{"id":1002,"date":"2022-09-05T06:00:14","date_gmt":"2022-09-05T04:00:14","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1002"},"modified":"2022-09-11T13:57:23","modified_gmt":"2022-09-11T11:57:23","slug":"si-r-es-un-anillo-y-a-%e2%88%88-r-entonces-0-a-0","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-r-es-un-anillo-y-a-%e2%88%88-r-entonces-0-a-0\/","title":{"rendered":"Si R es un anillo y a \u2208 R, entonces 0 * a = 0"},"content":{"rendered":"<p>Demostrar que si R es un anillo y a \u2208 R, entonces<\/p>\n<pre lang=\"text\">\n0 * a = 0.\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.ring\n\nvariables {R : Type*} [ring R]\nvariable (a : R)\n\nexample : 0 * a = 0 :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport algebra.ring\n\nvariables {R : Type*} [ring R]\nvariable (a : R)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 0 * a = 0 :=\nbegin\n  have h : 0 * a + 0 * a = 0 * a + 0,\n    calc 0 * a + 0 * a\n         = (0 + 0) * a : (add_mul 0 0 a).symm\n     ... = 0 * a       : congr_arg (\u03bb x, x * a) (add_zero 0)\n     ... = 0 * a + 0   : self_eq_add_right.mpr rfl,\n  rw add_left_cancel h\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 0 * a = 0 :=\nbegin\n  have h : 0 * a + 0 * a = 0 * a + 0,\n    calc 0 * a + 0 * a\n         = (0 + 0) * a : by rw add_mul\n     ... = 0 * a       : by rw add_zero\n     ... = 0 * a + 0   : by rw add_zero,\n  rw add_left_cancel h\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 0 * a = 0 :=\nbegin\n  have h : 0 * a + 0 * a = 0 * a + 0,\n  { rw [\u2190add_mul, add_zero, add_zero] },\n  rw add_left_cancel h\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 0 * a = 0 :=\nbegin\n  have h : 0 * a + 0 * a = 0 * a + 0,\n    calc 0 * a + 0 * a\n         = (0 + 0) * a : by simp\n     ... = 0 * a       : by simp\n     ... = 0 * a + 0   : by simp,\n  simp,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 0 * a = 0 :=\nby simp\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Producto_por_cero_por_a.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li>J. Avigad, K. Buzzard, R.Y. Lewis y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 12.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si R es un anillo y a \u2208 R, entonces 0 * a = 0. Para ello, completar la siguiente teor\u00eda de Lean: import algebra.ring variables {R : Type*} [ring R] variable (a : R) example : 0 * a = 0 := sorry<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[284],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1002"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/comments?post=1002"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1002\/revisions"}],"predecessor-version":[{"id":1087,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1002\/revisions\/1087"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1002"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1002"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1002"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}