lemma 7.60 Functional equation of the exponential

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1353 · p. 223

Rests on

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 $\R$ ¶
    - depends_on definition A.36 Cut ¶ ↺
  - proves proof app:A-long-proofs@proof-29 ¶
- proves proof app:A-long-proofs@proof-33 ¶
depends_on definition 7.59 The real exponential ¶
- 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 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 7.50 Power series; radius of convergence ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
  - depends_on proposition 7.46 Geometric series ¶ ↺
  - proves proof ch:05-real-analysis@proof-31 ¶
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 ¶
  - depends_on definition 7.4 Convergence ¶ ↺
  - depends_on proposition 7.3 Triangle inequality ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
    - proves proof ch:05-real-analysis@proof-1 ¶
  - proves proof ch:05-real-analysis@proof-3 ¶
- proves proof ch:05-real-analysis@proof-30 ¶
proves proof ch:05-real-analysis@proof-38 ¶

Supports

depends_on proposition 7.68 The natural base ¶
- depends_on remark 7.70 What the condition says ¶
depends_on proposition 7.61 The logarithm ¶
- depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
  - depends_on remark 7.70 What the condition says ¶ ↺
- depends_on definition 7.62 Real powers ¶
  - 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 proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 7.95 Where $\varphi$ is used ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
  - depends_on proposition 7.63 Laws of real powers ¶
    - depends_on definition 7.90 The golden ratio ¶ ↺
    - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶ ↺
    - 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 ¶ ↺
  - depends_on proposition 7.89 Irrationality of square roots ¶ ↺
- depends_on lemma 7.67 The limit over real exponents ¶ ↺
- depends_on proposition 7.68 The natural base ¶ ↺
- depends_on proposition 7.63 Laws of real powers ¶ ↺
depends_on proposition 7.63 Laws of real powers ¶ ↺

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`	→	Monotone convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:1361
`depends_on`	→	The real exponential	declared	parts/02-mathematical-methods/05-real-analysis.tex:1361
`depends_on`	→	eq:lin-binomial	declared	parts/02-mathematical-methods/05-real-analysis.tex:1361
`depends_on`	→	Cauchy product	declared	parts/02-mathematical-methods/05-real-analysis.tex:1361
`depends_on`	←	The natural base	declared	parts/02-mathematical-methods/05-real-analysis.tex:1628
`depends_on`	←	The logarithm	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	←	Laws of real powers	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`proves`	←	ch:05-real-analysis@proof-38	declared	parts/02-mathematical-methods/05-real-analysis.tex:1365