SYMSP_1: Construction of a bilinear antisymmetric form in symplectic vector space

definition

let F be Field;
attr c₂ is strict ;
struct SymStr of F -> RelStr , VectSpStr of F;
aggr SymStr(# carrier, addF, ZeroF, lmult, InternalRel #) -> SymStr of F;

end;

registration

let F be Field;
cluster non empty SymStr of F;
existence

proof end;

end;

notation

let F be Field;
let S be SymStr of F;
let a, b be Element of S;
synonym a _|_ b for S <= a;

end;

Lm2: now

defpred S₁[ set ] means ex a, b being Element of {0} st
( $1 = [a,b] & b = o );
set CV = [:{0},{0}:];
let F be Field; :: thesis:
consider mo being set such that
A1: for x being set holds
( x in mo iff ( x in [:{0},{0}:] & S₁[x] ) ) from XBOOLE_0:sch 1();
mo c= [:{0},{0}:]

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
thus ( not x in mo or x in [:{0},{0}:] ) by A1; :: thesis:

end;

then reconsider mo = mo as Relation of {0} ;
take mo = mo; :: thesis:
thus for x being set holds
( x in mo iff ( x in [:{0},{0}:] & ex a, b being Element of {0} st
( x = [a,b] & b = o ) ) ) by A1; :: thesis:

end;

registration

let F be Field;
let X be non empty set ;
let md be BinOp of X;
let o be Element of X;
let mF be Function of [: the carrier of F,X:],X;
let mo be Relation of X;
cluster SymStr(# X,md,o,mF,mo #) -> non empty ;
coherence ;

end;

registration

let F be Field;
cluster non empty right_complementable Abelian add-associative right_zeroed SymStr of F;
existence

proof end;

end;

Lm4: now

let F be Field; :: thesis:
let mF be Function of [: the carrier of F,{0}:],{0}; :: thesis:
assume A1: for a being Element of F
for x being Element of {0} holds mF . (a,x) = o ; :: thesis:
let mo be Relation of {0}; :: thesis:
let MPS be non empty right_complementable Abelian add-associative right_zeroed SymStr of F; :: thesis:
assume A2: MPS = SymStr(# {0},md,o,mF,mo #) ; :: thesis:
for x, y being Element of F
for v, w being Element of MPS holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ F) * v = v )

proof

let x, y be Element of F; :: thesis:
let v, w be Element of MPS; :: thesis:
A3: (x * y) * v = x * (y * v)

proof

set z = x * y;
A4: (x * y) * v = mF . ((x * y),v) by A2, VECTSP_1:def 24;
reconsider v = v as Element of MPS ;
reconsider v = v as Element of MPS ;
A5: (x * y) * v = o by A1, A2, A4;
reconsider v = v as Element of MPS ;
A6: mF . (y,v) = o by A1, A2;
reconsider v = v as Element of MPS ;
y * v = o by A2, A6, VECTSP_1:def 24;
then x * (y * v) = mF . (x,o) by A2, VECTSP_1:def 24;
hence (x * y) * v = x * (y * v) by A1, A5; :: thesis:

end;

A7: x * (v + w) = (x * v) + (x * w)

proof

reconsider v = v, w = w as Element of {0} by A2;
A8: md . (v,w) = o by Lm1;
reconsider v = v, w = w as Element of MPS ;
x * (v + w) = mF . (x,o) by A2, A8, VECTSP_1:def 24;
then A9: x * (v + w) = o by A1;
mF . (x,v) = o by A1;
then A10: x * v = o by A2, VECTSP_1:def 24;
mF . (x,w) = o by A1;
then A11: x * w = o by A2, VECTSP_1:def 24;
o = 0. MPS by A2;
hence x * (v + w) = (x * v) + (x * w) by A9, A10, A11, RLVECT_1:10; :: thesis:

end;

A12: (x + y) * v = (x * v) + (y * v)

proof

set z = x + y;
A13: (x + y) * v = mF . ((x + y),v) by A2, VECTSP_1:def 24;
reconsider v = v as Element of MPS ;
reconsider v = v as Element of MPS ;
A14: (x + y) * v = o by A1, A2, A13;
reconsider v = v as Element of MPS ;
A15: mF . (x,v) = o by A1, A2;
reconsider v = v as Element of MPS ;
A16: x * v = o by A2, A15, VECTSP_1:def 24;
reconsider v = v as Element of MPS ;
A17: mF . (y,v) = o by A1, A2;
A18: o = 0. MPS by A2;
reconsider v = v as Element of MPS ;
y * v = o by A2, A17, VECTSP_1:def 24;
hence (x + y) * v = (x * v) + (y * v) by A14, A16, A18, RLVECT_1:10; :: thesis:

end;

(1_ F) * v = v

proof

set one1 = 1_ F;
A19: (1_ F) * v = mF . ((1_ F),v) by A2, VECTSP_1:def 24;
reconsider v = v as Element of MPS ;
mF . ((1_ F),v) = o by A1, A2;
hence (1_ F) * v = v by A2, A19, TARSKI:def 1; :: thesis:

end;

hence ( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ F) * v = v ) by A7, A12, A3; :: thesis:

end;

hence ( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital ) by VECTSP_1:def 26, VECTSP_1:def 27, VECTSP_1:def 28, VECTSP_1:def 29; :: thesis:

end;

Lm5: now

set CV = [:{0},{0}:];
let F be Field; :: thesis:
let mF be Function of [: the carrier of F,{0}:],{0}; :: thesis:
let mo be Relation of {0}; :: thesis:
assume A1: for x being set holds
( x in mo iff ( x in [:{0},{0}:] & ex a, b being Element of {0} st
( x = [a,b] & b = o ) ) ) ; :: thesis:
let MPS be non empty right_complementable Abelian add-associative right_zeroed SymStr of F; :: thesis:
assume A2: MPS = SymStr(# {0},md,o,mF,mo #) ; :: thesis:
thus for a being Element of MPS st a <> 0. MPS holds
ex p being Element of MPS st not a _|_ by A2, TARSKI:def 1; :: thesis:
A3: for a, b being Element of MPS holds
( b _|_ iff ( [a,b] in [:{0},{0}:] & ex c, d being Element of {0} st
( [a,b] = [c,d] & d = o ) ) )

proof

let a, b be Element of MPS; :: thesis:
( b _|_ iff [a,b] in mo ) by A2, ORDERS_2:def 9;
hence ( b _|_ iff ( [a,b] in [:{0},{0}:] & ex c, d being Element of {0} st
( [a,b] = [c,d] & d = o ) ) ) by A1; :: thesis:

end;

A4: for a, b being Element of MPS holds
( b _|_ iff b = o )

proof

let a, b be Element of MPS; :: thesis:
A5: ( b = o implies b _|_ )

proof

consider c, d being Element of MPS such that
A6: ( c = a & d = b ) ;
assume A7: b = o ; :: thesis:
[a,b] = [c,d] by A6;
hence b _|_ by A2, A3, A7; :: thesis:

end;

( b _|_ implies b = o )

proof

assume b _|_ ; :: thesis:
then ex c, d being Element of {0} st
( [a,b] = [c,d] & d = o ) by A3;
hence b = o by ZFMISC_1:33; :: thesis:

end;

hence ( b _|_ iff b = o ) by A5; :: thesis:

end;

thus for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ :: thesis:

proof

let a, b be Element of MPS; :: thesis:
let l be Element of F; :: thesis:
assume b _|_ ; :: thesis:
then b = o by A4;
hence b _|_ by A4; :: thesis:

end;

thus for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ :: thesis:

proof

let a, b, c be Element of MPS; :: thesis:
assume that
A8: a _|_ and
a _|_ ; :: thesis:
a = o by A4, A8;
hence a _|_ by A4; :: thesis:

end;

thus for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ :: thesis:

proof

let a, b, x be Element of MPS; :: thesis:
assume A9: not a _|_ ; :: thesis:
assume for k being Element of F holds not a _|_ ; :: thesis:
a <> o by A4, A9;
hence contradiction by A2, TARSKI:def 1; :: thesis:

end;

let a, b, c be Element of MPS; :: thesis:
assume that
b + c _|_ and
c + a _|_ ; :: thesis:
assume not a + b _|_ ; :: thesis:
then a + b <> o by A4;
hence contradiction by A2, TARSKI:def 1; :: thesis:

end;

definition

let F be Field;
let IT be non empty right_complementable Abelian add-associative right_zeroed SymStr of F;
attr IT is SymSp-like means :Def1: :: SYMSP_1:def 1
for a, b, c, x being Element of IT
for l being Element of F holds
( ( a <> 0. IT implies ex y being Element of IT st not a _|_ ) & ( b _|_ implies b _|_ ) & ( a _|_ & a _|_ implies a _|_ ) & ( not a _|_ implies ex k being Element of F st a _|_ ) & ( b + c _|_ & c + a _|_ implies a + b _|_ ) );

end;

definition

let F be Field;
let S be SymSp of F;
let a, b, x be Element of S;
assume A1: not a _|_ ;
canceled;
canceled;
canceled;
func ProJ (a,b,x) -> Element of F means :Def5: :: SYMSP_1:def 5
for l being Element of F st a _|_ holds
it = l;
existence

proof end;

uniqueness

proof end;

end;

definition

let F be Field;
let S be SymSp of F;
let x, y, a, b be Element of S;
assume A1: ( not a _|_ & (1_ F) + (1_ F) <> 0. F ) ;
func PProJ (a,b,x,y) -> Element of F means :Def6: :: SYMSP_1:def 6
for q being Element of S st not a _|_ & not x _|_ holds
it = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) if ex p being Element of S st
( not a _|_ & not x _|_ )
otherwise it = 0. F;
existence

proof end;

uniqueness

proof end;

consistency ;

end;