reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;
reserve
  D for non empty doubleLoopStr,
  A for Subset of D;
reserve K for Field-like non degenerated
  associative add-associative right_zeroed right_complementable
  distributive Abelian non empty doubleLoopStr,
  a, b, c for Element of K;
reserve v for Valuation of K;

theorem Th74:
  K is having_valuation & v.a = 0 implies
  for I being Ideal of ValuatRing v holds
  a in I iff I = the carrier of ValuatRing v
  proof
    assume
A1: K is having_valuation;
    assume
A2: v.a = 0;
    let I be Ideal of ValuatRing v;
    thus a in I implies I = the carrier of ValuatRing v
    proof
      assume
A3:   a in I;
      thus I c= the carrier of ValuatRing v;
      let x be object;
      assume x in the carrier of ValuatRing v;
      then reconsider x as Element of ValuatRing v;
      reconsider b = x as Element of K by A1,Th51;
A4:   a <> 0.K by A1,A2,Def8;
      reconsider y = a, z = a" as Element of ValuatRing v
      by A1,A2,Th72;
A5:   y*(z*x) in I by A3,IDEAL_1:def 2;
      reconsider za = z*x as Element of K by A1,Th51;
      z*x = a"*b by A1,Th55;
      then y*(z*x) = a*(a"*b) by A1,Th55;
      hence thesis by A5,A4,Lm8;
    end;
    the carrier of ValuatRing v = NonNegElements v by A1,Def12;
    hence thesis by A2;
  end;
