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

definition

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

end;

registration

let F be Field;

cluster non empty for SymStr over F;

existence

proof end;

end;

notation

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

end;

Lm2: now :: thesis:

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 for SymStr over F;

existence

proof end;

end;

Lm4: now :: thesis:

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 over 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 12;
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 12;
then x * (y * v) = mF . (x,o) by A2, VECTSP_1:def 12;
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 12;
then A9: x * (v + w) = o by A1;
mF . (x,v) = o by A1;
then A10: x * v = o by A2, VECTSP_1:def 12;
mF . (x,w) = o by A1;
then A11: x * w = o by A2, VECTSP_1:def 12;
o = 0. MPS by A2;
hence x * (v + w) = (x * v) + (x * w) by A9, A10, A11, RLVECT_1:4; :: 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 12;
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 12;
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 12;
hence (x + y) * v = (x * v) + (y * v) by A14, A16, A18, RLVECT_1:4; :: thesis:

end;

(1_ F) * v = v

proof

set one1 = 1_ F;
A19: (1_ F) * v = mF . ((1_ F),v) by A2, VECTSP_1:def 12;
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 14, VECTSP_1:def 15, VECTSP_1:def 16, VECTSP_1:def 17; :: thesis:

end;

Lm5: now :: thesis:

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 over 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 5;
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 XTUPLE_0:1; :: 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 over 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;

:: deftheorem Def1 defines SymSp-like SYMSP_1:def 1 :

theorem Th1: :: SYMSP_1:1

for F being Field
for S being SymSp of F
for a being Element of S holds a _|_

proof end;

theorem Th2: :: SYMSP_1:2

for F being Field
for S being SymSp of F
for a, b being Element of S st b _|_ holds
a _|_

proof end;

theorem Th3: :: SYMSP_1:3

for F being Field
for S being SymSp of F
for a, b, c being Element of S st not b _|_ & b _|_ holds
not b _|_

proof end;

theorem Th4: :: SYMSP_1:4

for F being Field
for S being SymSp of F
for b, a, c being Element of S st not a _|_ & a _|_ holds
not a _|_

proof end;

theorem Th5: :: SYMSP_1:5

for F being Field
for S being SymSp of F
for b, a being Element of S
for l being Element of F st not a _|_ & not l = 0. F holds
( not a _|_ & not l * a _|_ )

proof end;

theorem Th6: :: SYMSP_1:6

for F being Field
for S being SymSp of F
for a, b being Element of S st b _|_ holds
b _|_

proof end;

theorem Th7: :: SYMSP_1:7

for F being Field
for S being SymSp of F
for a, c, b being Element of S holds
( c _|_ or not c _|_ or not c _|_ )

proof end;

theorem Th8: :: SYMSP_1:8

for F being Field
for S being SymSp of F
for a9, a, b, b9 being Element of S st not a _|_ & b _|_ & not b _|_ & a _|_ holds
( not a _|_ & not b _|_ )

proof end;

theorem Th9: :: SYMSP_1:9

for F being Field
for S being SymSp of F
for a, b being Element of S st a <> 0. S & b <> 0. S holds
ex p being Element of S st
( not a _|_ & not b _|_ )

proof end;

theorem Th10: :: SYMSP_1:10

for F being Field
for S being SymSp of F
for a, b, c being Element of S st (1_ F) + (1_ F) <> 0. F & a <> 0. S & b <> 0. S & c <> 0. S holds
ex p being Element of S st
( not a _|_ & not b _|_ & not c _|_ )

proof end;

theorem Th11: :: SYMSP_1:11

for F being Field
for S being SymSp of F
for a, b, d, c being Element of S st d _|_ & d _|_ holds
d _|_

proof end;

theorem Th12: :: SYMSP_1:12

for F being Field
for S being SymSp of F
for b, a, x being Element of S
for k, l being Element of F st not a _|_ & a _|_ & a _|_ holds
k = l

proof end;

theorem Th13: :: SYMSP_1:13

for F being Field
for S being SymSp of F
for a being Element of S st (1_ F) + (1_ F) <> 0. F holds
a _|_

proof end;

definition

let F be Field;
let S be SymSp of F;
let a, b, x be Element of S;
assume A1: not a _|_ ;

func ProJ (a,b,x) -> Element of F means :Def2: :: SYMSP_1:def 2
for l being Element of F st a _|_ holds
it = l;

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines ProJ SYMSP_1:def 2 :

theorem Th14: :: SYMSP_1:14

for F being Field
for S being SymSp of F
for b, a, x being Element of S st not a _|_ holds
a _|_

proof end;

theorem Th15: :: SYMSP_1:15

for F being Field
for S being SymSp of F
for b, a, x being Element of S
for l being Element of F st not a _|_ holds
ProJ (a,b,(l * x)) = l * (ProJ (a,b,x))

proof end;

theorem Th16: :: SYMSP_1:16

for F being Field
for S being SymSp of F
for b, a, x, y being Element of S st not a _|_ holds
ProJ (a,b,(x + y)) = (ProJ (a,b,x)) + (ProJ (a,b,y))

proof end;

theorem :: SYMSP_1:17

for F being Field
for S being SymSp of F
for b, a, x being Element of S
for l being Element of F st not a _|_ & l <> 0. F holds
ProJ (a,(l * b),x) = (l ") * (ProJ (a,b,x))

proof end;

theorem Th18: :: SYMSP_1:18

for F being Field
for S being SymSp of F
for b, a, x being Element of S
for l being Element of F st not a _|_ & l <> 0. F holds
ProJ ((l * a),b,x) = ProJ (a,b,x)

proof end;

theorem Th19: :: SYMSP_1:19

for F being Field
for S being SymSp of F
for b, a, p, c being Element of S st not a _|_ & a _|_ holds
( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) )

proof end;

theorem Th20: :: SYMSP_1:20

for F being Field
for S being SymSp of F
for b, a, p, c being Element of S st not a _|_ & b _|_ & c _|_ holds
ProJ ((a + p),b,c) = ProJ (a,b,c)

proof end;

theorem Th21: :: SYMSP_1:21

for F being Field
for S being SymSp of F
for b, a, c being Element of S st not a _|_ & a _|_ holds
ProJ (a,b,c) = 1_ F

proof end;

theorem Th22: :: SYMSP_1:22

for F being Field
for S being SymSp of F
for b, a being Element of S st not a _|_ holds
ProJ (a,b,b) = 1_ F

proof end;

theorem Th23: :: SYMSP_1:23

for F being Field
for S being SymSp of F
for b, a, x being Element of S st not a _|_ holds
( a _|_ iff ProJ (a,b,x) = 0. F )

proof end;

theorem Th24: :: SYMSP_1:24

for F being Field
for S being SymSp of F
for b, a, q, p being Element of S st not a _|_ & not a _|_ holds
(ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p)

proof end;

theorem Th25: :: SYMSP_1:25

for F being Field
for S being SymSp of F
for b, a, c being Element of S st not a _|_ & not a _|_ holds
ProJ (a,b,c) = (ProJ (a,c,b)) "

proof end;

theorem Th26: :: SYMSP_1:26

for F being Field
for S being SymSp of F
for b, a, c being Element of S st not a _|_ & c + a _|_ holds
ProJ (a,b,c) = ProJ (c,b,a)

proof end;

theorem Th27: :: SYMSP_1:27

for F being Field
for S being SymSp of F
for a, b, c being Element of S st not b _|_ & not b _|_ holds
ProJ (c,b,a) = (- ((ProJ (b,a,c)) ")) * (ProJ (a,b,c))

proof end;

theorem Th28: :: SYMSP_1:28

for F being Field
for S being SymSp of F
for a, p, q, b being Element of S st (1_ F) + (1_ F) <> 0. F & not p _|_ & not q _|_ & not q _|_ holds
(ProJ (a,p,q)) * (ProJ (b,q,p)) = (ProJ (p,a,b)) * (ProJ (q,b,a))

proof end;

theorem Th29: :: SYMSP_1:29

for F being Field
for S being SymSp of F
for p, a, x, q being Element of S st (1_ F) + (1_ F) <> 0. F & not a _|_ & not x _|_ & not a _|_ & not x _|_ holds
(ProJ (a,q,p)) * (ProJ (p,a,x)) = (ProJ (x,q,p)) * (ProJ (q,a,x))

proof end;

theorem Th30: :: SYMSP_1:30

for F being Field
for S being SymSp of F
for p, a, x, q, b, y being Element of S st (1_ F) + (1_ F) <> 0. F & not a _|_ & not x _|_ & not a _|_ & not x _|_ & not a _|_ holds
((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y))

proof end;

theorem Th31: :: SYMSP_1:31

for F being Field
for S being SymSp of F
for a, p, x, y being Element of S st not p _|_ & not p _|_ & not p _|_ holds
(ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x))

proof 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 :Def3: :: SYMSP_1:def 3
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;

:: deftheorem Def3 defines PProJ SYMSP_1:def 3 :

theorem Th32: :: SYMSP_1:32

for F being Field
for S being SymSp of F
for b, a, x, y being Element of S st (1_ F) + (1_ F) <> 0. F & not a _|_ & x = 0. S holds
PProJ (a,b,x,y) = 0. F

proof end;

theorem Th33: :: SYMSP_1:33

for F being Field
for S being SymSp of F
for b, a, x, y being Element of S st (1_ F) + (1_ F) <> 0. F & not a _|_ holds
( PProJ (a,b,x,y) = 0. F iff x _|_ )

proof end;

theorem :: SYMSP_1:34

for F being Field
for S being SymSp of F
for b, a, x, y being Element of S st (1_ F) + (1_ F) <> 0. F & not a _|_ holds
PProJ (a,b,x,y) = - (PProJ (a,b,y,x))

proof end;

theorem :: SYMSP_1:35

for F being Field
for S being SymSp of F
for b, a, x, y being Element of S
for l being Element of F st (1_ F) + (1_ F) <> 0. F & not a _|_ holds
PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))

proof end;

theorem :: SYMSP_1:36

for F being Field
for S being SymSp of F
for b, a, x, y, z being Element of S st (1_ F) + (1_ F) <> 0. F & not a _|_ holds
PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z))

proof end;