lemma A.501 Iterated integration over a box

open in the book · appendices/A-long-proofs.tex:24585 · p. 3038

Rests on

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 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - 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 definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-33 ¶
    - 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 ¶

Supports

depends_on corollary A.502 All iterated orders agree ¶
- depends_on lemma A.507 Coordinate permutations ¶
depends_on lemma A.509 Primitive maps have the substitution property ¶
- 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`	→	Darboux sums and the definite integral	declared	appendices/A-long-proofs.tex:24598
`depends_on`	→	Multiple integral	declared	appendices/A-long-proofs.tex:24598
`depends_on`	→	Continuous functions are integrable	declared	appendices/A-long-proofs.tex:24598
`depends_on`	←	All iterated orders agree	declared	appendices/A-long-proofs.tex:24637
`depends_on`	←	Primitive maps have the substitution property	declared	appendices/A-long-proofs.tex:24829
`proves`	←	app:A-long-proofs@proof-297	declared	appendices/A-long-proofs.tex:24602