:: ORTSP_1 semantic presentation
reconsider c1 = {0} as non empty set ;
reconsider c2 = 0 as Element of c1 by TARSKI:def 1;
deffunc H1( Element of c1, Element of c1) -> Element of c1 = c2;
consider c3 being BinOp of c1 such that
Lemma1:
for b1, b2 being Element of c1 holds c3 . b1,b2 = H1(b1,b2)
from BINOP_1:sch 4();
Lemma3:
for b1 being Field
for b2 being Function of [:the carrier of b1,c1:],c1
for b3 being Relation of c1 holds
( SymStr(# c1,c3,c2,b2,b3 #) is Abelian & SymStr(# c1,c3,c2,b2,b3 #) is add-associative & SymStr(# c1,c3,c2,b2,b3 #) is right_zeroed & SymStr(# c1,c3,c2,b2,b3 #) is right_complementable )
E4:
now
let c
4 be
Field;
let c
5 be
Function of
[:the carrier of c4,c1:],c
1;
assume E5:
for b
1 being
Element of c
4for b
2 being
Element of c
1 holds c
5 . [b1,b2] = c
2
;
let c
6 be
Relation of c
1;
let c
7 be non
empty Abelian add-associative right_zeroed right_complementable SymStr of c
4;
assume E6:
c
7 = SymStr(# c
1,c
3,c
2,c
5,c
6 #)
;
thus
c
7 is
VectSp-like
proof
for b
1, b
2 being
Element of c
4for b
3, b
4 being
Element of c
7 holds
( b
1 * (b3 + b4) = (b1 * b3) + (b1 * b4) &
(b1 + b2) * b
3 = (b1 * b3) + (b2 * b3) &
(b1 * b2) * b
3 = b
1 * (b2 * b3) &
(1. c4) * b
3 = b
3 )
proof
let c
8, c
9 be
Element of c
4;
let c
10, c
11 be
Element of c
7;
E7:
c
8 * (c10 + c11) = (c8 * c10) + (c8 * c11)
proof
reconsider c
12 = c
10, c
13 = c
11 as
Element of c
7 ;
E8:
c
12 + c
13 = c
3 . c
12,c
13
by E6, RLVECT_1:5;
reconsider c
14 = c
12, c
15 = c
13 as
Element of c
1 by E6;
E9:
c
3 . c
14,c
15 = c
2
by Lemma1;
reconsider c
16 = c
14, c
17 = c
15 as
Element of c
7 ;
E10:
c
8 * (c16 + c17) = c
5 . c
8,c
2
by E6, E8, E9, VECTSP_1:def 24;
E11:
c
8 * (c16 + c17) = c
2
by E5, E10;
reconsider c
18 = c
16 as
Element of c
7 ;
E12:
c
5 . c
8,c
18 = c
2
by E5;
reconsider c
19 = c
18 as
Element of c
7 ;
E13:
c
8 * c
19 = c
2
by E6, E12, VECTSP_1:def 24;
reconsider c
20 = c
17 as
Element of c
7 ;
E14:
c
5 . c
8,c
20 = c
2
by E5;
reconsider c
21 = c
20 as
Element of c
7 ;
E15:
c
8 * c
21 = c
2
by E6, E14, VECTSP_1:def 24;
c
2 = 0. c
7
by E6, RLVECT_1:def 2;
hence
c
8 * (c10 + c11) = (c8 * c10) + (c8 * c11)
by E11, E13, E15, RLVECT_1:10;
end;
E8:
(c8 + c9) * c
10 = (c8 * c10) + (c9 * c10)
proof
set c
12 = c
8 + c
9;
E9:
(c8 + c9) * c
10 = c
5 . (c8 + c9),c
10
by E6, VECTSP_1:def 24;
reconsider c
13 = c
10 as
Element of c
7 ;
reconsider c
14 = c
13 as
Element of c
7 ;
E10:
(c8 + c9) * c
14 = c
2
by E5, E6, E9;
reconsider c
15 = c
14 as
Element of c
7 ;
E11:
c
5 . c
8,c
15 = c
2
by E5, E6;
reconsider c
16 = c
15 as
Element of c
7 ;
E12:
c
8 * c
16 = c
2
by E6, E11, VECTSP_1:def 24;
reconsider c
17 = c
16 as
Element of c
7 ;
E13:
c
5 . c
9,c
17 = c
2
by E5, E6;
reconsider c
18 = c
17 as
Element of c
7 ;
E14:
c
9 * c
18 = c
2
by E6, E13, VECTSP_1:def 24;
c
2 = 0. c
7
by E6, RLVECT_1:def 2;
hence
(c8 + c9) * c
10 = (c8 * c10) + (c9 * c10)
by E10, E12, E14, RLVECT_1:10;
end;
E9:
(c8 * c9) * c
10 = c
8 * (c9 * c10)
proof
set c
12 = c
8 * c
9;
E10:
(c8 * c9) * c
10 = c
5 . (c8 * c9),c
10
by E6, VECTSP_1:def 24;
reconsider c
13 = c
10 as
Element of c
7 ;
reconsider c
14 = c
13 as
Element of c
7 ;
E11:
(c8 * c9) * c
14 = c
2
by E5, E6, E10;
reconsider c
15 = c
14 as
Element of c
7 ;
E12:
c
5 . c
9,c
15 = c
2
by E5, E6;
reconsider c
16 = c
15 as
Element of c
7 ;
c
9 * c
16 = c
2
by E6, E12, VECTSP_1:def 24;
then E13:
c
8 * (c9 * c16) = c
5 . c
8,c
2
by E6, VECTSP_1:def 24;
thus
(c8 * c9) * c
10 = c
8 * (c9 * c10)
by E5, E11, E13;
end;
(1. c4) * c
10 = c
10
hence
( c
8 * (c10 + c11) = (c8 * c10) + (c8 * c11) &
(c8 + c9) * c
10 = (c8 * c10) + (c9 * c10) &
(c8 * c9) * c
10 = c
8 * (c9 * c10) &
(1. c4) * c
10 = c
10 )
by E7, E8, E9;
end;
hence
c
7 is
VectSp-like
by VECTSP_1:def 26;
end;
end;
E5:
now
let c
4 be
Field;
let c
5 be
Function of
[:the carrier of c4,c1:],c
1;
assume
for b
1 being
Element of c
4for b
2 being
Element of c
1 holds c
5 . [b1,b2] = c
2
;
set c
6 =
[:c1,c1:];
let c
7 be
Relation of c
1;
assume E6:
for b
1 being
set holds
( b
1 in c
7 iff ( b
1 in [:c1,c1:] & ex b
2, b
3 being
Element of c
1 st
( b
1 = [b2,b3] & b
3 = c
2 ) ) )
;
let c
8 be non
empty Abelian add-associative right_zeroed right_complementable SymStr of c
4;
assume E7:
c
8 = SymStr(# c
1,c
3,c
2,c
5,c
7 #)
;
E8:
for b
1, b
2 being
Element of c
8 holds
( b
1 _|_ b
2 iff (
[b1,b2] in [:c1,c1:] & ex b
3, b
4 being
Element of c
1 st
(
[b1,b2] = [b3,b4] & b
4 = c
2 ) ) )
E9:
for b
1, b
2 being
Element of c
8 holds
( b
1 _|_ b
2 iff b
2 = c
2 )
proof
let c
9, c
10 be
Element of c
8;
E10:
( c
9 _|_ c
10 implies c
10 = c
2 )
( c
10 = c
2 implies c
9 _|_ c
10 )
proof
assume E11:
c
10 = c
2
;
consider c
11, c
12 being
Element of c
8 such that E12:
( c
11 = c
9 & c
12 = c
10 )
;
[c9,c10] = [c11,c12]
by E12;
hence
c
9 _|_ c
10
by E7, E8, E11;
end;
hence
( c
9 _|_ c
10 iff c
10 = c
2 )
by E10;
end;
0. c
8 = c
2
by E7, TARSKI:def 1;
hence
for b
1, b
2, b
3, b
4 being
Element of c
8 holds
not ( b
1 <> 0. c
8 & b
2 <> 0. c
8 & b
3 <> 0. c
8 & b
4 <> 0. c
8 & ( for b
5 being
Element of c
8 holds
not ( not b
5 _|_ b
1 & not b
5 _|_ b
2 & not b
5 _|_ b
3 & not b
5 _|_ b
4 ) ) )
by E7, TARSKI:def 1;
thus
for b
1, b
2 being
Element of c
8for b
3 being
Element of c
4 holds
( b
1 _|_ b
2 implies b
3 * b
1 _|_ b
2 )
thus
for b
1, b
2, b
3 being
Element of c
8 holds
( b
2 _|_ b
1 & b
3 _|_ b
1 implies b
2 + b
3 _|_ b
1 )
thus
for b
1, b
2, b
3 being
Element of c
8 holds
not ( not b
2 _|_ b
1 & ( for b
4 being
Element of c
4 holds
not b
3 - (b4 * b2) _|_ b
1 ) )
thus
for b
1, b
2, b
3 being
Element of c
8 holds
( b
1 _|_ b
2 - b
3 & b
2 _|_ b
3 - b
1 implies b
3 _|_ b
1 - b
2 )
end;
:: deftheorem Def1 ORTSP_1:def 1 :
canceled;
:: deftheorem Def2 defines OrtSp-like ORTSP_1:def 2 :
theorem Th1: :: ORTSP_1:1
canceled;
theorem Th2: :: ORTSP_1:2
canceled;
theorem Th3: :: ORTSP_1:3
canceled;
theorem Th4: :: ORTSP_1:4
canceled;
theorem Th5: :: ORTSP_1:5
canceled;
theorem Th6: :: ORTSP_1:6
canceled;
theorem Th7: :: ORTSP_1:7
canceled;
theorem Th8: :: ORTSP_1:8
canceled;
theorem Th9: :: ORTSP_1:9
canceled;
theorem Th10: :: ORTSP_1:10
canceled;
theorem Th11: :: ORTSP_1:11
theorem Th12: :: ORTSP_1:12
theorem Th13: :: ORTSP_1:13
theorem Th14: :: ORTSP_1:14
theorem Th15: :: ORTSP_1:15
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4 being
Element of b
2for b
5 being
Element of b
1 holds
( not b
3 _|_ b
4 & not b
5 = 0. b
1 implies ( not b
5 * b
3 _|_ b
4 & not b
3 _|_ b
5 * b
4 ) )
theorem Th16: :: ORTSP_1:16
theorem Th17: :: ORTSP_1:17
canceled;
theorem Th18: :: ORTSP_1:18
canceled;
theorem Th19: :: ORTSP_1:19
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5, b
6 being
Element of b
2 holds
( b
3 - b
4 _|_ b
5 & b
3 - b
6 _|_ b
5 implies b
4 - b
6 _|_ b
5 )
theorem Th20: :: ORTSP_1:20
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5 being
Element of b
2for b
6, b
7 being
Element of b
1 holds
( not b
3 _|_ b
4 & b
5 - (b6 * b3) _|_ b
4 & b
5 - (b7 * b3) _|_ b
4 implies b
6 = b
7 )
theorem Th21: :: ORTSP_1:21
theorem Th22: :: ORTSP_1:22
:: deftheorem Def3 ORTSP_1:def 3 :
canceled;
:: deftheorem Def4 ORTSP_1:def 4 :
canceled;
:: deftheorem Def5 ORTSP_1:def 5 :
canceled;
:: deftheorem Def6 defines ProJ ORTSP_1:def 6 :
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5 being
Element of b
2 holds
( not b
4 _|_ b
3 implies for b
6 being
Element of b
1 holds
( b
6 = ProJ b
3,b
4,b
5 iff for b
7 being
Element of b
1 holds
( b
5 - (b7 * b4) _|_ b
3 implies b
6 = b
7 ) ) );
theorem Th23: :: ORTSP_1:23
canceled;
theorem Th24: :: ORTSP_1:24
theorem Th25: :: ORTSP_1:25
theorem Th26: :: ORTSP_1:26
theorem Th27: :: ORTSP_1:27
theorem Th28: :: ORTSP_1:28
theorem Th29: :: ORTSP_1:29
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5, b
6 being
Element of b
2 holds
( not b
3 _|_ b
4 & b
5 _|_ b
4 implies (
ProJ b
4,
(b3 + b5),b
6 = ProJ b
4,b
3,b
6 &
ProJ b
4,b
3,
(b6 + b5) = ProJ b
4,b
3,b
6 ) )
theorem Th30: :: ORTSP_1:30
theorem Th31: :: ORTSP_1:31
theorem Th32: :: ORTSP_1:32
theorem Th33: :: ORTSP_1:33
theorem Th34: :: ORTSP_1:34
theorem Th35: :: ORTSP_1:35
theorem Th36: :: ORTSP_1:36
theorem Th37: :: ORTSP_1:37
theorem Th38: :: ORTSP_1:38
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5, b
6 being
Element of b
2 holds
( not b
3 _|_ b
4 & not b
3 _|_ b
5 & not b
6 _|_ b
4 & not b
6 _|_ b
5 implies
(ProJ b4,b6,b3) * (ProJ b3,b4,b5) = (ProJ b6,b4,b5) * (ProJ b5,b6,b3) )
theorem Th39: :: ORTSP_1:39
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
2 holds
( not b
3 _|_ b
4 & not b
3 _|_ b
5 & not b
6 _|_ b
4 & not b
6 _|_ b
5 & not b
7 _|_ b
4 implies
((ProJ b4,b7,b3) * (ProJ b3,b4,b5)) * (ProJ b5,b3,b8) = ((ProJ b4,b7,b6) * (ProJ b6,b4,b5)) * (ProJ b5,b6,b8) )
theorem Th40: :: ORTSP_1:40
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5, b
6 being
Element of b
2 holds
( not b
3 _|_ b
4 & not b
5 _|_ b
4 & not b
6 _|_ b
4 implies
(ProJ b4,b3,b5) * (ProJ b5,b4,b6) = (ProJ b4,b3,b6) * (ProJ b6,b4,b5) )
definition
let c
4 be
Field;
let c
5 be
OrtSp of c
4;
let c
6, c
7, c
8, c
9 be
Element of c
5;
assume E31:
not c
9 _|_ c
8
;
func PProJ c
5,c
6,c
3,c
4 -> Element of a
1 means :
Def7:
:: ORTSP_1:def 7
for b
1 being
Element of a
2 holds
( not b
1 _|_ a
5 & not b
1 _|_ a
3 implies a
7 = ((ProJ a5,a6,b1) * (ProJ b1,a5,a3)) * (ProJ a3,b1,a4) )
if ex b
1 being
Element of a
2 st
( not b
1 _|_ a
5 & not b
1 _|_ a
3 )
a
7 = 0. a
1 if for b
1 being
Element of a
2 holds
( b
1 _|_ a
5 or b
1 _|_ a
3 )
;
existence
( not ( ex b1 being Element of c5 st
( not b1 _|_ c8 & not b1 _|_ c6 ) & ( for b1 being Element of c4 holds
ex b2 being Element of c5 st
( not b2 _|_ c8 & not b2 _|_ c6 & not b1 = ((ProJ c8,c9,b2) * (ProJ b2,c8,c6)) * (ProJ c6,b2,c7) ) ) ) & not ( ( for b1 being Element of c5 holds
( b1 _|_ c8 or b1 _|_ c6 ) ) & ( for b1 being Element of c4 holds
not b1 = 0. c4 ) ) )
uniqueness
for b1, b2 being Element of c4 holds
( ( ex b3 being Element of c5 st
( not b3 _|_ c8 & not b3 _|_ c6 ) & ( for b3 being Element of c5 holds
( not b3 _|_ c8 & not b3 _|_ c6 implies b1 = ((ProJ c8,c9,b3) * (ProJ b3,c8,c6)) * (ProJ c6,b3,c7) ) ) & ( for b3 being Element of c5 holds
( not b3 _|_ c8 & not b3 _|_ c6 implies b2 = ((ProJ c8,c9,b3) * (ProJ b3,c8,c6)) * (ProJ c6,b3,c7) ) ) implies b1 = b2 ) & ( ( for b3 being Element of c5 holds
( b3 _|_ c8 or b3 _|_ c6 ) ) & b1 = 0. c4 & b2 = 0. c4 implies b1 = b2 ) )
consistency
for b1 being Element of c4 holds
( ex b2 being Element of c5 st
( not b2 _|_ c8 & not b2 _|_ c6 ) & ( for b2 being Element of c5 holds
( b2 _|_ c8 or b2 _|_ c6 ) ) implies ( ( for b2 being Element of c5 holds
( not b2 _|_ c8 & not b2 _|_ c6 implies b1 = ((ProJ c8,c9,b2) * (ProJ b2,c8,c6)) * (ProJ c6,b2,c7) ) ) iff b1 = 0. c4 ) )
;
end;
:: deftheorem Def7 defines PProJ ORTSP_1:def 7 :
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5, b
6 being
Element of b
2 holds
( not b
6 _|_ b
5 implies for b
7 being
Element of b
1 holds
( ( ex b
8 being
Element of b
2 st
( not b
8 _|_ b
5 & not b
8 _|_ b
3 ) implies ( b
7 = PProJ b
5,b
6,b
3,b
4 iff for b
8 being
Element of b
2 holds
( not b
8 _|_ b
5 & not b
8 _|_ b
3 implies b
7 = ((ProJ b5,b6,b8) * (ProJ b8,b5,b3)) * (ProJ b3,b8,b4) ) ) ) & ( ( for b
8 being
Element of b
2 holds
( b
8 _|_ b
5 or b
8 _|_ b
3 ) ) implies ( b
7 = PProJ b
5,b
6,b
3,b
4 iff b
7 = 0. b
1 ) ) ) );
theorem Th41: :: ORTSP_1:41
canceled;
theorem Th42: :: ORTSP_1:42
canceled;
theorem Th43: :: ORTSP_1:43
Lemma33:
for b1 being Field
for b2 being OrtSp of b1
for b3 being Element of b2 holds b3 _|_ 0. b2
theorem Th44: :: ORTSP_1:44
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5, b
6 being
Element of b
2 holds
( not b
3 _|_ b
4 implies (
PProJ b
4,b
3,b
5,b
6 = 0. b
1 iff b
6 _|_ b
5 ) )
theorem Th45: :: ORTSP_1:45
theorem Th46: :: ORTSP_1:46
theorem Th47: :: ORTSP_1:47
for b
1 being
Fieldfor b
2 being
OrtSp of b
1for b
3, b
4, b
5, b
6, b
7 being
Element of b
2 holds
( not b
3 _|_ b
4 implies
PProJ b
4,b
3,b
5,
(b6 + b7) = (PProJ b4,b3,b5,b6) + (PProJ b4,b3,b5,b7) )