lemma 7.72 Addition theorems

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1828 · p. 229

Rests on

depends_on equation 7.53 eq:ana-cos-sin-series ¶
depends_on equation 5.3 eq:lin-binomial ¶
depends_on proposition 7.49 Cauchy product ¶
- 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 7.45 Series ¶
  - depends_on definition 7.2 Absolute value ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
- depends_on proposition 7.6 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 ¶
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 corollary A.46 Monotone convergence ¶ ↺
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-34 ¶
    - 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 proposition 7.6 Algebra of limits ¶ ↺
proves proof ch:05-real-analysis@proof-47 ¶

Supports

depends_on proposition 7.79 The polygon recursion ¶
- depends_on corollary 7.81 The decimal expansion of $\pi$ ¶
  - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
- depends_on proposition 7.83 Viète's product ¶
  - depends_on remark 7.84 One construction, two faces ¶
- depends_on remark 7.82 What Archimedes proved ¶
- depends_on remark 7.80 Why these are the polygon perimeters ¶
- depends_on remark 7.84 One construction, two faces ¶ ↺
depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
- depends_on lemma 7.78 Chord, arc, tangent ¶
  - depends_on proposition 7.79 The polygon recursion ¶ ↺
- depends_on proposition 7.85 Irrationality of $\pi$ ¶
  - depends_on remark 7.94 Three numbers, two kinds ¶
  - depends_on remark 7.87 Priority, and what irrationality does not give ¶
- depends_on proposition 7.79 The polygon recursion ¶ ↺
- depends_on proposition 7.83 Viète's product ¶ ↺
- depends_on proposition 125.2 Crystallographic restriction ¶
  - depends_on remark 125.3 What the theorem does and does not forbid ¶
- depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶ ↺
depends_on proposition 7.83 Viète's product ¶ ↺
depends_on proposition 125.2 Crystallographic restriction ¶ ↺
depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶ ↺
depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺

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`	→	eq:ana-cos-sin-series	declared	parts/02-mathematical-methods/05-real-analysis.tex:1834
`depends_on`	→	eq:lin-binomial	declared	parts/02-mathematical-methods/05-real-analysis.tex:1834
`depends_on`	→	Cauchy product	declared	parts/02-mathematical-methods/05-real-analysis.tex:1834
`depends_on`	→	Ratio test	declared	parts/02-mathematical-methods/05-real-analysis.tex:1834
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:1834
`depends_on`	←	The polygon recursion	declared	parts/02-mathematical-methods/05-real-analysis.tex:2198
`depends_on`	←	Special values, periodicity, and the kernel	declared	parts/02-mathematical-methods/05-real-analysis.tex:2077
`depends_on`	←	Viète's product	declared	parts/02-mathematical-methods/05-real-analysis.tex:2412
`depends_on`	←	Crystallographic restriction	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	←	$\pi$ elsewhere in this treatise	declared	parts/02-mathematical-methods/05-real-analysis.tex:2623
`depends_on`	←	Why these are the polygon perimeters	declared	parts/02-mathematical-methods/05-real-analysis.tex:2312
`proves`	←	ch:05-real-analysis@proof-47	declared	parts/02-mathematical-methods/05-real-analysis.tex:1838