equal
deleted
inserted
replaced
488 lemma has_integral_mult_left: |
488 lemma has_integral_mult_left: |
489 fixes c :: "_ :: real_normed_algebra" |
489 fixes c :: "_ :: real_normed_algebra" |
490 shows "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) S" |
490 shows "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) S" |
491 using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def) |
491 using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def) |
492 |
492 |
493 text\<open>The case analysis eliminates the condition @{term "f integrable_on S"} at the cost |
493 text\<open>The case analysis eliminates the condition \<^term>\<open>f integrable_on S\<close> at the cost |
494 of the type class constraint \<open>division_ring\<close>\<close> |
494 of the type class constraint \<open>division_ring\<close>\<close> |
495 corollary integral_mult_left [simp]: |
495 corollary integral_mult_left [simp]: |
496 fixes c:: "'a::{real_normed_algebra,division_ring}" |
496 fixes c:: "'a::{real_normed_algebra,division_ring}" |
497 shows "integral S (\<lambda>x. f x * c) = integral S f * c" |
497 shows "integral S (\<lambda>x. f x * c) = integral S f * c" |
498 proof (cases "f integrable_on S \<or> c = 0") |
498 proof (cases "f integrable_on S \<or> c = 0") |