lemma A.504 Transitivity

open in the book · appendices/A-long-proofs.tex:24678 · p. 3039

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 lemma A.497 The determinant is multiplicative ¶
- depends_on equation 5.19 eq:lin-leibniz-det ¶
- proves proof app:A-long-proofs@proof-294 ¶
depends_on proposition 7.104 Chain rule in several variables ¶ ↺
proves proof app:A-long-proofs@proof-299 ¶

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type	direction	node	provenance	where
`depends_on`	→	The substitution property	declared	appendices/A-long-proofs.tex:24683
`depends_on`	→	The determinant is multiplicative	declared	appendices/A-long-proofs.tex:24683
`depends_on`	→	Chain rule in several variables	declared	appendices/A-long-proofs.tex:24683
`proves`	←	app:A-long-proofs@proof-299	declared	appendices/A-long-proofs.tex:24687