
theorem kerp:
for F being Field,
    p being non constant Element of the carrier of Polynom-Ring F
holds ker(poly_mod p) = {p}-Ideal
proof
let F be Field,
    p be non constant Element of the carrier of Polynom-Ring F;
set R = Polynom-Ring F, S = Polynom-Ring(p);
set f = poly_mod p;
reconsider p1 = p as Element of R;
A: now let x be object;
   assume A0: x in ker f;
   then x in {v where v is Element of R : f.v = 0.S} by VECTSP10:def 9;
   then consider y being Element of R such that A1: y = x & f.y = 0.S;
   reconsider q = x as Element of the carrier of R by A0;
   reconsider q1 = x as Element of R by A0;
   A2: q - (q div p) *' p = q mod p
                         .= 0.S by A1,dpm
                         .= 0_.(F) by defprfp;
   reconsider qp = q div p as Element of R by POLYNOM3:def 10;
   qp * p1 = (q div p) *' p by POLYNOM3:def 10;
   then A3: -(qp * p1) = -((q div p) *' p) by lemminuspoly;
   q1 - qp * p1 = q + -((q div p) *' p) by A3,POLYNOM3:def 10
               .= 0.R by A2,POLYNOM3:def 10;
   then qp * p1 = q1 by RLVECT_1:21;
   then q in the set of all p*r where r is Element of R;
   hence x in {p}-Ideal by IDEAL_1:64;
   end;
now let x be object;
  assume x in {p}-Ideal;
  then x in the set of all p*r where r is Element of R by IDEAL_1:64;
  then consider y being Element of R such that A1: x = p * y;
  reconsider q = y as Element of the carrier of R;
  p * y = p *' q by POLYNOM3:def 10; then
  f.x = (p *' q) mod p by A1,dpm
     .= 0_.(F) by T2,div2
     .= 0.S by defprfp;
  then x in {v where v is Element of R : f.v = 0.S} by A1;
  hence x in ker(poly_mod p) by VECTSP10:def 9;
  end;
hence thesis by A,TARSKI:2;
end;
