let R be domRing; :: thesis: for p being non zero Polynomial of R holds card () <= deg p
let p be non zero Polynomial of R; :: thesis: card () <= deg p
defpred S1[ Nat] means for p being non zero Polynomial of R st deg p = \$1 holds
card () <= deg p;
IA: S1[ 0 ]
proof
let p be non zero Polynomial of R; :: thesis: ( deg p = 0 implies card () <= deg p )
assume A1: deg p = 0 ; :: thesis: card () <= deg p
reconsider q = p as Element of the carrier of () by POLYNOM3:def 10;
consider a being Element of R such that
A2: q = a | R by ;
a <> 0. R by A2;
then reconsider a = a as non zero Element of R by STRUCT_0:def 12;
q = a | R by A2;
hence card () <= deg p by bag1; :: thesis: verum
end;
IS: now :: thesis: for k being Nat st S1[k] holds
S1[k + 1]
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume IV: S1[k] ; :: thesis: S1[k + 1]
now :: thesis: for p being non zero Polynomial of R st deg p = k + 1 holds
card () <= deg p
let p be non zero Polynomial of R; :: thesis: ( deg p = k + 1 implies card (BRoots b1) <= deg b1 )
assume A1: deg p = k + 1 ; :: thesis: card (BRoots b1) <= deg b1
per cases ( ex x being Element of R st x is_a_root_of p or for x being Element of R holds not x is_a_root_of p ) ;
suppose ex x being Element of R st x is_a_root_of p ; :: thesis: card (BRoots b1) <= deg b1
then consider x being Element of R such that
A2: x is_a_root_of p ;
consider q being Polynomial of R such that
A3: p = (rpoly (1,x)) *' q by ;
A4: q <> 0_. R by A3;
reconsider q = q as non zero Polynomial of R by A3;
BRoots p = (BRoots (rpoly (1,x))) + () by ;
then A6: card () = (card (BRoots (rpoly (1,x)))) + (card ()) by UPROOTS:15
.= 1 + (card ()) by bag2 ;
deg p = (deg q) + (deg (rpoly (1,x))) by
.= (deg q) + 1 by HURWITZ:27 ;
hence card () <= deg p by ; :: thesis: verum
end;
suppose A2: for x being Element of R holds not x is_a_root_of p ; :: thesis: card (BRoots b1) <= deg b1
now :: thesis: ( Roots p <> {} implies for x being Element of Roots p holds contradiction )
assume A3: Roots p <> {} ; :: thesis: for x being Element of Roots p holds contradiction
let x be Element of Roots p; :: thesis: contradiction
x in Roots p by A3;
then reconsider x = x as Element of R ;
x is_a_root_of p by ;
hence contradiction by A2; :: thesis: verum
end;
then support () = {} by UPROOTS:def 9;
hence card () <= deg p by bag1a; :: thesis: verum
end;
end;
end;
hence S1[k + 1] ; :: thesis: verum
end;
I: for k being Nat holds S1[k] from NAT_1:sch 2(IA, IS);
p <> 0_. R ;
then deg p is Element of NAT by T8;
hence card () <= deg p by I; :: thesis: verum