:: ORTSP_1 semantic presentation
begin
reconsider X = {0} as non empty set ;
reconsider o = 0 as Element of X by TARSKI:def_1;
deffunc H1( Element of X, Element of X) -> Element of X = o;
consider md being BinOp of X such that
Lm1: for x, y being Element of X holds md . (x,y) = H1(x,y) from BINOP_1:sch_4();
Lm2: now__::_thesis:_for_F_being_Field_ex_mo_being_Relation_of_X_st_
for_x_being_set_holds_
(_x_in_mo_iff_(_x_in_[:X,X:]_&_ex_a,_b_being_Element_of_X_st_
(_x_=_[a,b]_&_b_=_o_)_)_)
defpred S1[ set ] means ex a, b being Element of X st
( $1 = [a,b] & b = o );
set CV = [:X,X:];
let F be Field; ::_thesis: ex mo being Relation of X st
for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) )
consider mo being set such that
A1: for x being set holds
( x in mo iff ( x in [:X,X:] & S1[x] ) ) from XBOOLE_0:sch_1();
mo c= [:X,X:]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in mo or x in [:X,X:] )
thus ( not x in mo or x in [:X,X:] ) by A1; ::_thesis: verum
end;
then reconsider mo = mo as Relation of X ;
take mo = mo; ::_thesis: for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) )
thus for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) ) by A1; ::_thesis: verum
end;
Lm3: for F being Field
for mF being Function of [: the carrier of F,X:],X
for mo being Relation of X holds
( SymStr(# X,md,o,mF,mo #) is Abelian & SymStr(# X,md,o,mF,mo #) is add-associative & SymStr(# X,md,o,mF,mo #) is right_zeroed & SymStr(# X,md,o,mF,mo #) is right_complementable )
proof
let F be Field; ::_thesis: for mF being Function of [: the carrier of F,X:],X
for mo being Relation of X holds
( SymStr(# X,md,o,mF,mo #) is Abelian & SymStr(# X,md,o,mF,mo #) is add-associative & SymStr(# X,md,o,mF,mo #) is right_zeroed & SymStr(# X,md,o,mF,mo #) is right_complementable )
let mF be Function of [: the carrier of F,X:],X; ::_thesis: for mo being Relation of X holds
( SymStr(# X,md,o,mF,mo #) is Abelian & SymStr(# X,md,o,mF,mo #) is add-associative & SymStr(# X,md,o,mF,mo #) is right_zeroed & SymStr(# X,md,o,mF,mo #) is right_complementable )
let mo be Relation of X; ::_thesis: ( SymStr(# X,md,o,mF,mo #) is Abelian & SymStr(# X,md,o,mF,mo #) is add-associative & SymStr(# X,md,o,mF,mo #) is right_zeroed & SymStr(# X,md,o,mF,mo #) is right_complementable )
set H = SymStr(# X,md,o,mF,mo #);
A1: for x being Element of SymStr(# X,md,o,mF,mo #) holds x + (0. SymStr(# X,md,o,mF,mo #)) = x
proof
let x be Element of SymStr(# X,md,o,mF,mo #); ::_thesis: x + (0. SymStr(# X,md,o,mF,mo #)) = x
md . (x,(0. SymStr(# X,md,o,mF,mo #))) = o by Lm1;
hence x + (0. SymStr(# X,md,o,mF,mo #)) = x by TARSKI:def_1; ::_thesis: verum
end;
A2: SymStr(# X,md,o,mF,mo #) is right_complementable
proof
let x be Element of SymStr(# X,md,o,mF,mo #); :: according to ALGSTR_0:def_16 ::_thesis: x is right_complementable
take - x ; :: according to ALGSTR_0:def_11 ::_thesis: x + (- x) = 0. SymStr(# X,md,o,mF,mo #)
0. SymStr(# X,md,o,mF,mo #) = o by STRUCT_0:def_6;
hence x + (- x) = 0. SymStr(# X,md,o,mF,mo #) by Lm1; ::_thesis: verum
end;
A3: for x, y, z being Element of SymStr(# X,md,o,mF,mo #) holds (x + y) + z = x + (y + z)
proof
let x, y, z be Element of SymStr(# X,md,o,mF,mo #); ::_thesis: (x + y) + z = x + (y + z)
reconsider x = x, y = y, z = z as Element of X ;
md . (x,(md . (y,z))) = o by Lm1;
hence (x + y) + z = x + (y + z) by Lm1; ::_thesis: verum
end;
for x, y being Element of SymStr(# X,md,o,mF,mo #) holds x + y = y + x
proof
let x, y be Element of SymStr(# X,md,o,mF,mo #); ::_thesis: x + y = y + x
md . (x,y) = o by Lm1;
hence x + y = y + x by Lm1; ::_thesis: verum
end;
hence ( SymStr(# X,md,o,mF,mo #) is Abelian & SymStr(# X,md,o,mF,mo #) is add-associative & SymStr(# X,md,o,mF,mo #) is right_zeroed & SymStr(# X,md,o,mF,mo #) is right_complementable ) by A3, A1, A2, RLVECT_1:def_2, RLVECT_1:def_3, RLVECT_1:def_4; ::_thesis: verum
end;
Lm4: now__::_thesis:_for_F_being_Field
for_mF_being_Function_of_[:_the_carrier_of_F,X:],X_st_(_for_a_being_Element_of_F
for_x_being_Element_of_X_holds_mF_._[a,x]_=_o_)_holds_
for_mo_being_Relation_of_X
for_MPS_being_non_empty_right_complementable_Abelian_add-associative_right_zeroed_SymStr_over_F_st_MPS_=_SymStr(#_X,md,o,mF,mo_#)_holds_
(_MPS_is_vector-distributive_&_MPS_is_scalar-distributive_&_MPS_is_scalar-associative_&_MPS_is_scalar-unital_)
let F be Field; ::_thesis: for mF being Function of [: the carrier of F,X:],X st ( for a being Element of F
for x being Element of X holds mF . [a,x] = o ) holds
for mo being Relation of X
for MPS being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st MPS = SymStr(# X,md,o,mF,mo #) holds
( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital )
let mF be Function of [: the carrier of F,X:],X; ::_thesis: ( ( for a being Element of F
for x being Element of X holds mF . [a,x] = o ) implies for mo being Relation of X
for MPS being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st MPS = SymStr(# X,md,o,mF,mo #) holds
( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital ) )
assume A1: for a being Element of F
for x being Element of X holds mF . [a,x] = o ; ::_thesis: for mo being Relation of X
for MPS being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st MPS = SymStr(# X,md,o,mF,mo #) holds
( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital )
let mo be Relation of X; ::_thesis: for MPS being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st MPS = SymStr(# X,md,o,mF,mo #) holds
( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital )
let MPS be non empty right_complementable Abelian add-associative right_zeroed SymStr over F; ::_thesis: ( MPS = SymStr(# X,md,o,mF,mo #) implies ( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital ) )
assume A2: MPS = SymStr(# X,md,o,mF,mo #) ; ::_thesis: ( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital )
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: 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 )
let v, w be Element of MPS; ::_thesis: ( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ F) * v = v )
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: verum
end;
A7: x * (v + w) = (x * v) + (x * w)
proof
reconsider v = v, w = w as Element of MPS ;
A8: o = 0. MPS by A2, STRUCT_0:def_6;
reconsider v = v, w = w as Element of X by A2;
A9: md . (v,w) = o by Lm1;
reconsider v = v, w = w as Element of MPS ;
x * (v + w) = mF . (x,o) by A2, A9, VECTSP_1:def_12;
then A10: x * (v + w) = o by A1;
reconsider w = w as Element of MPS ;
reconsider v = v as Element of MPS ;
A11: mF . (x,v) = o by A1;
reconsider v = v as Element of MPS ;
A12: mF . (x,w) = o by A1;
reconsider w = w as Element of MPS ;
A13: x * w = o by A2, A12, VECTSP_1:def_12;
x * v = o by A2, A11, VECTSP_1:def_12;
hence x * (v + w) = (x * v) + (x * w) by A10, A13, A8, RLVECT_1:4; ::_thesis: verum
end;
A14: (x + y) * v = (x * v) + (y * v)
proof
set z = x + y;
A15: (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 ;
A16: (x + y) * v = o by A1, A2, A15;
reconsider v = v as Element of MPS ;
A17: mF . (x,v) = o by A1, A2;
reconsider v = v as Element of MPS ;
A18: x * v = o by A2, A17, VECTSP_1:def_12;
reconsider v = v as Element of MPS ;
A19: mF . (y,v) = o by A1, A2;
A20: o = 0. MPS by A2, STRUCT_0:def_6;
reconsider v = v as Element of MPS ;
y * v = o by A2, A19, VECTSP_1:def_12;
hence (x + y) * v = (x * v) + (y * v) by A16, A18, A20, RLVECT_1:4; ::_thesis: verum
end;
(1_ F) * v = v
proof
set one1 = 1_ F;
A21: (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, A21, TARSKI:def_1; ::_thesis: verum
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, A14, A3; ::_thesis: verum
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: verum
end;
Lm5: now__::_thesis:_for_F_being_Field
for_mF_being_Function_of_[:_the_carrier_of_F,X:],X
for_mo_being_Relation_of_X_st_(_for_x_being_set_holds_
(_x_in_mo_iff_(_x_in_[:X,X:]_&_ex_a,_b_being_Element_of_X_st_
(_x_=_[a,b]_&_b_=_o_)_)_)_)_holds_
for_MPS_being_non_empty_right_complementable_Abelian_add-associative_right_zeroed_SymStr_over_F_st_MPS_=_SymStr(#_X,md,o,mF,mo_#)_holds_
(_(_for_a,_b,_c,_d_being_Element_of_MPS_st_a_<>_0._MPS_&_b_<>_0._MPS_&_c_<>_0._MPS_&_d_<>_0._MPS_holds_
ex_p_being_Element_of_MPS_st_
(_not_a__|__&_not_b__|__&_not_c__|__&_not_d__|__)_)_&_(_for_a,_b_being_Element_of_MPS
for_l_being_Element_of_F_st_b__|__holds_
b__|__)_&_(_for_a,_b,_c_being_Element_of_MPS_st_a__|__&_a__|__holds_
a__|__)_&_(_for_a,_b,_x_being_Element_of_MPS_st_not_a__|__holds_
ex_k_being_Element_of_F_st_a__|__)_&_(_for_a,_b,_c_being_Element_of_MPS_st_b_-_c__|__&_c_-_a__|__holds_
a_-_b__|__)_)
set CV = [:X,X:];
let F be Field; ::_thesis: for mF being Function of [: the carrier of F,X:],X
for mo being Relation of X st ( for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) ) ) holds
for MPS being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st MPS = SymStr(# X,md,o,mF,mo #) holds
( ( for a, b, c, d being Element of MPS st a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS holds
ex p being Element of MPS st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ ) & ( for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ) & ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) )
let mF be Function of [: the carrier of F,X:],X; ::_thesis: for mo being Relation of X st ( for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) ) ) holds
for MPS being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st MPS = SymStr(# X,md,o,mF,mo #) holds
( ( for a, b, c, d being Element of MPS st a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS holds
ex p being Element of MPS st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ ) & ( for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ) & ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) )
let mo be Relation of X; ::_thesis: ( ( for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) ) ) implies for MPS being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st MPS = SymStr(# X,md,o,mF,mo #) holds
( ( for a, b, c, d being Element of MPS st a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS holds
ex p being Element of MPS st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ ) & ( for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ) & ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) ) )
assume A1: for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) ) ; ::_thesis: for MPS being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st MPS = SymStr(# X,md,o,mF,mo #) holds
( ( for a, b, c, d being Element of MPS st a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS holds
ex p being Element of MPS st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ ) & ( for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ) & ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) )
let MPS be non empty right_complementable Abelian add-associative right_zeroed SymStr over F; ::_thesis: ( MPS = SymStr(# X,md,o,mF,mo #) implies ( ( for a, b, c, d being Element of MPS st a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS holds
ex p being Element of MPS st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ ) & ( for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ) & ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) ) )
assume A2: MPS = SymStr(# X,md,o,mF,mo #) ; ::_thesis: ( ( for a, b, c, d being Element of MPS st a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS holds
ex p being Element of MPS st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ ) & ( for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ) & ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) )
0. MPS = o by A2, TARSKI:def_1;
hence for a, b, c, d being Element of MPS st a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS holds
ex p being Element of MPS st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) by A2, TARSKI:def_1; ::_thesis: ( ( for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ ) & ( for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ) & ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) )
A3: for a, b being Element of MPS holds
( b _|_ iff ( [a,b] in [:X,X:] & ex c, d being Element of X st
( [a,b] = [c,d] & d = o ) ) )
proof
let a, b be Element of MPS; ::_thesis: ( b _|_ iff ( [a,b] in [:X,X:] & ex c, d being Element of X st
( [a,b] = [c,d] & d = o ) ) )
( b _|_ iff [a,b] in mo ) by A2, ORDERS_2:def_5;
hence ( b _|_ iff ( [a,b] in [:X,X:] & ex c, d being Element of X st
( [a,b] = [c,d] & d = o ) ) ) by A1; ::_thesis: verum
end;
A4: for a, b being Element of MPS holds
( b _|_ iff b = o )
proof
let a, b be Element of MPS; ::_thesis: ( b _|_ iff b = o )
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: b _|_
[a,b] = [c,d] by A6;
hence b _|_ by A2, A3, A7; ::_thesis: verum
end;
( b _|_ implies b = o )
proof
assume b _|_ ; ::_thesis: b = o
then ex c, d being Element of X st
( [a,b] = [c,d] & d = o ) by A3;
hence b = o by XTUPLE_0:1; ::_thesis: verum
end;
hence ( b _|_ iff b = o ) by A5; ::_thesis: verum
end;
thus for a, b being Element of MPS
for l being Element of F st b _|_ holds
b _|_ ::_thesis: ( ( for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ) & ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) )
proof
let a, b be Element of MPS; ::_thesis: for l being Element of F st b _|_ holds
b _|_
let l be Element of F; ::_thesis: ( b _|_ implies b _|_ )
assume b _|_ ; ::_thesis: b _|_
then b = o by A4;
hence b _|_ by A4; ::_thesis: verum
end;
thus for a, b, c being Element of MPS st a _|_ & a _|_ holds
a _|_ ::_thesis: ( ( for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ) & ( for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ) )
proof
let a, b, c be Element of MPS; ::_thesis: ( a _|_ & a _|_ implies a _|_ )
assume that
A8: a _|_ and
a _|_ ; ::_thesis: a _|_
a = o by A4, A8;
hence a _|_ by A4; ::_thesis: verum
end;
thus for a, b, x being Element of MPS st not a _|_ holds
ex k being Element of F st a _|_ ::_thesis: for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_
proof
let a, b, x be Element of MPS; ::_thesis: ( not a _|_ implies ex k being Element of F st a _|_ )
assume A9: not a _|_ ; ::_thesis: ex k being Element of F st a _|_
assume for k being Element of F holds not a _|_ ; ::_thesis: contradiction
a <> o by A4, A9;
hence contradiction by A2, TARSKI:def_1; ::_thesis: verum
end;
thus for a, b, c being Element of MPS st b - c _|_ & c - a _|_ holds
a - b _|_ ::_thesis: verum
proof
let a, b, c be Element of MPS; ::_thesis: ( b - c _|_ & c - a _|_ implies a - b _|_ )
assume that
b - c _|_ and
c - a _|_ ; ::_thesis: a - b _|_
assume not a - b _|_ ; ::_thesis: contradiction
then a - b <> o by A4;
hence contradiction by A2, TARSKI:def_1; ::_thesis: verum
end;
end;
definition
let F be Field;
let IT be non empty right_complementable Abelian add-associative right_zeroed SymStr over F;
attrIT is OrtSp-like means :Def1: :: ORTSP_1:def 1
for a, b, c, d, x being Element of IT
for l being Element of F holds
( ( a <> 0. IT & b <> 0. IT & c <> 0. IT & d <> 0. IT implies ex p being Element of IT st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( 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 OrtSp-like ORTSP_1:def_1_:_
for F being Field
for IT being non empty right_complementable Abelian add-associative right_zeroed SymStr over F holds
( IT is OrtSp-like iff for a, b, c, d, x being Element of IT
for l being Element of F holds
( ( a <> 0. IT & b <> 0. IT & c <> 0. IT & d <> 0. IT implies ex p being Element of IT st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( 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 _|_ ) ) );
registration
let F be Field;
cluster non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict OrtSp-like for SymStr over F;
existence
ex b1 being non empty right_complementable Abelian add-associative right_zeroed SymStr over F st
( b1 is OrtSp-like & b1 is vector-distributive & b1 is scalar-distributive & b1 is scalar-associative & b1 is scalar-unital & b1 is strict )
proof
reconsider mF = [: the carrier of F,X:] --> o as Function of [: the carrier of F,X:],X ;
consider mo being Relation of X such that
A1: for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) ) by Lm2;
reconsider MPS = SymStr(# X,md,o,mF,mo #) as non empty right_complementable Abelian add-associative right_zeroed SymStr over F by Lm3;
take MPS ; ::_thesis: ( MPS is OrtSp-like & MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital & MPS is strict )
thus for a, b, c, d, x being Element of MPS
for l being Element of F holds
( ( a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS implies ex p being Element of MPS st
( not a _|_ & not b _|_ & not c _|_ & not d _|_ ) ) & ( 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 _|_ ) ) by A1, Lm5; :: according to ORTSP_1:def_1 ::_thesis: ( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital & MPS is strict )
for a being Element of F
for x being Element of X holds mF . [a,x] = o by FUNCOP_1:7;
hence ( MPS is vector-distributive & MPS is scalar-distributive & MPS is scalar-associative & MPS is scalar-unital ) by Lm4; ::_thesis: MPS is strict
thus MPS is strict ; ::_thesis: verum
end;
end;
definition
let F be Field;
mode OrtSp of F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital OrtSp-like SymStr over F;
end;
theorem Th1: :: ORTSP_1:1
for F being Field
for S being OrtSp of F
for a being Element of S holds a _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F
for a being Element of S holds a _|_
let S be OrtSp of F; ::_thesis: for a being Element of S holds a _|_
let a be Element of S; ::_thesis: a _|_
set 0F = 0. F;
set 0V = 0. S;
A1: now__::_thesis:_(_not_a__|__implies_a__|__)
assume not a _|_ ; ::_thesis: a _|_
then consider m being Element of F such that
A2: a _|_ by Def1;
a _|_ by A2, Def1;
hence a _|_ by VECTSP_1:14; ::_thesis: verum
end;
now__::_thesis:_(_a__|__implies_a__|__)
assume a _|_ ; ::_thesis: a _|_
then a _|_ by Def1;
hence a _|_ by VECTSP_1:14; ::_thesis: verum
end;
hence a _|_ by A1; ::_thesis: verum
end;
theorem Th2: :: ORTSP_1:2
for F being Field
for S being OrtSp of F
for a, b being Element of S st b _|_ holds
a _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F
for a, b being Element of S st b _|_ holds
a _|_
let S be OrtSp of F; ::_thesis: for a, b being Element of S st b _|_ holds
a _|_
let a, b be Element of S; ::_thesis: ( b _|_ implies a _|_ )
set 0V = 0. S;
assume b _|_ ; ::_thesis: a _|_
then - (- b) _|_ by RLVECT_1:17;
then A1: (0. S) - (- b) _|_ by RLVECT_1:4;
(- b) - a _|_ by Th1;
then a - (0. S) _|_ by A1, Def1;
then a _|_ by VECTSP_1:18;
then a _|_ by Def1;
then a _|_ by VECTSP_1:14;
hence a _|_ by RLVECT_1:17; ::_thesis: verum
end;
theorem Th3: :: ORTSP_1:3
for F being Field
for S being OrtSp of F
for a, b, c being Element of S st not b _|_ & b _|_ holds
not b _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F
for a, b, c being Element of S st not b _|_ & b _|_ holds
not b _|_
let S be OrtSp of F; ::_thesis: for a, b, c being Element of S st not b _|_ & b _|_ holds
not b _|_
let a, b, c be Element of S; ::_thesis: ( not b _|_ & b _|_ implies not b _|_ )
assume that
A1: not b _|_ and
A2: b _|_ ; ::_thesis: not b _|_
assume b _|_ ; ::_thesis: contradiction
then b _|_ by Def1;
then b _|_ by VECTSP_1:14;
then b _|_ by A2, Def1;
then b _|_ by RLVECT_1:def_3;
then b _|_ by RLVECT_1:5;
hence contradiction by A1, RLVECT_1:4; ::_thesis: verum
end;
theorem Th4: :: ORTSP_1:4
for F being Field
for S being OrtSp of F
for b, a, c being Element of S st not a _|_ & a _|_ holds
not a _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, c being Element of S st not a _|_ & a _|_ holds
not a _|_
let S be OrtSp of F; ::_thesis: for b, a, c being Element of S st not a _|_ & a _|_ holds
not a _|_
let b, a, c be Element of S; ::_thesis: ( not a _|_ & a _|_ implies not a _|_ )
assume that
A1: not a _|_ and
A2: a _|_ ; ::_thesis: not a _|_
a _|_ by A2, Def1;
then A3: a _|_ by VECTSP_1:14;
assume a _|_ ; ::_thesis: contradiction
then a _|_ by A3, Def1;
then a _|_ by RLVECT_1:def_3;
then a _|_ by RLVECT_1:5;
hence contradiction by A1, RLVECT_1:4; ::_thesis: verum
end;
theorem Th5: :: ORTSP_1:5
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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 _|_ )
let S be OrtSp of F; ::_thesis: 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 _|_ )
let b, a be Element of S; ::_thesis: for l being Element of F st not a _|_ & not l = 0. F holds
( not a _|_ & not l * a _|_ )
let l be Element of F; ::_thesis: ( not a _|_ & not l = 0. F implies ( not a _|_ & not l * a _|_ ) )
set 1F = 1_ F;
assume that
A1: not a _|_ and
A2: not l = 0. F ; ::_thesis: ( not a _|_ & not l * a _|_ )
A3: now__::_thesis:_not_l_*_a__|_
consider k being Element of F such that
A4: k * l = 1_ F by A2, VECTSP_1:def_9;
assume l * a _|_ ; ::_thesis: contradiction
then b _|_ by Th2;
then b _|_ by Def1;
then b _|_ by A4, VECTSP_1:def_16;
then b _|_ by VECTSP_1:def_17;
hence contradiction by A1, Th2; ::_thesis: verum
end;
now__::_thesis:_not_a__|_
consider k being Element of F such that
A5: k * l = 1_ F by A2, VECTSP_1:def_9;
assume a _|_ ; ::_thesis: contradiction
then a _|_ by Def1;
then a _|_ by A5, VECTSP_1:def_16;
hence contradiction by A1, VECTSP_1:def_17; ::_thesis: verum
end;
hence ( not a _|_ & not l * a _|_ ) by A3; ::_thesis: verum
end;
theorem Th6: :: ORTSP_1:6
for F being Field
for S being OrtSp of F
for a, b being Element of S st b _|_ holds
b _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F
for a, b being Element of S st b _|_ holds
b _|_
let S be OrtSp of F; ::_thesis: for a, b being Element of S st b _|_ holds
b _|_
let a, b be Element of S; ::_thesis: ( b _|_ implies b _|_ )
assume b _|_ ; ::_thesis: b _|_
then b _|_ by Def1;
hence b _|_ by VECTSP_1:14; ::_thesis: verum
end;
theorem Th7: :: ORTSP_1:7
for F being Field
for S being OrtSp of F
for a, b, d, c being Element of S st d _|_ & d _|_ holds
d _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F
for a, b, d, c being Element of S st d _|_ & d _|_ holds
d _|_
let S be OrtSp of F; ::_thesis: for a, b, d, c being Element of S st d _|_ & d _|_ holds
d _|_
let a, b, d, c be Element of S; ::_thesis: ( d _|_ & d _|_ implies d _|_ )
assume that
A1: d _|_ and
A2: d _|_ ; ::_thesis: d _|_
d _|_ by A1, Th6;
then d _|_ by VECTSP_1:17;
then d _|_ by A2, Def1;
then d _|_ by RLVECT_1:def_3;
then d _|_ by RLVECT_1:def_3;
then d _|_ by RLVECT_1:5;
hence d _|_ by RLVECT_1:4; ::_thesis: verum
end;
theorem Th8: :: ORTSP_1:8
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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
let S be OrtSp of F; ::_thesis: for b, a, x being Element of S
for k, l being Element of F st not a _|_ & a _|_ & a _|_ holds
k = l
let b, a, x be Element of S; ::_thesis: for k, l being Element of F st not a _|_ & a _|_ & a _|_ holds
k = l
let k, l be Element of F; ::_thesis: ( not a _|_ & a _|_ & a _|_ implies k = l )
set 1F = 1_ F;
assume that
A1: not a _|_ and
A2: ( a _|_ & a _|_ ) ; ::_thesis: k = l
a _|_ by A2, Th7;
then a _|_ by VECTSP_1:14;
then a _|_ by VECTSP_1:def_16;
then a _|_ by VECTSP_1:9;
then a _|_ by VECTSP_1:def_8;
then a _|_ by VECTSP_1:def_15;
then a _|_ by Def1;
then A3: a _|_ by VECTSP_1:def_16;
assume not k = l ; ::_thesis: contradiction
then k - l <> 0. F by RLVECT_1:21;
then a _|_ by A3, VECTSP_1:def_10;
hence contradiction by A1, VECTSP_1:def_17; ::_thesis: verum
end;
theorem Th9: :: ORTSP_1:9
for F being Field
for S being OrtSp of F
for a, b being Element of S st a _|_ & b _|_ holds
a - b _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F
for a, b being Element of S st a _|_ & b _|_ holds
a - b _|_
let S be OrtSp of F; ::_thesis: for a, b being Element of S st a _|_ & b _|_ holds
a - b _|_
let a, b be Element of S; ::_thesis: ( a _|_ & b _|_ implies a - b _|_ )
set 0V = 0. S;
assume that
A1: a _|_ and
A2: b _|_ ; ::_thesis: a - b _|_
a _|_ by A1, Def1;
then a _|_ by VECTSP_1:14;
then - a _|_ by Th2;
then (0. S) + (- a) _|_ by RLVECT_1:4;
then (b + (- b)) + (- a) _|_ by RLVECT_1:5;
then b + ((- b) - a) _|_ by RLVECT_1:def_3;
then A3: b - (a + b) _|_ by VECTSP_1:17;
b + (0. S) _|_ by A2, RLVECT_1:4;
then b + (a + (- a)) _|_ by RLVECT_1:5;
then (a + b) - a _|_ by RLVECT_1:def_3;
hence a - b _|_ by A3, Def1; ::_thesis: verum
end;
theorem :: ORTSP_1:10
for F being Field
for S being OrtSp of F st (1_ F) + (1_ F) <> 0. F & ex a being Element of S st a <> 0. S holds
ex b being Element of S st not b _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F st (1_ F) + (1_ F) <> 0. F & ex a being Element of S st a <> 0. S holds
ex b being Element of S st not b _|_
let S be OrtSp of F; ::_thesis: ( (1_ F) + (1_ F) <> 0. F & ex a being Element of S st a <> 0. S implies ex b being Element of S st not b _|_ )
set 1F = 1_ F;
set 0V = 0. S;
assume that
A1: (1_ F) + (1_ F) <> 0. F and
A2: ex a being Element of S st a <> 0. S ; ::_thesis: not for b being Element of S holds b _|_
consider a being Element of S such that
A3: a <> 0. S by A2;
assume A4: for b being Element of S holds b _|_ ; ::_thesis: contradiction
A5: for c, d being Element of S holds d _|_
proof
let c, d be Element of S; ::_thesis: d _|_
( d _|_ & c _|_ ) by A4;
then d - c _|_ by Th9;
then A6: d + c _|_ by Th2;
d + c _|_ by A4;
then d + c _|_ by A6, Def1;
then d + c _|_ by RLVECT_1:def_3;
then d + c _|_ by RLVECT_1:def_3;
then d + c _|_ by RLVECT_1:5;
then d + c _|_ by RLVECT_1:4;
then d + c _|_ by VECTSP_1:def_17;
then d + c _|_ by VECTSP_1:def_17;
then d + c _|_ by VECTSP_1:def_15;
then d + c _|_ by Def1;
then d + c _|_ by VECTSP_1:def_16;
then d + c _|_ by A1, VECTSP_1:def_10;
then d + c _|_ by VECTSP_1:def_17;
then A7: d _|_ by Th2;
d _|_ by A4, Th6;
then d _|_ by A7, Def1;
then d _|_ by RLVECT_1:def_3;
then d _|_ by RLVECT_1:5;
hence d _|_ by RLVECT_1:4; ::_thesis: verum
end;
ex c being Element of S st
( not a _|_ & not a _|_ & not a _|_ & not a _|_ ) by A3, Def1;
hence contradiction by A5; ::_thesis: verum
end;
definition
let F be Field;
let S be OrtSp of F;
let a, b, x be Element of S;
assume A1: not a _|_ ;
func ProJ (a,b,x) -> Element of F means :Def2: :: ORTSP_1:def 2
for l being Element of F st a _|_ holds
it = l;
existence
ex b1 being Element of F st
for l being Element of F st a _|_ holds
b1 = l
proof
consider k being Element of F such that
A2: a _|_ by A1, Def1;
take k ; ::_thesis: for l being Element of F st a _|_ holds
k = l
let l be Element of F; ::_thesis: ( a _|_ implies k = l )
assume a _|_ ; ::_thesis: k = l
hence k = l by A1, A2, Th8; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of F st ( for l being Element of F st a _|_ holds
b1 = l ) & ( for l being Element of F st a _|_ holds
b2 = l ) holds
b1 = b2
proof
let l1, l2 be Element of F; ::_thesis: ( ( for l being Element of F st a _|_ holds
l1 = l ) & ( for l being Element of F st a _|_ holds
l2 = l ) implies l1 = l2 )
assume that
A3: for l being Element of F st a _|_ holds
l1 = l and
A4: for l being Element of F st a _|_ holds
l2 = l ; ::_thesis: l1 = l2
consider k being Element of F such that
A5: a _|_ by A1, Def1;
l1 = k by A3, A5;
hence l1 = l2 by A4, A5; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines ProJ ORTSP_1:def_2_:_
for F being Field
for S being OrtSp of F
for a, b, x being Element of S st not a _|_ holds
for b6 being Element of F holds
( b6 = ProJ (a,b,x) iff for l being Element of F st a _|_ holds
b6 = l );
theorem Th11: :: ORTSP_1:11
for F being Field
for S being OrtSp of F
for b, a, x being Element of S st not a _|_ holds
a _|_
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, x being Element of S st not a _|_ holds
a _|_
let S be OrtSp of F; ::_thesis: for b, a, x being Element of S st not a _|_ holds
a _|_
let b, a, x be Element of S; ::_thesis: ( not a _|_ implies a _|_ )
assume A1: not a _|_ ; ::_thesis: a _|_
then ex l being Element of F st a _|_ by Def1;
hence a _|_ by A1, Def2; ::_thesis: verum
end;
theorem Th12: :: ORTSP_1:12
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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))
let S be OrtSp of F; ::_thesis: 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))
let b, a, x be Element of S; ::_thesis: for l being Element of F st not a _|_ holds
ProJ (a,b,(l * x)) = l * (ProJ (a,b,x))
let l be Element of F; ::_thesis: ( not a _|_ implies ProJ (a,b,(l * x)) = l * (ProJ (a,b,x)) )
set L = x - ((ProJ (a,b,x)) * b);
A1: l * (x - ((ProJ (a,b,x)) * b)) = (l * x) - (l * ((ProJ (a,b,x)) * b)) by VECTSP_1:23
.= (l * x) - ((l * (ProJ (a,b,x))) * b) by VECTSP_1:def_16 ;
assume A2: not a _|_ ; ::_thesis: ProJ (a,b,(l * x)) = l * (ProJ (a,b,x))
then A3: a _|_ by Th11;
a _|_ by A2, Th11;
then a _|_ by A1, Def1;
hence ProJ (a,b,(l * x)) = l * (ProJ (a,b,x)) by A2, A3, Th8; ::_thesis: verum
end;
theorem Th13: :: ORTSP_1:13
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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))
let S be OrtSp of F; ::_thesis: 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))
let b, a, x, y be Element of S; ::_thesis: ( not a _|_ implies ProJ (a,b,(x + y)) = (ProJ (a,b,x)) + (ProJ (a,b,y)) )
set 1F = 1_ F;
set L = (x - ((ProJ (a,b,x)) * b)) + (y - ((ProJ (a,b,y)) * b));
A1: (x - ((ProJ (a,b,x)) * b)) + (y - ((ProJ (a,b,y)) * b)) = (x + (- ((ProJ (a,b,x)) * b))) + (y - ((ProJ (a,b,y)) * b))
.= (x + (- ((ProJ (a,b,x)) * b))) + (y + (- ((ProJ (a,b,y)) * b)))
.= (((- ((ProJ (a,b,x)) * b)) + x) + y) + (- ((ProJ (a,b,y)) * b)) by RLVECT_1:def_3
.= ((x + y) + (- ((ProJ (a,b,x)) * b))) + (- ((ProJ (a,b,y)) * b)) by RLVECT_1:def_3
.= (x + y) + ((- ((ProJ (a,b,x)) * b)) + (- ((ProJ (a,b,y)) * b))) by RLVECT_1:def_3
.= (x + y) + (((1_ F) * (- ((ProJ (a,b,x)) * b))) + (- ((ProJ (a,b,y)) * b))) by VECTSP_1:def_17
.= (x + y) + (((1_ F) * (- ((ProJ (a,b,x)) * b))) + ((1_ F) * (- ((ProJ (a,b,y)) * b)))) by VECTSP_1:def_17
.= (x + y) + (((1_ F) * ((- (ProJ (a,b,x))) * b)) + ((1_ F) * (- ((ProJ (a,b,y)) * b)))) by VECTSP_1:21
.= (x + y) + (((1_ F) * ((- (ProJ (a,b,x))) * b)) + ((1_ F) * ((- (ProJ (a,b,y))) * b))) by VECTSP_1:21
.= (x + y) + ((((1_ F) * (- (ProJ (a,b,x)))) * b) + ((1_ F) * ((- (ProJ (a,b,y))) * b))) by VECTSP_1:def_16
.= (x + y) + ((((- (ProJ (a,b,x))) * (1_ F)) * b) + (((1_ F) * (- (ProJ (a,b,y)))) * b)) by VECTSP_1:def_16
.= (x + y) + (((- (ProJ (a,b,x))) * b) + (((- (ProJ (a,b,y))) * (1_ F)) * b)) by VECTSP_1:def_8
.= (x + y) + (((- (ProJ (a,b,x))) * b) + ((- (ProJ (a,b,y))) * b)) by VECTSP_1:def_8
.= (x + y) + (((- (ProJ (a,b,x))) + (- (ProJ (a,b,y)))) * b) by VECTSP_1:def_15
.= (x + y) + ((- ((ProJ (a,b,x)) + (ProJ (a,b,y)))) * b) by RLVECT_1:31
.= (x + y) - (((ProJ (a,b,x)) + (ProJ (a,b,y))) * b) by VECTSP_1:21 ;
assume A2: not a _|_ ; ::_thesis: ProJ (a,b,(x + y)) = (ProJ (a,b,x)) + (ProJ (a,b,y))
then ( a _|_ & a _|_ ) by Th11;
then A3: a _|_ by Def1;
a _|_ by A2, Th11;
hence ProJ (a,b,(x + y)) = (ProJ (a,b,x)) + (ProJ (a,b,y)) by A2, A1, A3, Th8; ::_thesis: verum
end;
theorem :: ORTSP_1:14
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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))
let S be OrtSp of F; ::_thesis: 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))
let b, a, x be Element of S; ::_thesis: for l being Element of F st not a _|_ & l <> 0. F holds
ProJ (a,(l * b),x) = (l ") * (ProJ (a,b,x))
let l be Element of F; ::_thesis: ( not a _|_ & l <> 0. F implies ProJ (a,(l * b),x) = (l ") * (ProJ (a,b,x)) )
assume that
A1: not a _|_ and
A2: l <> 0. F ; ::_thesis: ProJ (a,(l * b),x) = (l ") * (ProJ (a,b,x))
set L = x - ((ProJ (a,(l * b),x)) * (l * b));
not a _|_ by A1, A2, Th5;
then A3: a _|_ by Th11;
A4: x - ((ProJ (a,(l * b),x)) * (l * b)) = x - (((ProJ (a,(l * b),x)) * l) * b) by VECTSP_1:def_16;
a _|_ by A1, Th11;
then (ProJ (a,b,x)) * (l ") = ((ProJ (a,(l * b),x)) * l) * (l ") by A1, A3, A4, Th8;
then (ProJ (a,b,x)) * (l ") = (ProJ (a,(l * b),x)) * (l * (l ")) by GROUP_1:def_3;
then (l ") * (ProJ (a,b,x)) = (ProJ (a,(l * b),x)) * (1_ F) by A2, VECTSP_1:def_10;
hence ProJ (a,(l * b),x) = (l ") * (ProJ (a,b,x)) by VECTSP_1:def_8; ::_thesis: verum
end;
theorem Th15: :: ORTSP_1:15
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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)
let S be OrtSp of F; ::_thesis: 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)
let b, a, x be Element of S; ::_thesis: for l being Element of F st not a _|_ & l <> 0. F holds
ProJ ((l * a),b,x) = ProJ (a,b,x)
let l be Element of F; ::_thesis: ( not a _|_ & l <> 0. F implies ProJ ((l * a),b,x) = ProJ (a,b,x) )
assume that
A1: not a _|_ and
A2: l <> 0. F ; ::_thesis: ProJ ((l * a),b,x) = ProJ (a,b,x)
not l * a _|_ by A1, A2, Th5;
then l * a _|_ by Th11;
then x - ((ProJ ((l * a),b,x)) * b) _|_ by Th2;
then x - ((ProJ ((l * a),b,x)) * b) _|_ by Def1;
then x - ((ProJ ((l * a),b,x)) * b) _|_ by VECTSP_1:def_16;
then x - ((ProJ ((l * a),b,x)) * b) _|_ by A2, VECTSP_1:def_10;
then x - ((ProJ ((l * a),b,x)) * b) _|_ by VECTSP_1:def_17;
then A3: a _|_ by Th2;
a _|_ by A1, Th11;
hence ProJ ((l * a),b,x) = ProJ (a,b,x) by A1, A3, Th8; ::_thesis: verum
end;
theorem :: ORTSP_1:16
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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) )
let S be OrtSp of F; ::_thesis: 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) )
let b, a, p, c be Element of S; ::_thesis: ( not a _|_ & a _|_ implies ( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) ) )
set 0V = 0. S;
assume that
A1: not a _|_ and
A2: a _|_ ; ::_thesis: ( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) )
not a _|_ by A1, A2, Th4;
then a _|_ by Th11;
then a _|_ by VECTSP_1:def_14;
then A3: a _|_ by VECTSP_1:17;
( a _|_ & a _|_ ) by A1, A2, Th6, Th11;
then a _|_ by Def1;
then a _|_ by RLVECT_1:def_3;
then a _|_ by RLVECT_1:def_3;
then a _|_ by RLVECT_1:5;
then A4: a _|_ by RLVECT_1:4;
a _|_ by A2, Def1;
then a _|_ by A3, Def1;
then a _|_ by RLVECT_1:def_3;
then a _|_ by RLVECT_1:5;
then A5: a _|_ by RLVECT_1:4;
a _|_ by A1, Th11;
hence ( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) ) by A1, A5, A4, Th8; ::_thesis: verum
end;
theorem :: ORTSP_1:17
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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)
let S be OrtSp of F; ::_thesis: for b, a, p, c being Element of S st not a _|_ & b _|_ & c _|_ holds
ProJ ((a + p),b,c) = ProJ (a,b,c)
let b, a, p, c be Element of S; ::_thesis: ( not a _|_ & b _|_ & c _|_ implies ProJ ((a + p),b,c) = ProJ (a,b,c) )
assume that
A1: not a _|_ and
A2: b _|_ and
A3: c _|_ ; ::_thesis: ProJ ((a + p),b,c) = ProJ (a,b,c)
p _|_ by A2, Th2;
then p _|_ by Def1;
then A4: p _|_ by Th6;
p _|_ by A3, Th2;
then p _|_ by A4, Def1;
then A5: c - ((ProJ (a,b,c)) * b) _|_ by Th2;
a _|_ by A1, Th11;
then c - ((ProJ (a,b,c)) * b) _|_ by Th2;
then c - ((ProJ (a,b,c)) * b) _|_ by A5, Def1;
then A6: a + p _|_ by Th2;
not b _|_ by A1, Th2;
then not b _|_ by A2, Th4;
then A7: not a + p _|_ by Th2;
then a + p _|_ by Th11;
hence ProJ ((a + p),b,c) = ProJ (a,b,c) by A7, A6, Th8; ::_thesis: verum
end;
theorem Th18: :: ORTSP_1:18
for F being Field
for S being OrtSp of F
for b, a, c being Element of S st not a _|_ & a _|_ holds
ProJ (a,b,c) = 1_ F
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, c being Element of S st not a _|_ & a _|_ holds
ProJ (a,b,c) = 1_ F
let S be OrtSp of F; ::_thesis: for b, a, c being Element of S st not a _|_ & a _|_ holds
ProJ (a,b,c) = 1_ F
let b, a, c be Element of S; ::_thesis: ( not a _|_ & a _|_ implies ProJ (a,b,c) = 1_ F )
assume that
A1: not a _|_ and
A2: a _|_ ; ::_thesis: ProJ (a,b,c) = 1_ F
( a _|_ & a _|_ ) by A1, A2, Th11, VECTSP_1:def_17;
hence ProJ (a,b,c) = 1_ F by A1, Th8; ::_thesis: verum
end;
theorem Th19: :: ORTSP_1:19
for F being Field
for S being OrtSp of F
for b, a being Element of S st not a _|_ holds
ProJ (a,b,b) = 1_ F
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a being Element of S st not a _|_ holds
ProJ (a,b,b) = 1_ F
let S be OrtSp of F; ::_thesis: for b, a being Element of S st not a _|_ holds
ProJ (a,b,b) = 1_ F
let b, a be Element of S; ::_thesis: ( not a _|_ implies ProJ (a,b,b) = 1_ F )
A1: b - b = 0. S by RLVECT_1:5;
assume not a _|_ ; ::_thesis: ProJ (a,b,b) = 1_ F
hence ProJ (a,b,b) = 1_ F by A1, Th1, Th18; ::_thesis: verum
end;
theorem Th20: :: ORTSP_1:20
for F being Field
for S being OrtSp of F
for b, a, x being Element of S st not a _|_ holds
( a _|_ iff ProJ (a,b,x) = 0. F )
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, x being Element of S st not a _|_ holds
( a _|_ iff ProJ (a,b,x) = 0. F )
let S be OrtSp of F; ::_thesis: for b, a, x being Element of S st not a _|_ holds
( a _|_ iff ProJ (a,b,x) = 0. F )
let b, a, x be Element of S; ::_thesis: ( not a _|_ implies ( a _|_ iff ProJ (a,b,x) = 0. F ) )
set 0F = 0. F;
set 0V = 0. S;
A1: now__::_thesis:_(_not_a__|__&_a__|__implies_ProJ_(a,b,x)_=_0._F_)
assume that
A2: not a _|_ and
A3: a _|_ ; ::_thesis: ProJ (a,b,x) = 0. F
a _|_ by A3, RLVECT_1:4;
then a _|_ by RLVECT_1:12;
then A4: a _|_ by VECTSP_1:14;
a _|_ by A2, Th11;
hence ProJ (a,b,x) = 0. F by A2, A4, Th8; ::_thesis: verum
end;
now__::_thesis:_(_not_a__|__&_ProJ_(a,b,x)_=_0._F_implies_a__|__)
assume ( not a _|_ & ProJ (a,b,x) = 0. F ) ; ::_thesis: a _|_
then a _|_ by Th11;
then a _|_ by VECTSP_1:14;
then a _|_ by RLVECT_1:12;
hence a _|_ by RLVECT_1:4; ::_thesis: verum
end;
hence ( not a _|_ implies ( a _|_ iff ProJ (a,b,x) = 0. F ) ) by A1; ::_thesis: verum
end;
theorem Th21: :: ORTSP_1:21
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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)
let S be OrtSp of F; ::_thesis: 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)
let b, a, q, p be Element of S; ::_thesis: ( not a _|_ & not a _|_ implies (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p) )
assume that
A1: not a _|_ and
A2: not a _|_ ; ::_thesis: (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p)
( a _|_ & a _|_ ) by A1, A2, Th11;
then a _|_ by Def1;
then a _|_ by RLVECT_1:def_3;
then a _|_ by RLVECT_1:def_3;
then a _|_ by RLVECT_1:5;
then a _|_ by RLVECT_1:4;
then A3: a _|_ by A1, Th12;
a _|_ by A1, Th11;
then (ProJ (a,q,p)) * (ProJ (a,b,q)) = ProJ (a,b,p) by A1, A3, Th8;
then A4: (ProJ (a,q,p)) * ((ProJ (a,b,q)) * ((ProJ (a,b,q)) ")) = (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") by GROUP_1:def_3;
ProJ (a,b,q) <> 0. F by A1, A2, Th20;
then (ProJ (a,q,p)) * (1_ F) = (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") by A4, VECTSP_1:def_10;
hence (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p) by VECTSP_1:def_8; ::_thesis: verum
end;
theorem Th22: :: ORTSP_1:22
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, c being Element of S st not a _|_ & not a _|_ holds
ProJ (a,b,c) = (ProJ (a,c,b)) "
let S be OrtSp of F; ::_thesis: for b, a, c being Element of S st not a _|_ & not a _|_ holds
ProJ (a,b,c) = (ProJ (a,c,b)) "
let b, a, c be Element of S; ::_thesis: ( not a _|_ & not a _|_ implies ProJ (a,b,c) = (ProJ (a,c,b)) " )
set 1F = 1_ F;
set 0F = 0. F;
assume that
A1: not a _|_ and
A2: not a _|_ ; ::_thesis: ProJ (a,b,c) = (ProJ (a,c,b)) "
A3: ProJ (a,c,b) <> 0. F by A1, A2, Th20;
(ProJ (a,b,b)) * ((ProJ (a,b,c)) ") = ProJ (a,c,b) by A1, A2, Th21;
then ((ProJ (a,b,c)) ") * (1_ F) = ProJ (a,c,b) by A1, Th19;
then A4: (ProJ (a,b,c)) " = ProJ (a,c,b) by VECTSP_1:def_8;
ProJ (a,b,c) <> 0. F by A1, A2, Th20;
then 1_ F = (ProJ (a,c,b)) * (ProJ (a,b,c)) by A4, VECTSP_1:def_10;
then (ProJ (a,c,b)) " = ((ProJ (a,c,b)) ") * ((ProJ (a,c,b)) * (ProJ (a,b,c))) by VECTSP_1:def_8
.= (((ProJ (a,c,b)) ") * (ProJ (a,c,b))) * (ProJ (a,b,c)) by GROUP_1:def_3
.= (ProJ (a,b,c)) * (1_ F) by A3, VECTSP_1:def_10 ;
hence ProJ (a,b,c) = (ProJ (a,c,b)) " by VECTSP_1:def_8; ::_thesis: verum
end;
theorem Th23: :: ORTSP_1:23
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, c being Element of S st not a _|_ & c + a _|_ holds
ProJ (a,b,c) = - (ProJ (c,b,a))
let S be OrtSp of F; ::_thesis: for b, a, c being Element of S st not a _|_ & c + a _|_ holds
ProJ (a,b,c) = - (ProJ (c,b,a))
let b, a, c be Element of S; ::_thesis: ( not a _|_ & c + a _|_ implies ProJ (a,b,c) = - (ProJ (c,b,a)) )
set 1F = 1_ F;
assume that
A1: not a _|_ and
A2: c + a _|_ ; ::_thesis: ProJ (a,b,c) = - (ProJ (c,b,a))
a _|_ by A1, Th11;
then a _|_ by VECTSP_1:21;
then a _|_ by Def1;
then a _|_ by VECTSP_1:def_14;
then a _|_ by VECTSP_1:def_16;
then a _|_ by VECTSP_1:10;
then a _|_ by VECTSP_1:def_8;
then a _|_ by VECTSP_1:14;
then ((ProJ (a,b,c)) * b) - c _|_ by Th2;
then A3: ((ProJ (a,b,c)) * b) - c _|_ by Th6;
c + a _|_ by A2, Def1;
then c - (- a) _|_ by RLVECT_1:17;
then (- a) - ((ProJ (a,b,c)) * b) _|_ by A3, Def1;
then A4: c _|_ by Th2;
( not b _|_ & b _|_ ) by A1, A2, Th2;
then A5: not b _|_ by Th3;
then A6: not c _|_ by Th2;
then c _|_ by Th11;
then ProJ (a,b,c) = ProJ (c,b,(- a)) by A6, A4, Th8
.= ProJ (c,b,((- (1_ F)) * a)) by VECTSP_1:14
.= (- (1_ F)) * (ProJ (c,b,a)) by A5, Th2, Th12
.= - ((ProJ (c,b,a)) * (1_ F)) by VECTSP_1:9 ;
hence ProJ (a,b,c) = - (ProJ (c,b,a)) by VECTSP_1:def_8; ::_thesis: verum
end;
theorem Th24: :: ORTSP_1:24
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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))
let S be OrtSp of F; ::_thesis: 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))
let a, b, c be Element of S; ::_thesis: ( not b _|_ & not b _|_ implies ProJ (c,b,a) = ((ProJ (b,a,c)) ") * (ProJ (a,b,c)) )
set 1F = 1_ F;
assume that
A1: not b _|_ and
A2: not b _|_ ; ::_thesis: ProJ (c,b,a) = ((ProJ (b,a,c)) ") * (ProJ (a,b,c))
ProJ (b,a,c) <> 0. F by A1, A2, Th20;
then A3: - ((ProJ (b,a,c)) ") <> 0. F by VECTSP_1:25;
ProJ (b,a,c) <> 0. F by A1, A2, Th20;
then (- (1_ F)) * (((ProJ (b,a,c)) ") * (ProJ (b,a,c))) = (- (1_ F)) * (1_ F) by VECTSP_1:def_10;
then ((- (1_ F)) * ((ProJ (b,a,c)) ")) * (ProJ (b,a,c)) = (- (1_ F)) * (1_ F) by GROUP_1:def_3;
then ((- (1_ F)) * ((ProJ (b,a,c)) ")) * (ProJ (b,a,c)) = - (1_ F) by VECTSP_1:def_8;
then (- (((ProJ (b,a,c)) ") * (1_ F))) * (ProJ (b,a,c)) = - (1_ F) by VECTSP_1:9;
then (- ((ProJ (b,a,c)) ")) * (ProJ (b,a,c)) = - (1_ F) by VECTSP_1:def_8;
then ProJ (b,a,((- ((ProJ (b,a,c)) ")) * c)) = - (1_ F) by A1, Th12;
then b _|_ by A1, Th11;
then b _|_ by VECTSP_1:14;
then b _|_ by RLVECT_1:17;
then A4: ((- ((ProJ (b,a,c)) ")) * c) + a _|_ by Th2;
not a _|_ by A1, Th2;
then ProJ (a,b,((- ((ProJ (b,a,c)) ")) * c)) = - (ProJ (((- ((ProJ (b,a,c)) ")) * c),b,a)) by A4, Th23;
then ProJ (a,b,((- ((ProJ (b,a,c)) ")) * c)) = - (ProJ (c,b,a)) by A2, A3, Th2, Th15;
then (- ((ProJ (b,a,c)) ")) * (ProJ (a,b,c)) = - (ProJ (c,b,a)) by A1, Th2, Th12;
then (- (((ProJ (b,a,c)) ") * (ProJ (a,b,c)))) * (- (1_ F)) = (- (ProJ (c,b,a))) * (- (1_ F)) by VECTSP_1:9;
then (((ProJ (b,a,c)) ") * (ProJ (a,b,c))) * (1_ F) = (- (ProJ (c,b,a))) * (- (1_ F)) by VECTSP_1:10;
then (((ProJ (b,a,c)) ") * (ProJ (a,b,c))) * (1_ F) = (ProJ (c,b,a)) * (1_ F) by VECTSP_1:10;
then ((ProJ (b,a,c)) ") * (ProJ (a,b,c)) = (ProJ (c,b,a)) * (1_ F) by VECTSP_1:def_8;
hence ProJ (c,b,a) = ((ProJ (b,a,c)) ") * (ProJ (a,b,c)) by VECTSP_1:def_8; ::_thesis: verum
end;
theorem Th25: :: ORTSP_1:25
for F being Field
for S being OrtSp of F
for p, a, x, q being Element of S st not a _|_ & not x _|_ & not a _|_ & not x _|_ holds
(ProJ (a,q,p)) * (ProJ (p,a,x)) = (ProJ (q,a,x)) * (ProJ (x,q,p))
proof
let F be Field; ::_thesis: for S being OrtSp of F
for p, a, x, q being Element of S st not a _|_ & not x _|_ & not a _|_ & not x _|_ holds
(ProJ (a,q,p)) * (ProJ (p,a,x)) = (ProJ (q,a,x)) * (ProJ (x,q,p))
let S be OrtSp of F; ::_thesis: for p, a, x, q being Element of S st not a _|_ & not x _|_ & not a _|_ & not x _|_ holds
(ProJ (a,q,p)) * (ProJ (p,a,x)) = (ProJ (q,a,x)) * (ProJ (x,q,p))
let p, a, x, q be Element of S; ::_thesis: ( not a _|_ & not x _|_ & not a _|_ & not x _|_ implies (ProJ (a,q,p)) * (ProJ (p,a,x)) = (ProJ (q,a,x)) * (ProJ (x,q,p)) )
set 0F = 0. F;
set 1F = 1_ F;
assume that
A1: not a _|_ and
A2: not x _|_ and
A3: not a _|_ and
A4: not x _|_ ; ::_thesis: (ProJ (a,q,p)) * (ProJ (p,a,x)) = (ProJ (q,a,x)) * (ProJ (x,q,p))
A5: ( p <> 0. S & q <> 0. S ) by A1, A3, Th1;
A6: not p _|_ by A1, Th2;
( a <> 0. S & x <> 0. S ) by A1, A2, Th1, Th2;
then consider r being Element of S such that
A7: not p _|_ and
A8: not q _|_ and
A9: not a _|_ and
A10: not x _|_ by A5, Def1;
A11: not r _|_ by A8, Th2;
A12: not r _|_ by A10, Th2;
A13: not r _|_ by A7, Th2;
then A14: ProJ (r,q,p) <> 0. F by A11, Th20;
A15: not r _|_ by A9, Th2;
A16: not q _|_ by A4, Th2;
A17: not p _|_ by A2, Th2;
A18: not q _|_ by A3, Th2;
((ProJ (a,q,p)) * (ProJ (p,a,x))) * ((ProJ (q,a,x)) ") = (((ProJ (a,r,p)) * ((ProJ (a,r,q)) ")) * (ProJ (p,a,x))) * ((ProJ (q,a,x)) ") by A3, A9, Th21
.= (((ProJ (a,r,p)) * ((ProJ (a,r,q)) ")) * ((ProJ (p,r,x)) * ((ProJ (p,r,a)) "))) * ((ProJ (q,a,x)) ") by A6, A7, Th21
.= (((ProJ (a,r,p)) * ((ProJ (a,r,q)) ")) * ((ProJ (p,r,x)) * ((ProJ (p,r,a)) "))) * (ProJ (q,x,a)) by A18, A16, Th22
.= (((ProJ (a,r,p)) * ((ProJ (a,r,q)) ")) * ((ProJ (p,r,x)) * ((ProJ (p,r,a)) "))) * ((ProJ (q,r,a)) * ((ProJ (q,r,x)) ")) by A16, A8, Th21
.= ((((ProJ (a,p,r)) ") * ((ProJ (a,r,q)) ")) * ((ProJ (p,r,x)) * ((ProJ (p,r,a)) "))) * ((ProJ (q,r,a)) * ((ProJ (q,r,x)) ")) by A1, A9, Th22
.= ((((ProJ (a,p,r)) ") * ((ProJ (a,r,q)) ")) * ((ProJ (p,r,x)) * (ProJ (p,a,r)))) * ((ProJ (q,r,a)) * ((ProJ (q,r,x)) ")) by A6, A7, Th22
.= ((((ProJ (a,p,r)) ") * (ProJ (a,q,r))) * ((ProJ (p,r,x)) * (ProJ (p,a,r)))) * ((ProJ (q,r,a)) * ((ProJ (q,r,x)) ")) by A3, A9, Th22
.= ((((ProJ (a,p,r)) ") * (ProJ (a,q,r))) * ((ProJ (p,r,x)) * (ProJ (p,a,r)))) * (((ProJ (q,a,r)) ") * ((ProJ (q,r,x)) ")) by A18, A8, Th22
.= ((((ProJ (a,p,r)) ") * (ProJ (a,q,r))) * (((ProJ (p,x,r)) ") * (ProJ (p,a,r)))) * (((ProJ (q,a,r)) ") * ((ProJ (q,r,x)) ")) by A17, A7, Th22
.= ((((ProJ (p,x,r)) ") * (ProJ (p,a,r))) * (((ProJ (a,p,r)) ") * (ProJ (a,q,r)))) * (((ProJ (q,a,r)) ") * (ProJ (q,x,r))) by A16, A8, Th22
.= (((ProJ (p,x,r)) ") * ((((ProJ (a,p,r)) ") * (ProJ (a,q,r))) * (ProJ (p,a,r)))) * (((ProJ (q,a,r)) ") * (ProJ (q,x,r))) by GROUP_1:def_3
.= (((ProJ (p,x,r)) ") * ((((ProJ (a,p,r)) ") * (ProJ (p,a,r))) * (ProJ (a,q,r)))) * (((ProJ (q,a,r)) ") * (ProJ (q,x,r))) by GROUP_1:def_3
.= (((ProJ (p,x,r)) ") * ((ProJ (r,a,p)) * (ProJ (a,q,r)))) * (((ProJ (q,a,r)) ") * (ProJ (q,x,r))) by A1, A9, Th24
.= ((((ProJ (p,x,r)) ") * (ProJ (r,a,p))) * (ProJ (a,q,r))) * (((ProJ (q,a,r)) ") * (ProJ (q,x,r))) by GROUP_1:def_3
.= (((ProJ (p,x,r)) ") * (ProJ (r,a,p))) * ((ProJ (a,q,r)) * (((ProJ (q,a,r)) ") * (ProJ (q,x,r)))) by GROUP_1:def_3
.= (((ProJ (p,x,r)) ") * (ProJ (r,a,p))) * ((((ProJ (q,a,r)) ") * (ProJ (a,q,r))) * (ProJ (q,x,r))) by GROUP_1:def_3
.= (((ProJ (p,x,r)) ") * (ProJ (r,a,p))) * ((ProJ (r,q,a)) * (ProJ (q,x,r))) by A18, A8, Th24
.= ((ProJ (p,x,r)) ") * ((ProJ (r,a,p)) * ((ProJ (r,q,a)) * (ProJ (q,x,r)))) by GROUP_1:def_3
.= ((ProJ (p,x,r)) ") * (((ProJ (r,a,p)) * (ProJ (r,q,a))) * (ProJ (q,x,r))) by GROUP_1:def_3
.= ((ProJ (p,x,r)) ") * (((ProJ (r,a,p)) * ((ProJ (r,a,q)) ")) * (ProJ (q,x,r))) by A11, A15, Th22
.= ((ProJ (p,x,r)) ") * ((ProJ (r,q,p)) * (ProJ (q,x,r))) by A11, A15, Th21
.= (ProJ (p,r,x)) * ((ProJ (r,q,p)) * (ProJ (q,x,r))) by A17, A7, Th22
.= (((ProJ (r,x,p)) ") * (ProJ (x,r,p))) * ((ProJ (r,q,p)) * (ProJ (q,x,r))) by A13, A12, Th24
.= (((ProJ (r,x,p)) ") * (ProJ (x,r,p))) * ((ProJ (r,q,p)) * (((ProJ (x,r,q)) ") * (ProJ (r,x,q)))) by A4, A10, Th24
.= (((ProJ (r,x,p)) ") * (ProJ (x,r,p))) * ((ProJ (r,x,q)) * ((ProJ (r,q,p)) * ((ProJ (x,r,q)) "))) by GROUP_1:def_3
.= (((ProJ (x,r,p)) * ((ProJ (r,x,p)) ")) * (ProJ (r,x,q))) * ((ProJ (r,q,p)) * ((ProJ (x,r,q)) ")) by GROUP_1:def_3
.= ((ProJ (x,r,p)) * ((ProJ (r,x,q)) * ((ProJ (r,x,p)) "))) * ((ProJ (r,q,p)) * ((ProJ (x,r,q)) ")) by GROUP_1:def_3
.= ((ProJ (x,r,p)) * (ProJ (r,p,q))) * ((ProJ (r,q,p)) * ((ProJ (x,r,q)) ")) by A13, A12, Th21
.= (ProJ (x,r,p)) * ((ProJ (r,p,q)) * ((ProJ (r,q,p)) * ((ProJ (x,r,q)) "))) by GROUP_1:def_3
.= (ProJ (x,r,p)) * (((ProJ (r,p,q)) * (ProJ (r,q,p))) * ((ProJ (x,r,q)) ")) by GROUP_1:def_3
.= (ProJ (x,r,p)) * ((((ProJ (r,q,p)) ") * (ProJ (r,q,p))) * ((ProJ (x,r,q)) ")) by A13, A11, Th22
.= (ProJ (x,r,p)) * (((ProJ (x,r,q)) ") * (1_ F)) by A14, VECTSP_1:def_10
.= (ProJ (x,r,p)) * ((ProJ (x,r,q)) ") by VECTSP_1:def_8
.= ProJ (x,q,p) by A4, A10, Th21 ;
then A19: ((ProJ (q,a,x)) * ((ProJ (q,a,x)) ")) * ((ProJ (a,q,p)) * (ProJ (p,a,x))) = (ProJ (q,a,x)) * (ProJ (x,q,p)) by GROUP_1:def_3;
ProJ (q,a,x) <> 0. F by A18, A16, Th20;
then ((ProJ (a,q,p)) * (ProJ (p,a,x))) * (1_ F) = (ProJ (q,a,x)) * (ProJ (x,q,p)) by A19, VECTSP_1:def_10;
hence (ProJ (a,q,p)) * (ProJ (p,a,x)) = (ProJ (q,a,x)) * (ProJ (x,q,p)) by VECTSP_1:def_8; ::_thesis: verum
end;
theorem Th26: :: ORTSP_1:26
for F being Field
for S being OrtSp of F
for p, a, x, q, b, y being Element of S st 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
let F be Field; ::_thesis: for S being OrtSp of F
for p, a, x, q, b, y being Element of S st 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))
let S be OrtSp of F; ::_thesis: for p, a, x, q, b, y being Element of S st 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))
let p, a, x, q, b, y be Element of S; ::_thesis: ( not a _|_ & not x _|_ & not a _|_ & not x _|_ & not a _|_ implies ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) )
set 0F = 0. F;
set 1F = 1_ F;
assume that
A1: not a _|_ and
A2: not x _|_ and
A3: not a _|_ and
A4: not x _|_ and
A5: not a _|_ ; ::_thesis: ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y))
A6: now__::_thesis:_(_not_x__|__implies_((ProJ_(a,b,p))_*_(ProJ_(p,a,x)))_*_(ProJ_(x,p,y))_=_((ProJ_(a,b,q))_*_(ProJ_(q,a,x)))_*_(ProJ_(x,q,y))_)
A7: ProJ (a,b,q) <> 0. F by A3, A5, Th20;
assume A8: not x _|_ ; ::_thesis: ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y))
(ProJ (a,q,p)) * (ProJ (p,a,x)) = (ProJ (x,q,p)) * (ProJ (q,a,x)) by A1, A2, A3, A4, Th25;
then ((ProJ (a,b,p)) * ((ProJ (a,b,q)) ")) * (ProJ (p,a,x)) = (ProJ (x,q,p)) * (ProJ (q,a,x)) by A3, A5, Th21;
then (((ProJ (a,b,q)) ") * (ProJ (a,b,p))) * (ProJ (p,a,x)) = ((ProJ (x,y,p)) * ((ProJ (x,y,q)) ")) * (ProJ (q,a,x)) by A4, A8, Th21;
then (ProJ (a,b,q)) * (((ProJ (a,b,q)) ") * ((ProJ (a,b,p)) * (ProJ (p,a,x)))) = (ProJ (a,b,q)) * (((ProJ (x,y,p)) * ((ProJ (x,y,q)) ")) * (ProJ (q,a,x))) by GROUP_1:def_3;
then ((ProJ (a,b,q)) * ((ProJ (a,b,q)) ")) * ((ProJ (a,b,p)) * (ProJ (p,a,x))) = (ProJ (a,b,q)) * (((ProJ (x,y,p)) * ((ProJ (x,y,q)) ")) * (ProJ (q,a,x))) by GROUP_1:def_3;
then ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (1_ F) = (ProJ (a,b,q)) * (((ProJ (x,y,p)) * ((ProJ (x,y,q)) ")) * (ProJ (q,a,x))) by A7, VECTSP_1:def_10;
then (ProJ (a,b,p)) * (ProJ (p,a,x)) = (ProJ (a,b,q)) * (((ProJ (x,y,p)) * ((ProJ (x,y,q)) ")) * (ProJ (q,a,x))) by VECTSP_1:def_8;
then (ProJ (a,b,p)) * (ProJ (p,a,x)) = (ProJ (a,b,q)) * ((((ProJ (x,y,q)) ") * ((ProJ (x,p,y)) ")) * (ProJ (q,a,x))) by A2, A8, Th22;
then (ProJ (a,b,p)) * (ProJ (p,a,x)) = ((ProJ (a,b,q)) * (((ProJ (x,y,q)) ") * ((ProJ (x,p,y)) "))) * (ProJ (q,a,x)) by GROUP_1:def_3;
then (ProJ (a,b,p)) * (ProJ (p,a,x)) = (ProJ (q,a,x)) * (((ProJ (a,b,q)) * ((ProJ (x,y,q)) ")) * ((ProJ (x,p,y)) ")) by GROUP_1:def_3;
then ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = (((ProJ (q,a,x)) * ((ProJ (a,b,q)) * ((ProJ (x,y,q)) "))) * ((ProJ (x,p,y)) ")) * (ProJ (x,p,y)) by GROUP_1:def_3;
then A9: ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (q,a,x)) * ((ProJ (a,b,q)) * ((ProJ (x,y,q)) "))) * (((ProJ (x,p,y)) ") * (ProJ (x,p,y))) by GROUP_1:def_3;
ProJ (x,p,y) <> 0. F by A2, A8, Th20;
then ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (q,a,x)) * ((ProJ (a,b,q)) * ((ProJ (x,y,q)) "))) * (1_ F) by A9, VECTSP_1:def_10;
then ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = (ProJ (q,a,x)) * ((ProJ (a,b,q)) * ((ProJ (x,y,q)) ")) by VECTSP_1:def_8;
then ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = (ProJ (q,a,x)) * ((ProJ (a,b,q)) * (ProJ (x,q,y))) by A4, A8, Th22;
hence ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) by GROUP_1:def_3; ::_thesis: verum
end;
now__::_thesis:_(_x__|__implies_((ProJ_(a,b,p))_*_(ProJ_(p,a,x)))_*_(ProJ_(x,p,y))_=_((ProJ_(a,b,q))_*_(ProJ_(q,a,x)))_*_(ProJ_(x,q,y))_)
assume A10: x _|_ ; ::_thesis: ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y))
then ProJ (x,p,y) = 0. F by A2, Th20;
then A11: ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = 0. F by VECTSP_1:7;
ProJ (x,q,y) = 0. F by A4, A10, Th20;
hence ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) by A11, VECTSP_1:7; ::_thesis: verum
end;
hence ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) by A6; ::_thesis: verum
end;
theorem Th27: :: ORTSP_1:27
for F being Field
for S being OrtSp 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
let F be Field; ::_thesis: for S being OrtSp 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))
let S be OrtSp of F; ::_thesis: 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))
let a, p, x, y be Element of S; ::_thesis: ( not p _|_ & not p _|_ & not p _|_ implies (ProJ (p,a,x)) * (ProJ (x,p,y)) = (ProJ (p,a,y)) * (ProJ (y,p,x)) )
set 0F = 0. F;
set 1F = 1_ F;
assume that
A1: not p _|_ and
A2: not p _|_ and
A3: not p _|_ ; ::_thesis: (ProJ (p,a,x)) * (ProJ (x,p,y)) = (ProJ (p,a,y)) * (ProJ (y,p,x))
A4: not y _|_ by A3, Th2;
A5: not x _|_ by A2, Th2;
A6: now__::_thesis:_(_not_x__|__implies_(ProJ_(p,a,x))_*_(ProJ_(x,p,y))_=_(ProJ_(p,a,y))_*_(ProJ_(y,p,x))_)
A7: ProJ (p,a,x) <> 0. F by A1, A2, Th20;
assume A8: not x _|_ ; ::_thesis: (ProJ (p,a,x)) * (ProJ (x,p,y)) = (ProJ (p,a,y)) * (ProJ (y,p,x))
then A9: not y _|_ by Th2;
(ProJ (p,a,y)) * ((ProJ (p,a,x)) ") = ProJ (p,x,y) by A1, A2, Th21;
then ((ProJ (p,a,y)) * ((ProJ (p,a,x)) ")) * (ProJ (p,a,x)) = (((ProJ (x,y,p)) ") * (ProJ (y,x,p))) * (ProJ (p,a,x)) by A5, A8, Th24;
then (ProJ (p,a,y)) * (((ProJ (p,a,x)) ") * (ProJ (p,a,x))) = (((ProJ (x,y,p)) ") * (ProJ (y,x,p))) * (ProJ (p,a,x)) by GROUP_1:def_3;
then (ProJ (p,a,y)) * (1_ F) = (((ProJ (x,y,p)) ") * (ProJ (y,x,p))) * (ProJ (p,a,x)) by A7, VECTSP_1:def_10;
then ProJ (p,a,y) = ((ProJ (y,x,p)) * ((ProJ (x,y,p)) ")) * (ProJ (p,a,x)) by VECTSP_1:def_8;
then ProJ (p,a,y) = (ProJ (y,x,p)) * (((ProJ (x,y,p)) ") * (ProJ (p,a,x))) by GROUP_1:def_3;
then (ProJ (y,p,x)) * (ProJ (p,a,y)) = (ProJ (y,p,x)) * (((ProJ (y,p,x)) ") * (((ProJ (x,y,p)) ") * (ProJ (p,a,x)))) by A4, A9, Th22;
then A10: (ProJ (y,p,x)) * (ProJ (p,a,y)) = ((ProJ (y,p,x)) * ((ProJ (y,p,x)) ")) * (((ProJ (x,y,p)) ") * (ProJ (p,a,x))) by GROUP_1:def_3;
ProJ (y,p,x) <> 0. F by A4, A9, Th20;
then (ProJ (y,p,x)) * (ProJ (p,a,y)) = (((ProJ (x,y,p)) ") * (ProJ (p,a,x))) * (1_ F) by A10, VECTSP_1:def_10
.= ((ProJ (x,y,p)) ") * (ProJ (p,a,x)) by VECTSP_1:def_8 ;
hence (ProJ (p,a,x)) * (ProJ (x,p,y)) = (ProJ (p,a,y)) * (ProJ (y,p,x)) by A5, A8, Th22; ::_thesis: verum
end;
now__::_thesis:_(_x__|__implies_(ProJ_(p,a,x))_*_(ProJ_(x,p,y))_=_(ProJ_(p,a,y))_*_(ProJ_(y,p,x))_)
assume A11: x _|_ ; ::_thesis: (ProJ (p,a,x)) * (ProJ (x,p,y)) = (ProJ (p,a,y)) * (ProJ (y,p,x))
then ProJ (x,p,y) = 0. F by A5, Th20;
then A12: (ProJ (p,a,x)) * (ProJ (x,p,y)) = 0. F by VECTSP_1:7;
y _|_ by A11, Th2;
then ProJ (y,p,x) = 0. F by A4, Th20;
hence (ProJ (p,a,x)) * (ProJ (x,p,y)) = (ProJ (p,a,y)) * (ProJ (y,p,x)) by A12, VECTSP_1:7; ::_thesis: verum
end;
hence (ProJ (p,a,x)) * (ProJ (x,p,y)) = (ProJ (p,a,y)) * (ProJ (y,p,x)) by A6; ::_thesis: verum
end;
definition
let F be Field;
let S be OrtSp of F;
let x, y, a, b be Element of S;
assume A1: not a _|_ ;
func PProJ (a,b,x,y) -> Element of F means :Def3: :: ORTSP_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
( ( ex p being Element of S st
( not a _|_ & not x _|_ ) implies ex b1 being Element of F st
for q being Element of S st not a _|_ & not x _|_ holds
b1 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) & ( ( for p being Element of S holds
( a _|_ or x _|_ ) ) implies ex b1 being Element of F st b1 = 0. F ) )
proof
thus ( ex p being Element of S st
( not a _|_ & not x _|_ ) implies ex IT being Element of F st
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)) ) ::_thesis: ( ( for p being Element of S holds
( a _|_ or x _|_ ) ) implies ex b1 being Element of F st b1 = 0. F )
proof
given p being Element of S such that A2: ( not a _|_ & not x _|_ ) ; ::_thesis: ex IT being Element of F st
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))
take ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) ; ::_thesis: for q being Element of S st not a _|_ & not x _|_ 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))
let q be Element of S; ::_thesis: ( not a _|_ & not x _|_ implies ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) )
assume ( not a _|_ & not x _|_ ) ; ::_thesis: ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y))
hence ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) by A1, A2, Th26; ::_thesis: verum
end;
thus ( ( for p being Element of S holds
( a _|_ or x _|_ ) ) implies ex b1 being Element of F st b1 = 0. F ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of F holds
( ( ex p being Element of S st
( not a _|_ & not x _|_ ) & ( for q being Element of S st not a _|_ & not x _|_ holds
b1 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) & ( for q being Element of S st not a _|_ & not 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
( a _|_ or x _|_ ) ) & b1 = 0. F & b2 = 0. F implies b1 = b2 ) )
proof
let IT1, IT2 be Element of F; ::_thesis: ( ( ex p being Element of S st
( not a _|_ & not x _|_ ) & ( for q being Element of S st not a _|_ & not x _|_ holds
IT1 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) & ( for q being Element of S st not a _|_ & not x _|_ holds
IT2 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) implies IT1 = IT2 ) & ( ( for p being Element of S holds
( a _|_ or x _|_ ) ) & IT1 = 0. F & IT2 = 0. F implies IT1 = IT2 ) )
thus ( ex p being Element of S st
( not a _|_ & not x _|_ ) & ( for q being Element of S st not a _|_ & not x _|_ holds
IT1 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) & ( for q being Element of S st not a _|_ & not x _|_ holds
IT2 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) implies IT1 = IT2 ) ::_thesis: ( ( for p being Element of S holds
( a _|_ or x _|_ ) ) & IT1 = 0. F & IT2 = 0. F implies IT1 = IT2 )
proof
given p being Element of S such that A3: ( not a _|_ & not x _|_ ) ; ::_thesis: ( ex q being Element of S st
( not a _|_ & not x _|_ & not IT1 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) or ex q being Element of S st
( not a _|_ & not x _|_ & not IT2 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) or IT1 = IT2 )
consider r being Element of S such that
A4: ( not a _|_ & not x _|_ ) by A3;
assume that
A5: for q being Element of S st not a _|_ & not x _|_ holds
IT1 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) and
A6: for q being Element of S st not a _|_ & not x _|_ holds
IT2 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ; ::_thesis: IT1 = IT2
IT1 = ((ProJ (a,b,r)) * (ProJ (r,a,x))) * (ProJ (x,r,y)) by A5, A4;
hence IT1 = IT2 by A6, A4; ::_thesis: verum
end;
thus ( ( for p being Element of S holds
( a _|_ or x _|_ ) ) & IT1 = 0. F & IT2 = 0. F implies IT1 = IT2 ) ; ::_thesis: verum
end;
consistency
for b1 being Element of F holds verum ;
end;
:: deftheorem Def3 defines PProJ ORTSP_1:def_3_:_
for F being Field
for S being OrtSp of F
for x, y, a, b being Element of S st not a _|_ holds
for b7 being Element of F holds
( ( ex p being Element of S st
( not a _|_ & not x _|_ ) implies ( b7 = PProJ (a,b,x,y) iff for q being Element of S st not a _|_ & not x _|_ holds
b7 = ((ProJ (a,b,q)) * (ProJ (q,a,x))) * (ProJ (x,q,y)) ) ) & ( ( for p being Element of S holds
( a _|_ or x _|_ ) ) implies ( b7 = PProJ (a,b,x,y) iff b7 = 0. F ) ) );
theorem Th28: :: ORTSP_1:28
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S st not a _|_ & x = 0. S holds
PProJ (a,b,x,y) = 0. F
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, x, y being Element of S st not a _|_ & x = 0. S holds
PProJ (a,b,x,y) = 0. F
let S be OrtSp of F; ::_thesis: for b, a, x, y being Element of S st not a _|_ & x = 0. S holds
PProJ (a,b,x,y) = 0. F
let b, a, x, y be Element of S; ::_thesis: ( not a _|_ & x = 0. S implies PProJ (a,b,x,y) = 0. F )
assume that
A1: not a _|_ and
A2: x = 0. S ; ::_thesis: PProJ (a,b,x,y) = 0. F
for p being Element of S holds
( a _|_ or x _|_ ) by A2, Th1, Th2;
hence PProJ (a,b,x,y) = 0. F by A1, Def3; ::_thesis: verum
end;
theorem Th29: :: ORTSP_1:29
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S st not a _|_ holds
( PProJ (a,b,x,y) = 0. F iff x _|_ )
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, x, y being Element of S st not a _|_ holds
( PProJ (a,b,x,y) = 0. F iff x _|_ )
let S be OrtSp of F; ::_thesis: for b, a, x, y being Element of S st not a _|_ holds
( PProJ (a,b,x,y) = 0. F iff x _|_ )
let b, a, x, y be Element of S; ::_thesis: ( not a _|_ implies ( PProJ (a,b,x,y) = 0. F iff x _|_ ) )
set 0F = 0. F;
assume A1: not a _|_ ; ::_thesis: ( PProJ (a,b,x,y) = 0. F iff x _|_ )
then A2: a <> 0. S by Th1, Th2;
A3: ( PProJ (a,b,x,y) = 0. F implies x _|_ )
proof
assume A4: PProJ (a,b,x,y) = 0. F ; ::_thesis: x _|_
A5: now__::_thesis:_(_ex_p_being_Element_of_S_st_
(_not_a__|__&_not_x__|__)_implies_x__|__)
given p being Element of S such that A6: not a _|_ and
A7: not x _|_ ; ::_thesis: x _|_
A8: now__::_thesis:_not_ProJ_(p,a,x)_=_0._F
assume A9: ProJ (p,a,x) = 0. F ; ::_thesis: contradiction
not p _|_ by A6, Th2;
then p _|_ by A9, Th20;
hence contradiction by A7, Th2; ::_thesis: verum
end;
((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = 0. F by A1, A4, A6, A7, Def3;
then ( (ProJ (a,b,p)) * (ProJ (p,a,x)) = 0. F or ProJ (x,p,y) = 0. F ) by VECTSP_1:12;
then ( ProJ (a,b,p) = 0. F or ProJ (p,a,x) = 0. F or ProJ (x,p,y) = 0. F ) by VECTSP_1:12;
hence x _|_ by A1, A6, A7, A8, Th20; ::_thesis: verum
end;
now__::_thesis:_(_(_for_p_being_Element_of_S_holds_
(_a__|__or_x__|__)_)_implies_x__|__)
assume A10: for p being Element of S holds
( a _|_ or x _|_ ) ; ::_thesis: x _|_
x = 0. S
proof
assume not x = 0. S ; ::_thesis: contradiction
then ex p being Element of S st
( not a _|_ & not x _|_ & not a _|_ & not x _|_ ) by A2, Def1;
hence contradiction by A10; ::_thesis: verum
end;
hence x _|_ by Th1, Th2; ::_thesis: verum
end;
hence x _|_ by A5; ::_thesis: verum
end;
( x _|_ implies PProJ (a,b,x,y) = 0. F )
proof
assume A11: x _|_ ; ::_thesis: PProJ (a,b,x,y) = 0. F
A12: now__::_thesis:_(_x_<>_0._S_implies_PProJ_(a,b,x,y)_=_0._F_)
assume x <> 0. S ; ::_thesis: PProJ (a,b,x,y) = 0. F
then ex p being Element of S st
( not a _|_ & not x _|_ & not a _|_ & not x _|_ ) by A2, Def1;
then consider p being Element of S such that
A13: not a _|_ and
A14: not x _|_ ;
ProJ (x,p,y) = 0. F by A11, A14, Th20;
then ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) = 0. F by VECTSP_1:7;
hence PProJ (a,b,x,y) = 0. F by A1, A13, A14, Def3; ::_thesis: verum
end;
now__::_thesis:_(_x_=_0._S_implies_PProJ_(a,b,x,y)_=_0._F_)
assume x = 0. S ; ::_thesis: PProJ (a,b,x,y) = 0. F
then for p being Element of S holds
( a _|_ or x _|_ ) by Th1, Th2;
hence PProJ (a,b,x,y) = 0. F by A1, Def3; ::_thesis: verum
end;
hence PProJ (a,b,x,y) = 0. F by A12; ::_thesis: verum
end;
hence ( PProJ (a,b,x,y) = 0. F iff x _|_ ) by A3; ::_thesis: verum
end;
theorem :: ORTSP_1:30
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S st not a _|_ holds
PProJ (a,b,x,y) = PProJ (a,b,y,x)
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, x, y being Element of S st not a _|_ holds
PProJ (a,b,x,y) = PProJ (a,b,y,x)
let S be OrtSp of F; ::_thesis: for b, a, x, y being Element of S st not a _|_ holds
PProJ (a,b,x,y) = PProJ (a,b,y,x)
let b, a, x, y be Element of S; ::_thesis: ( not a _|_ implies PProJ (a,b,x,y) = PProJ (a,b,y,x) )
assume A1: not a _|_ ; ::_thesis: PProJ (a,b,x,y) = PProJ (a,b,y,x)
A2: now__::_thesis:_(_not_y__|__implies_PProJ_(a,b,x,y)_=_PProJ_(a,b,y,x)_)
assume not y _|_ ; ::_thesis: PProJ (a,b,x,y) = PProJ (a,b,y,x)
then A3: ( x <> 0. S & y <> 0. S ) by Th1, Th2;
a <> 0. S by A1, Th1, Th2;
then ex r being Element of S st
( not a _|_ & not x _|_ & not y _|_ & not a _|_ ) by A3, Def1;
then consider r being Element of S such that
A4: not a _|_ and
A5: not x _|_ and
A6: not y _|_ ;
A7: not r _|_ by A6, Th2;
PProJ (a,b,y,x) = ((ProJ (a,b,r)) * (ProJ (r,a,y))) * (ProJ (y,r,x)) by A1, A4, A6, Def3;
then A8: PProJ (a,b,y,x) = (ProJ (a,b,r)) * ((ProJ (r,a,y)) * (ProJ (y,r,x))) by GROUP_1:def_3;
( not r _|_ & not r _|_ ) by A4, A5, Th2;
then A9: PProJ (a,b,y,x) = (ProJ (a,b,r)) * ((ProJ (r,a,x)) * (ProJ (x,r,y))) by A7, A8, Th27;
PProJ (a,b,x,y) = ((ProJ (a,b,r)) * (ProJ (r,a,x))) * (ProJ (x,r,y)) by A1, A4, A5, Def3;
hence PProJ (a,b,x,y) = PProJ (a,b,y,x) by A9, GROUP_1:def_3; ::_thesis: verum
end;
now__::_thesis:_(_y__|__implies_PProJ_(a,b,x,y)_=_PProJ_(a,b,y,x)_)
assume y _|_ ; ::_thesis: PProJ (a,b,x,y) = PProJ (a,b,y,x)
then ( x _|_ & PProJ (a,b,y,x) = 0. F ) by A1, Th2, Th29;
hence PProJ (a,b,x,y) = PProJ (a,b,y,x) by A1, Th29; ::_thesis: verum
end;
hence PProJ (a,b,x,y) = PProJ (a,b,y,x) by A2; ::_thesis: verum
end;
theorem :: ORTSP_1:31
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S
for l being Element of F st not a _|_ holds
PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, x, y being Element of S
for l being Element of F st not a _|_ holds
PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
let S be OrtSp of F; ::_thesis: for b, a, x, y being Element of S
for l being Element of F st not a _|_ holds
PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
let b, a, x, y be Element of S; ::_thesis: for l being Element of F st not a _|_ holds
PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
let l be Element of F; ::_thesis: ( not a _|_ implies PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y)) )
set 0F = 0. F;
assume A1: not a _|_ ; ::_thesis: PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
A2: now__::_thesis:_(_not_y__|__implies_PProJ_(a,b,x,(l_*_y))_=_l_*_(PProJ_(a,b,x,y))_)
assume not y _|_ ; ::_thesis: PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
then A3: x <> 0. S by Th1;
a <> 0. S by A1, Th1, Th2;
then ex p being Element of S st
( not a _|_ & not x _|_ & not a _|_ & not x _|_ ) by A3, Def1;
then consider p being Element of S such that
A4: not a _|_ and
A5: not x _|_ ;
PProJ (a,b,x,(l * y)) = ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,(l * y))) by A1, A4, A5, Def3;
then A6: PProJ (a,b,x,(l * y)) = (l * (ProJ (x,p,y))) * ((ProJ (a,b,p)) * (ProJ (p,a,x))) by A5, Th12;
PProJ (a,b,x,y) = ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) by A1, A4, A5, Def3;
hence PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y)) by A6, GROUP_1:def_3; ::_thesis: verum
end;
now__::_thesis:_(_y__|__implies_PProJ_(a,b,x,(l_*_y))_=_l_*_(PProJ_(a,b,x,y))_)
assume A7: y _|_ ; ::_thesis: PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
then x _|_ by Th2;
then x _|_ by Def1;
then A8: PProJ (a,b,x,(l * y)) = 0. F by A1, Th29;
x _|_ by A7, Th2;
then l * (PProJ (a,b,x,y)) = l * (0. F) by A1, Th29;
hence PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y)) by A8, VECTSP_1:7; ::_thesis: verum
end;
hence PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y)) by A2; ::_thesis: verum
end;
theorem :: ORTSP_1:32
for F being Field
for S being OrtSp of F
for b, a, x, y, z being Element of S st not a _|_ holds
PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z))
proof
let F be Field; ::_thesis: for S being OrtSp of F
for b, a, x, y, z being Element of S st not a _|_ holds
PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z))
let S be OrtSp of F; ::_thesis: for b, a, x, y, z being Element of S st not a _|_ holds
PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z))
let b, a, x, y, z be Element of S; ::_thesis: ( not a _|_ implies PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z)) )
set 0F = 0. F;
assume A1: not a _|_ ; ::_thesis: PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z))
A2: now__::_thesis:_(_x_<>_0._S_implies_PProJ_(a,b,x,(y_+_z))_=_(PProJ_(a,b,x,y))_+_(PProJ_(a,b,x,z))_)
assume A3: x <> 0. S ; ::_thesis: PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z))
a <> 0. S by A1, Th1, Th2;
then ex p being Element of S st
( not a _|_ & not x _|_ & not a _|_ & not x _|_ ) by A3, Def1;
then consider p being Element of S such that
A4: ( not a _|_ & not x _|_ ) ;
A5: ( PProJ (a,b,x,(y + z)) = ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,(y + z))) & PProJ (a,b,x,y) = ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) ) by A1, A4, Def3;
( PProJ (a,b,x,z) = ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,z)) & ProJ (x,p,(y + z)) = (ProJ (x,p,y)) + (ProJ (x,p,z)) ) by A1, A4, Def3, Th13;
hence PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z)) by A5, VECTSP_1:def_7; ::_thesis: verum
end;
now__::_thesis:_(_x_=_0._S_implies_PProJ_(a,b,x,(y_+_z))_=_(PProJ_(a,b,x,y))_+_(PProJ_(a,b,x,z))_)
assume A6: x = 0. S ; ::_thesis: PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z))
then A7: PProJ (a,b,x,z) = 0. F by A1, Th28;
( PProJ (a,b,x,(y + z)) = 0. F & PProJ (a,b,x,y) = 0. F ) by A1, A6, Th28;
hence PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z)) by A7, RLVECT_1:4; ::_thesis: verum
end;
hence PProJ (a,b,x,(y + z)) = (PProJ (a,b,x,y)) + (PProJ (a,b,x,z)) by A2; ::_thesis: verum
end;