lemma A.509 Primitive maps have the substitution property

open in the book · appendices/A-long-proofs.tex:24826 · p. 3040

Rests on

depends_on definition A.508 Primitive map ¶
- 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 equation 5.19 eq:lin-leibniz-det ¶
depends_on equation 7.27 eq:ana-substitution ¶
depends_on lemma A.501 Iterated integration over a box ¶
- depends_on definition 7.39 Darboux sums and the definite integral ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
- 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 theorem 7.40 Continuous functions are integrable ¶
  - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
  - depends_on theorem 7.24 Extreme value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-9 ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on proposition 7.22 Sequential characterization ¶ ↺
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
    - proves proof ch:05-real-analysis@proof-10 ¶
  - proves proof ch:05-real-analysis@proof-23 ¶
- proves proof app:A-long-proofs@proof-297 ¶
proves proof app:A-long-proofs@proof-303 ¶

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`	→	Primitive map	declared	appendices/A-long-proofs.tex:24829
`depends_on`	→	eq:ana-substitution	declared	appendices/A-long-proofs.tex:24829
`depends_on`	→	Iterated integration over a box	declared	appendices/A-long-proofs.tex:24829
`depends_on`	←	The substitution property is universal	declared	appendices/A-long-proofs.tex:24974
`proves`	←	app:A-long-proofs@proof-303	declared	appendices/A-long-proofs.tex:24833