:: RLVECT_1 semantic presentation
:: deftheorem Def1 defines in RLVECT_1:def 1 :
theorem Th1: :: RLVECT_1:1
canceled;
theorem Th2: :: RLVECT_1:2
canceled;
theorem Th3: :: RLVECT_1:3
:: deftheorem Def2 defines 0. RLVECT_1:def 2 :
:: deftheorem Def3 defines + RLVECT_1:def 3 :
:: deftheorem Def4 defines * RLVECT_1:def 4 :
theorem Th4: :: RLVECT_1:4
canceled;
theorem Th5: :: RLVECT_1:5
E2:
now
take c
1 =
{0};
reconsider c
2 = 0 as
Element of c
1 by TARSKI:def 1;
take c
3 = c
2;
deffunc H
1(
Element of c
1,
Element of c
1)
-> Element of c
1 = c
3;
consider c
4 being
BinOp of c
1 such that E3:
for b
1, b
2 being
Element of c
1 holds c
4 . b
1,b
2 = H
1(b
1,b
2)
from BINOP_1:sch 4();
deffunc H
2(
Element of
REAL ,
Element of c
1)
-> Element of c
1 = c
3;
consider c
5 being
Function of
[:REAL ,c1:],c
1 such that E4:
for b
1 being
Element of
REAL for b
2 being
Element of c
1 holds c
5 . b
1,b
2 = H
2(b
1,b
2)
from BINOP_1:sch 4();
take c
6 = c
4;
take c
7 = c
5;
set c
8 =
RLSStruct(# c
1,c
3,c
6,c
7 #);
thus
for b
1, b
2 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) holds b
1 + b
2 = b
2 + b
1
proof
let c
9, c
10 be
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #);
E5:
( c
9 + c
10 = c
6 . c
9,c
10 & c
10 + c
9 = c
6 . c
10,c
9 )
;
reconsider c
11 = c
9, c
12 = c
10 as
Element of c
1 ;
( c
9 + c
10 = H
1(c
11,c
12) & c
10 + c
9 = H
1(c
12,c
11) )
by E3, E5;
hence
c
9 + c
10 = c
10 + c
9
;
end;
thus
for b
1, b
2, b
3 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) holds
(b1 + b2) + b
3 = b
1 + (b2 + b3)
proof
let c
9, c
10, c
11 be
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #);
E5:
(
(c9 + c10) + c
11 = c
6 . (c9 + c10),c
11 & c
9 + (c10 + c11) = c
6 . c
9,
(c10 + c11) )
;
reconsider c
12 = c
9, c
13 = c
10, c
14 = c
11 as
Element of c
1 ;
(
(c9 + c10) + c
11 = H
1(H
1(c
12,c
13),c
14) & c
9 + (c10 + c11) = H
1(c
12,H
1(c
13,c
14)) )
by E3, E5;
hence
(c9 + c10) + c
11 = c
9 + (c10 + c11)
;
end;
thus
for b
1 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) holds b
1 + (0. RLSStruct(# c1,c3,c6,c7 #)) = b
1
proof
let c
9 be
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #);
reconsider c
10 = c
9 as
Element of c
1 ;
c
9 + (0. RLSStruct(# c1,c3,c6,c7 #)) =
c
6 . c
9,
(0. RLSStruct(# c1,c3,c6,c7 #))
.=
H
1(c
10,c
3)
by E3
;
hence
c
9 + (0. RLSStruct(# c1,c3,c6,c7 #)) = c
9
by TARSKI:def 1;
end;
thus
for b
1 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) holds
ex b
2 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) st b
1 + b
2 = 0. RLSStruct(# c
1,c
3,c
6,c
7 #)
proof
let c
9 be
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #);
reconsider c
10 = c
3 as
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) ;
take
c
10
;
thus c
9 + c
10 =
c
6 . c
9,c
10
.=
0. RLSStruct(# c
1,c
3,c
6,c
7 #)
by E3
;
end;
thus
for b
1 being
Realfor b
2, b
3 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) holds b
1 * (b2 + b3) = (b1 * b2) + (b1 * b3)
proof
let c
9 be
Real;
let c
10, c
11 be
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #);
reconsider c
12 = c
10, c
13 = c
11 as
Element of c
1 ;
(c9 * c10) + (c9 * c11) =
c
6 . (c9 * c10),
(c9 * c11)
.=
c
3
by E3
.=
c
7 . c
9,
(c6 . c12,c13)
by E4
;
hence
c
9 * (c10 + c11) = (c9 * c10) + (c9 * c11)
;
end;
thus
for b
1, b
2 being
Realfor b
3 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) holds
(b1 + b2) * b
3 = (b1 * b3) + (b2 * b3)
proof
let c
9, c
10 be
Real;
let c
11 be
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #);
set c
12 = c
9 + c
10;
reconsider c
13 = c
11 as
Element of c
1 ;
E5:
(c9 + c10) * c
11 =
c
7 . (c9 + c10),c
11
.=
H
2(c
9 + c
10,c
13)
by E4
;
(c9 * c11) + (c10 * c11) =
c
6 . (c9 * c11),
(c10 * c11)
.=
H
1(H
2(c
9,c
13),H
2(c
10,c
13))
by E3
;
hence
(c9 + c10) * c
11 = (c9 * c11) + (c10 * c11)
by E5;
end;
thus
for b
1, b
2 being
Realfor b
3 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) holds
(b1 * b2) * b
3 = b
1 * (b2 * b3)
proof
let c
9, c
10 be
Real;
let c
11 be
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #);
set c
12 = c
9 * c
10;
reconsider c
13 = c
11 as
Element of c
1 ;
E5:
(c9 * c10) * c
11 =
c
7 . (c9 * c10),c
11
.=
H
2(c
9 * c
10,c
13)
by E4
;
c
9 * (c10 * c11) =
c
7 . c
9,
(c10 * c11)
.=
H
2(c
9,H
2(c
10,c
13))
by E4
;
hence
(c9 * c10) * c
11 = c
9 * (c10 * c11)
by E5;
end;
thus
for b
1 being
VECTOR of
RLSStruct(# c
1,c
3,c
6,c
7 #) holds 1
* b
1 = b
1
end;
:: deftheorem Def5 defines Abelian RLVECT_1:def 5 :
:: deftheorem Def6 defines add-associative RLVECT_1:def 6 :
:: deftheorem Def7 defines right_zeroed RLVECT_1:def 7 :
:: deftheorem Def8 defines right_complementable RLVECT_1:def 8 :
:: deftheorem Def9 defines RealLinearSpace-like RLVECT_1:def 9 :
theorem Th6: :: RLVECT_1:6
canceled;
theorem Th7: :: RLVECT_1:7
for b
1 being non
empty RLSStruct holds
( ( for b
2, b
3 being
VECTOR of b
1 holds b
2 + b
3 = b
3 + b
2 ) & ( for b
2, b
3, b
4 being
VECTOR of b
1 holds
(b2 + b3) + b
4 = b
2 + (b3 + b4) ) & ( for b
2 being
VECTOR of b
1 holds b
2 + (0. b1) = b
2 ) & ( for b
2 being
VECTOR of b
1 holds
ex b
3 being
VECTOR of b
1 st b
2 + b
3 = 0. b
1 ) & ( for b
2 being
Realfor b
3, b
4 being
VECTOR of b
1 holds b
2 * (b3 + b4) = (b2 * b3) + (b2 * b4) ) & ( for b
2, b
3 being
Realfor b
4 being
VECTOR of b
1 holds
(b2 + b3) * b
4 = (b2 * b4) + (b3 * b4) ) & ( for b
2, b
3 being
Realfor b
4 being
VECTOR of b
1 holds
(b2 * b3) * b
4 = b
2 * (b3 * b4) ) & ( for b
2 being
VECTOR of b
1 holds 1
* b
2 = b
2 ) implies b
1 is
RealLinearSpace )
by Def5, Def6, Def7, Def8, Def9;
Lemma8:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Element of b1 holds
( b2 + b3 = 0. b1 implies b3 + b2 = 0. b1 )
theorem Th8: :: RLVECT_1:8
canceled;
theorem Th9: :: RLVECT_1:9
canceled;
theorem Th10: :: RLVECT_1:10
:: deftheorem Def10 defines - RLVECT_1:def 10 :
Lemma11:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Element of b1 holds
ex b4 being Element of b1 st b2 + b4 = b3
:: deftheorem Def11 defines - RLVECT_1:def 11 :
theorem Th11: :: RLVECT_1:11
canceled;
theorem Th12: :: RLVECT_1:12
canceled;
theorem Th13: :: RLVECT_1:13
canceled;
theorem Th14: :: RLVECT_1:14
canceled;
theorem Th15: :: RLVECT_1:15
canceled;
theorem Th16: :: RLVECT_1:16
theorem Th17: :: RLVECT_1:17
canceled;
theorem Th18: :: RLVECT_1:18
canceled;
theorem Th19: :: RLVECT_1:19
theorem Th20: :: RLVECT_1:20
theorem Th21: :: RLVECT_1:21
theorem Th22: :: RLVECT_1:22
theorem Th23: :: RLVECT_1:23
theorem Th24: :: RLVECT_1:24
theorem Th25: :: RLVECT_1:25
theorem Th26: :: RLVECT_1:26
theorem Th27: :: RLVECT_1:27
theorem Th28: :: RLVECT_1:28
theorem Th29: :: RLVECT_1:29
theorem Th30: :: RLVECT_1:30
theorem Th31: :: RLVECT_1:31
theorem Th32: :: RLVECT_1:32
canceled;
theorem Th33: :: RLVECT_1:33
theorem Th34: :: RLVECT_1:34
theorem Th35: :: RLVECT_1:35
theorem Th36: :: RLVECT_1:36
theorem Th37: :: RLVECT_1:37
theorem Th38: :: RLVECT_1:38
theorem Th39: :: RLVECT_1:39
theorem Th40: :: RLVECT_1:40
Lemma27:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Element of b1 holds - (b2 + b3) = (- b3) + (- b2)
theorem Th41: :: RLVECT_1:41
theorem Th42: :: RLVECT_1:42
theorem Th43: :: RLVECT_1:43
theorem Th44: :: RLVECT_1:44
theorem Th45: :: RLVECT_1:45
theorem Th46: :: RLVECT_1:46
theorem Th47: :: RLVECT_1:47
theorem Th48: :: RLVECT_1:48
theorem Th49: :: RLVECT_1:49
theorem Th50: :: RLVECT_1:50
theorem Th51: :: RLVECT_1:51
:: deftheorem Def12 defines Sum RLVECT_1:def 12 :
Lemma33:
for b1 being non empty LoopStr holds Sum (<*> the carrier of b1) = 0. b1
Lemma34:
for b1 being non empty LoopStr
for b2 being FinSequence of the carrier of b1 holds
( len b2 = 0 implies Sum b2 = 0. b1 )
theorem Th52: :: RLVECT_1:52
canceled;
theorem Th53: :: RLVECT_1:53
canceled;
theorem Th54: :: RLVECT_1:54
theorem Th55: :: RLVECT_1:55
theorem Th56: :: RLVECT_1:56
theorem Th57: :: RLVECT_1:57
Lemma37:
for b1 being natural number holds
( b1 < 1 implies b1 = 0 )
by NAT_1:39;
theorem Th58: :: RLVECT_1:58
Lemma39:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2 being Element of b1 holds Sum <*b2*> = b2
theorem Th59: :: RLVECT_1:59
theorem Th60: :: RLVECT_1:60
theorem Th61: :: RLVECT_1:61
theorem Th62: :: RLVECT_1:62
theorem Th63: :: RLVECT_1:63
theorem Th64: :: RLVECT_1:64
theorem Th65: :: RLVECT_1:65
canceled;
theorem Th66: :: RLVECT_1:66
theorem Th67: :: RLVECT_1:67
theorem Th68: :: RLVECT_1:68
theorem Th69: :: RLVECT_1:69
theorem Th70: :: RLVECT_1:70
theorem Th71: :: RLVECT_1:71
theorem Th72: :: RLVECT_1:72
theorem Th73: :: RLVECT_1:73
theorem Th74: :: RLVECT_1:74
theorem Th75: :: RLVECT_1:75
theorem Th76: :: RLVECT_1:76
theorem Th77: :: RLVECT_1:77
theorem Th78: :: RLVECT_1:78
theorem Th79: :: RLVECT_1:79
theorem Th80: :: RLVECT_1:80
theorem Th81: :: RLVECT_1:81
theorem Th82: :: RLVECT_1:82
theorem Th83: :: RLVECT_1:83
theorem Th84: :: RLVECT_1:84
theorem Th85: :: RLVECT_1:85
theorem Th86: :: RLVECT_1:86
theorem Th87: :: RLVECT_1:87
theorem Th88: :: RLVECT_1:88
canceled;
theorem Th89: :: RLVECT_1:89
theorem Th90: :: RLVECT_1:90
theorem Th91: :: RLVECT_1:91
theorem Th92: :: RLVECT_1:92
theorem Th93: :: RLVECT_1:93
theorem Th94: :: RLVECT_1:94
theorem Th95: :: RLVECT_1:95
theorem Th96: :: RLVECT_1:96
theorem Th97: :: RLVECT_1:97
:: deftheorem Def13 defines non-zero RLVECT_1:def 13 :