lemma 7.116 Functions vanishing on a regular zero set

Rests on

• depends_on corollary 7.113 Inverse function theorem ¶
  • depends_on definition 7.98 Functions of class $C^{1}$ ¶
    • depends_on definition 7.20 Continuity 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 equation 7.7 eq:ana-limit-left ¶
      • depends_on equation 7.5 eq:ana-limit-right ¶
    • depends_on definition 7.97 Partial derivative; gradient ¶
      • 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 definition 5.15 Basis ¶
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.14 Linear independence ¶
  • depends_on proposition 7.104 Chain rule in several variables ¶
    • depends_on definition 7.13 Composite function ¶
      • depends_on definition 7.9 Real function ¶ ↺
    • depends_on definition 7.99 Differentiability at a point ¶
      • depends_on definition 7.16 Limit ¶ ↺
      • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on equation 6.12 eq:top-euclidean-metric ¶
    • depends_on definition 7.102 Differential of a function ¶
      • depends_on definition 7.99 Differentiability at a point ¶ ↺
      • depends_on definition 7.97 Partial derivative; gradient ¶ ↺
    • proves proof ch:05-real-analysis@proof-63 ¶
  • depends_on theorem 7.112 Implicit function theorem ¶
    • depends_on definition 7.99 Differentiability at a point ¶ ↺
    • depends_on proposition 7.104 Chain rule in several variables ¶ ↺
    • depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      • depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
      • depends_on definition 7.99 Differentiability at a point ¶ ↺
      • 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-62 ¶
    • proves proof ch:05-real-analysis@prooflink-1 ¶
  • proves proof ch:05-real-analysis@prooflink-2 ¶
• 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 ¶ ↺
      • proves proof ch:05-real-analysis@proof-20 ¶
    • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  • 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 ¶
• depends_on theorem 7.109 Leibniz integral rule ¶
  • 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 7.25 Heine–Cantor: uniform continuity ¶ ↺
  • depends_on theorem 7.35 Mean value theorem ¶ ↺
  • proves proof ch:05-real-analysis@proof-67 ¶
• proves proof ch:05-real-analysis@proof-71 ¶

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