-proofs.tex:24683