src/HOL/Algebra/More_Ring.thy
author wenzelm
Tue Oct 10 19:23:03 2017 +0200 (2017-10-10)
changeset 66831 29ea2b900a05
parent 66760 d44ea023ac09
permissions -rw-r--r--
tuned: each session has at most one defining entry;
     1 (*  Title:      HOL/Algebra/More_Ring.thy
     2     Author:     Jeremy Avigad
     3 *)
     4 
     5 section \<open>More on rings etc.\<close>
     6 
     7 theory More_Ring
     8   imports Ring
     9 begin
    10 
    11 lemma (in cring) field_intro2: "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub> \<Longrightarrow> \<forall>x \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}. x \<in> Units R \<Longrightarrow> field R"
    12   apply (unfold_locales)
    13     apply (use cring_axioms in auto)
    14    apply (rule trans)
    15     apply (subgoal_tac "a = (a \<otimes> b) \<otimes> inv b")
    16      apply assumption
    17     apply (subst m_assoc)
    18        apply auto
    19   apply (unfold Units_def)
    20   apply auto
    21   done
    22 
    23 lemma (in monoid) inv_char:
    24   "x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> x \<otimes> y = \<one> \<Longrightarrow> y \<otimes> x = \<one> \<Longrightarrow> inv x = y"
    25   apply (subgoal_tac "x \<in> Units G")
    26    apply (subgoal_tac "y = inv x \<otimes> \<one>")
    27     apply simp
    28    apply (erule subst)
    29    apply (subst m_assoc [symmetric])
    30       apply auto
    31   apply (unfold Units_def)
    32   apply auto
    33   done
    34 
    35 lemma (in comm_monoid) comm_inv_char: "x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> x \<otimes> y = \<one> \<Longrightarrow> inv x = y"
    36   apply (rule inv_char)
    37      apply auto
    38   apply (subst m_comm, auto)
    39   done
    40 
    41 lemma (in ring) inv_neg_one [simp]: "inv (\<ominus> \<one>) = \<ominus> \<one>"
    42   apply (rule inv_char)
    43      apply (auto simp add: l_minus r_minus)
    44   done
    45 
    46 lemma (in monoid) inv_eq_imp_eq: "x \<in> Units G \<Longrightarrow> y \<in> Units G \<Longrightarrow> inv x = inv y \<Longrightarrow> x = y"
    47   apply (subgoal_tac "inv (inv x) = inv (inv y)")
    48    apply (subst (asm) Units_inv_inv)+
    49     apply auto
    50   done
    51 
    52 lemma (in ring) Units_minus_one_closed [intro]: "\<ominus> \<one> \<in> Units R"
    53   apply (unfold Units_def)
    54   apply auto
    55   apply (rule_tac x = "\<ominus> \<one>" in bexI)
    56    apply auto
    57   apply (simp add: l_minus r_minus)
    58   done
    59 
    60 lemma (in monoid) inv_one [simp]: "inv \<one> = \<one>"
    61   apply (rule inv_char)
    62      apply auto
    63   done
    64 
    65 lemma (in ring) inv_eq_neg_one_eq: "x \<in> Units R \<Longrightarrow> inv x = \<ominus> \<one> \<longleftrightarrow> x = \<ominus> \<one>"
    66   apply auto
    67   apply (subst Units_inv_inv [symmetric])
    68    apply auto
    69   done
    70 
    71 lemma (in monoid) inv_eq_one_eq: "x \<in> Units G \<Longrightarrow> inv x = \<one> \<longleftrightarrow> x = \<one>"
    72   by (metis Units_inv_inv inv_one)
    73 
    74 end