:: SYMSP_1 semantic presentation
:: deftheorem Def1 defines _|_ SYMSP_1:def 1 :
set X = {0};
reconsider o = 0 as Element of {0} by TARSKI:def 1;
deffunc H1( Element of {0}, Element of {0}) -> Element of {0} = o;
consider md being BinOp of {0} such that
Lm1:
for x, y being Element of {0} holds md . x,y = H1(x,y)
from BINOP_1:sch 4();
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
not SymStr(# X,md,o,mF,mo #) is empty
end;
Lm3:
for F being Field
for mF being Function of [:the carrier of F,{0}:],{0}
for mo being Relation of {0} holds
( SymStr(# {0},md,o,mF,mo #) is Abelian & SymStr(# {0},md,o,mF,mo #) is add-associative & SymStr(# {0},md,o,mF,mo #) is right_zeroed & SymStr(# {0},md,o,mF,mo #) is right_complementable )
Lm4:
now
let F be
Field;
let mF be
Function of
[:the carrier of F,{0}:],
{0};
assume A1:
for
a being
Element of
F for
x being
Element of
{0} holds
mF . a,
x = o
;
let mo be
Relation of
{0};
let MPS be non
empty Abelian add-associative right_zeroed right_complementable SymStr of
F;
assume A2:
MPS = SymStr(#
{0},
md,
o,
mF,
mo #)
;
thus
MPS is
VectSp-like
proof
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;
let v,
w be
Element of
MPS;
A3:
x * (v + w) = (x * v) + (x * w)
proof
A4:
v + w = md . v,
w
by A2, RLVECT_1:5;
reconsider v =
v,
w =
w as
Element of
{0} by A2;
A5:
md . v,
w = o
by Lm1;
reconsider v =
v,
w =
w as
Element of
MPS ;
x * (v + w) = mF . x,
o
by A2, A4, A5, VECTSP_1:def 24;
then A6:
x * (v + w) = o
by A1;
mF . x,
v = o
by A1;
then A7:
x * v = o
by A2, VECTSP_1:def 24;
mF . x,
w = o
by A1;
then A8:
x * w = o
by A2, VECTSP_1:def 24;
o = 0. MPS
by A2;
hence
x * (v + w) = (x * v) + (x * w)
by A6, A7, A8, RLVECT_1:10;
end;
A9:
(x + y) * v = (x * v) + (y * v)
proof
set z =
x + y;
A10:
(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 ;
A11:
(x + y) * v = o
by A1, A2, A10;
reconsider v =
v as
Element of
MPS ;
A12:
mF . x,
v = o
by A1, A2;
reconsider v =
v as
Element of
MPS ;
A13:
x * v = o
by A2, A12, VECTSP_1:def 24;
reconsider v =
v as
Element of
MPS ;
A14:
mF . y,
v = o
by A1, A2;
reconsider v =
v as
Element of
MPS ;
A15:
y * v = o
by A2, A14, VECTSP_1:def 24;
o = 0. MPS
by A2;
hence
(x + y) * v = (x * v) + (y * v)
by A11, A13, A15, RLVECT_1:10;
end;
A16:
(x * y) * v = x * (y * v)
proof
set z =
x * y;
A17:
(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 ;
A18:
(x * y) * v = o
by A1, A2, A17;
reconsider v =
v as
Element of
MPS ;
A19:
mF . y,
v = o
by A1, A2;
reconsider v =
v as
Element of
MPS ;
y * v = o
by A2, A19, 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, A18;
end;
(1_ F) * v = v
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 A3, A9, A16;
end;
hence
MPS is
VectSp-like
by VECTSP_1:def 26;
end;
end;
Lm5:
now
let F be
Field;
let mF be
Function of
[:the carrier of F,{0}:],
{0};
assume
for
a being
Element of
F for
x being
Element of
{0} holds
mF . a,
x = o
;
set CV =
[:{0},{0}:];
let mo be
Relation of
{0};
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 ) ) )
;
let MPS be non
empty Abelian add-associative right_zeroed right_complementable SymStr of
F;
assume A2:
MPS = SymStr(#
{0},
md,
o,
mF,
mo #)
;
A3:
for
a,
b being
Element of
MPS holds
(
a _|_ b iff (
[a,b] in [:{0},{0}:] & ex
c,
d being
Element of
{0} st
(
[a,b] = [c,d] &
d = o ) ) )
A4:
for
a,
b being
Element of
MPS holds
(
a _|_ b iff
b = o )
thus
for
a being
Element of
MPS st
a <> 0. MPS holds
ex
p being
Element of
MPS st not
p _|_ a
by A2, TARSKI:def 1;
thus
for
a,
b being
Element of
MPS for
l being
Element of
F st
a _|_ b holds
l * a _|_ b
thus
for
a,
b,
c being
Element of
MPS st
b _|_ a &
c _|_ a holds
b + c _|_ a
thus
for
a,
b,
x being
Element of
MPS st not
b _|_ a holds
ex
k being
Element of
F st
x - (k * b) _|_ a
let a,
b,
c be
Element of
MPS;
assume
(
a _|_ b + c &
b _|_ c + a )
;
assume
not
c _|_ a + b
;
then
a + b <> o
by A4;
hence
contradiction
by A2, TARSKI:def 1;
end;
:: deftheorem Def2 defines SymSp-like SYMSP_1:def 2 :
theorem :: SYMSP_1:1
canceled;
theorem :: SYMSP_1:2
canceled;
theorem :: SYMSP_1:3
canceled;
theorem :: SYMSP_1:4
canceled;
theorem :: SYMSP_1:5
canceled;
theorem :: SYMSP_1:6
canceled;
theorem :: SYMSP_1:7
canceled;
theorem :: SYMSP_1:8
canceled;
theorem :: SYMSP_1:9
canceled;
theorem :: SYMSP_1:10
canceled;
theorem Th11: :: SYMSP_1:11
theorem Th12: :: SYMSP_1:12
theorem Th13: :: SYMSP_1:13
theorem Th14: :: SYMSP_1:14
theorem Th15: :: SYMSP_1:15
theorem Th16: :: SYMSP_1:16
theorem :: SYMSP_1:17
canceled;
theorem :: SYMSP_1:18
canceled;
theorem Th19: :: SYMSP_1:19
theorem Th20: :: SYMSP_1:20
theorem Th21: :: SYMSP_1:21
theorem Th22: :: SYMSP_1:22
theorem Th23: :: SYMSP_1:23
theorem Th24: :: SYMSP_1:24
theorem Th25: :: SYMSP_1:25
:: deftheorem SYMSP_1:def 3 :
canceled;
:: deftheorem SYMSP_1:def 4 :
canceled;
:: deftheorem SYMSP_1:def 5 :
canceled;
:: deftheorem Def6 defines ProJ SYMSP_1:def 6 :
theorem :: SYMSP_1:26
canceled;
theorem Th27: :: SYMSP_1:27
theorem Th28: :: SYMSP_1:28
theorem Th29: :: SYMSP_1:29
theorem :: SYMSP_1:30
theorem Th31: :: SYMSP_1:31
theorem Th32: :: SYMSP_1:32
for
F being
Field for
S being
SymSp of
F for
b,
a,
p,
c being
Element of
S st not
b _|_ a &
p _|_ a holds
(
ProJ a,
(b + p),
c = ProJ a,
b,
c &
ProJ a,
b,
(c + p) = ProJ a,
b,
c )
theorem Th33: :: SYMSP_1:33
theorem Th34: :: SYMSP_1:34
theorem Th35: :: SYMSP_1:35
theorem Th36: :: SYMSP_1:36
theorem Th37: :: SYMSP_1:37
theorem Th38: :: SYMSP_1:38
theorem Th39: :: SYMSP_1:39
theorem Th40: :: SYMSP_1:40
theorem Th41: :: SYMSP_1:41
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
a _|_ p & not
a _|_ q & not
b _|_ p & not
b _|_ q holds
(ProJ a,p,q) * (ProJ b,q,p) = (ProJ p,a,b) * (ProJ q,b,a)
theorem Th42: :: SYMSP_1:42
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
p _|_ a & not
p _|_ x & not
q _|_ a & not
q _|_ x holds
(ProJ a,q,p) * (ProJ p,a,x) = (ProJ x,q,p) * (ProJ q,a,x)
theorem Th43: :: SYMSP_1:43
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
p _|_ a & not
p _|_ x & not
q _|_ a & not
q _|_ x & not
b _|_ 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)
theorem Th44: :: SYMSP_1:44
for
F being
Field for
S being
SymSp of
F for
a,
p,
x,
y being
Element of
S st not
a _|_ p & not
x _|_ p & not
y _|_ p holds
(ProJ p,a,x) * (ProJ x,p,y) = (- (ProJ p,a,y)) * (ProJ y,p,x)
definition
let F be
Field;
let S be
SymSp of
F;
let x,
y,
a,
b be
Element of
S;
assume that A1:
not
b _|_ a
and A2:
(1_ F) + (1_ F) <> 0. F
;
func PProJ a,
b,
x,
y -> Element of
F means :
Def7:
:: SYMSP_1:def 7
for
q being
Element of
S st not
q _|_ a & not
q _|_ x holds
it = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) if ex
p being
Element of
S st
( not
p _|_ a & not
p _|_ x )
it = 0. F if for
p being
Element of
S holds
(
p _|_ a or
p _|_ x )
;
existence
( ( ex p being Element of S st
( not p _|_ a & not p _|_ x ) implies ex b1 being Element of F st
for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) & ( ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) implies ex b1 being Element of F st b1 = 0. F ) )
uniqueness
for b1, b2 being Element of F holds
( ( ex p being Element of S st
( not p _|_ a & not p _|_ x ) & ( for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) & ( for q being Element of S st not q _|_ a & not q _|_ x holds
b2 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) implies b1 = b2 ) & ( ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) & b1 = 0. F & b2 = 0. F implies b1 = b2 ) )
consistency
for b1 being Element of F st ex p being Element of S st
( not p _|_ a & not p _|_ x ) & ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) holds
( ( for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) iff b1 = 0. F )
;
end;
:: deftheorem Def7 defines PProJ SYMSP_1:def 7 :
for
F being
Field for
S being
SymSp of
F for
x,
y,
a,
b being
Element of
S st not
b _|_ a &
(1_ F) + (1_ F) <> 0. F holds
for
b7 being
Element of
F holds
( ( ex
p being
Element of
S st
( not
p _|_ a & not
p _|_ x ) implies (
b7 = PProJ a,
b,
x,
y iff for
q being
Element of
S st not
q _|_ a & not
q _|_ x holds
b7 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) ) & ( ( for
p being
Element of
S holds
(
p _|_ a or
p _|_ x ) ) implies (
b7 = PProJ a,
b,
x,
y iff
b7 = 0. F ) ) );
theorem :: SYMSP_1:45
canceled;
theorem :: SYMSP_1:46
canceled;
theorem Th47: :: SYMSP_1:47
Lm6:
for F being Field
for S being SymSp of F
for x being Element of S holds x _|_ 0. S
theorem Th48: :: SYMSP_1:48
theorem :: SYMSP_1:49
theorem :: SYMSP_1:50
theorem :: SYMSP_1:51
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
b _|_ a holds
PProJ a,
b,
x,
(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)