lemma A.72 Iterated integral inequality

open in the book · appendices/A-long-proofs.tex:4601 · p. 2828

Rests on

depends_on definition 7.53 $\ee$ ¶
- depends_on definition 7.45 Series ¶
  - depends_on definition 7.2 Absolute value ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
- depends_on proposition 7.48 Ratio test ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-33 ¶
    - depends_on definition 7.45 Series ¶ ↺
    - depends_on theorem 7.8 Cauchy criterion ¶
      - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-5 ¶
    - proves proof ch:05-real-analysis@proof-28 ¶
  - depends_on proposition 7.46 Geometric series ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶ ↺
    - proves proof ch:05-real-analysis@proof-27 ¶
  - depends_on theorem 7.8 Cauchy criterion ¶ ↺
  - proves proof ch:05-real-analysis@proof-29 ¶
depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
- depends_on definition 7.41 Antiderivative ¶
  - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    - 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-20 ¶
  - depends_on definition 7.26 Derivative of a function 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 definition 7.9 Real function ¶ ↺
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
    - 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 theorem 7.7 Bolzano–Weierstrass ¶
      - 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 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-24 ¶
- proves proof ch:05-real-analysis@proof-25 ¶
proves proof app:A-long-proofs@proof-52 ¶

Supports

depends_on theorem A.74 Flow of a time-dependent vector field ¶
- depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
  - depends_on lemma A.548 The joint flow is a translation action on the level set ¶
    - depends_on corollary A.553 Quasi-periodic motion ¶
    - depends_on lemma A.549 The action is transitive ¶
      - depends_on proposition A.552 The component is a torus ¶
    - depends_on proposition A.552 The component is a torus ¶ ↺
  - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
    - depends_on theorem A.568 The radical is the isotropy orbit ¶
      - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
  - depends_on proposition 13.131 Commuting fields have commuting flows ¶
    - depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
    - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.133 Frobenius ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
    - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶ ↺
  - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶ ↺
  - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶ ↺

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`	→	$\ee$	declared	appendices/A-long-proofs.tex:4608
`depends_on`	→	Fundamental theorem of calculus, II	declared	appendices/A-long-proofs.tex:4608
`depends_on`	←	Flow of a time-dependent vector field	declared	appendices/A-long-proofs.tex:4682
`proves`	←	app:A-long-proofs@proof-52	declared	appendices/A-long-proofs.tex:4611