proposition 7.63 Laws of real powers

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1436 · p. 224

Rests on

depends_on definition 7.62 Real powers ¶
- depends_on proposition 7.61 The logarithm ¶
  - 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 corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-33 ¶
  - depends_on equation 7.42 eq:ana-exp-derivative ¶
  - depends_on lemma 7.60 Functional equation of the exponential ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.59 The real exponential ¶
      - depends_on definition 7.45 Series ¶
      - depends_on proposition 7.48 Ratio test ¶
      - depends_on theorem 7.50 Power series; radius of convergence ¶
    - depends_on equation 5.3 eq:lin-binomial ¶
    - depends_on proposition 7.49 Cauchy product ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-30 ¶
    - proves proof ch:05-real-analysis@proof-38 ¶
  - depends_on proposition 7.32 Derivative of the inverse function ¶
    - depends_on definition 7.14 Inverse function ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 3.47 Bijective map ¶
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - depends_on theorem 7.23 Intermediate value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - proves proof ch:05-real-analysis@proof-8 ¶
    - proves proof ch:05-real-analysis@proof-16 ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶ ↺
    - 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 proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-11 ¶
  - depends_on theorem 7.23 Intermediate value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-39 ¶
depends_on equation 7.42 eq:ana-exp-derivative ¶ ↺
depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
depends_on proposition 7.31 Chain rule ¶
- depends_on definition 7.13 Composite function ¶ ↺
- depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-15 ¶
depends_on proposition 7.61 The logarithm ¶ ↺
proves proof ch:05-real-analysis@proof-40 ¶

Supports

depends_on definition 7.90 The golden ratio ¶
- depends_on proposition 7.91 The golden ratio ¶
  - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
    - depends_on remark 7.95 Where $\varphi$ is used ¶
  - depends_on proposition 125.2 Crystallographic restriction ¶
    - depends_on remark 125.3 What the theorem does and does not forbid ¶
  - depends_on remark 7.95 Where $\varphi$ is used ¶ ↺
  - depends_on remark 7.94 Three numbers, two kinds ¶
depends_on lemma 7.67 The limit over real exponents ¶
- depends_on proposition 7.68 The natural base ¶
  - depends_on remark 7.70 What the condition says ¶
depends_on proposition 7.68 The natural base ¶ ↺
depends_on proposition 7.91 The golden ratio ¶ ↺
depends_on proposition 7.89 Irrationality of square roots ¶
- depends_on proposition 7.91 The golden ratio ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Real powers	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`depends_on`	→	eq:ana-exp-derivative	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`depends_on`	→	Functional equation of the exponential	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`depends_on`	→	Chain rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`depends_on`	→	The logarithm	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`depends_on`	←	The golden ratio	declared	parts/02-mathematical-methods/05-real-analysis.tex:2681
`depends_on`	←	The limit over real exponents	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	←	The natural base	declared	parts/02-mathematical-methods/05-real-analysis.tex:1628
`depends_on`	←	The golden ratio	declared	parts/02-mathematical-methods/05-real-analysis.tex:2697
`depends_on`	←	Irrationality of square roots	declared	parts/02-mathematical-methods/05-real-analysis.tex:2649
`proves`	←	ch:05-real-analysis@proof-40	declared	parts/02-mathematical-methods/05-real-analysis.tex:1453