equal
deleted
inserted
replaced
806 lemma has_contour_integral_bound_linepath: |
806 lemma has_contour_integral_bound_linepath: |
807 assumes "(f has_contour_integral i) (linepath a b)" |
807 assumes "(f has_contour_integral i) (linepath a b)" |
808 "0 \<le> B" and B: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B" |
808 "0 \<le> B" and B: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B" |
809 shows "norm i \<le> B * norm(b - a)" |
809 shows "norm i \<le> B * norm(b - a)" |
810 proof - |
810 proof - |
811 have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))" |
811 have "norm i \<le> (B * norm (b - a)) * measure lborel (cbox 0 (1::real))" |
812 proof (rule has_integral_bound |
812 proof (rule has_integral_bound |
813 [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"]) |
813 [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"]) |
814 show "cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})) |
814 show "cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})) |
815 \<le> B * cmod (b - a)" |
815 \<le> B * cmod (b - a)" |
816 if "x \<in> cbox 0 1" for x::real |
816 if "x \<in> cbox 0 1" for x::real |