:: VECTSP_1 semantic presentation
:: deftheorem VECTSP_1:def 1 :
canceled;
:: deftheorem VECTSP_1:def 2 :
canceled;
:: deftheorem VECTSP_1:def 3 :
canceled;
:: deftheorem VECTSP_1:def 4 :
canceled;
:: deftheorem VECTSP_1:def 5 :
canceled;
:: deftheorem defines G_Real VECTSP_1:def 6 :
theorem :: VECTSP_1:1
canceled;
theorem :: VECTSP_1:2
canceled;
theorem :: VECTSP_1:3
canceled;
theorem :: VECTSP_1:4
canceled;
theorem :: VECTSP_1:5
canceled;
theorem :: VECTSP_1:6
:: deftheorem VECTSP_1:def 7 :
canceled;
:: deftheorem VECTSP_1:def 8 :
canceled;
:: deftheorem VECTSP_1:def 9 :
canceled;
:: deftheorem VECTSP_1:def 10 :
canceled;
:: deftheorem Def11 defines right-distributive VECTSP_1:def 11 :
:: deftheorem Def12 defines left-distributive VECTSP_1:def 12 :
:: deftheorem Def13 defines right_unital VECTSP_1:def 13 :
:: deftheorem VECTSP_1:def 14 :
canceled;
:: deftheorem defines F_Real VECTSP_1:def 15 :
:: deftheorem Def16 defines well-unital VECTSP_1:def 16 :
Lx1:
for L being non empty multLoopStr st L is well-unital holds
L is unital
Lx2:
for L being non empty multLoopStr st L is well-unital holds
1. L = 1_ L
:: deftheorem VECTSP_1:def 17 :
canceled;
:: deftheorem Def18 defines distributive VECTSP_1:def 18 :
:: deftheorem Def19 defines left_unital VECTSP_1:def 19 :
:: deftheorem Def20 defines almost_left_invertible VECTSP_1:def 20 :
:: deftheorem Def21 defines degenerated VECTSP_1:def 21 :
set FR = F_Real ;
Lm1:
1_ F_Real = 1
Lm2:
for L being non empty doubleLoopStr st L is distributive holds
( L is right-distributive & L is left-distributive )
theorem :: VECTSP_1:7
canceled;
theorem :: VECTSP_1:8
canceled;
theorem :: VECTSP_1:9
canceled;
theorem :: VECTSP_1:10
canceled;
theorem :: VECTSP_1:11
canceled;
theorem :: VECTSP_1:12
canceled;
theorem :: VECTSP_1:13
canceled;
theorem :: VECTSP_1:14
canceled;
theorem :: VECTSP_1:15
canceled;
theorem :: VECTSP_1:16
canceled;
theorem :: VECTSP_1:17
canceled;
theorem :: VECTSP_1:18
canceled;
theorem :: VECTSP_1:19
canceled;
theorem :: VECTSP_1:20
theorem :: VECTSP_1:21
theorem :: VECTSP_1:22
theorem :: VECTSP_1:23
canceled;
theorem :: VECTSP_1:24
canceled;
theorem :: VECTSP_1:25
canceled;
theorem :: VECTSP_1:26
canceled;
theorem :: VECTSP_1:27
canceled;
theorem :: VECTSP_1:28
canceled;
theorem :: VECTSP_1:29
canceled;
theorem :: VECTSP_1:30
canceled;
theorem :: VECTSP_1:31
canceled;
theorem :: VECTSP_1:32
canceled;
theorem Th33: :: VECTSP_1:33
:: deftheorem Def22 defines " VECTSP_1:def 22 :
:: deftheorem defines / VECTSP_1:def 23 :
theorem :: VECTSP_1:34
canceled;
theorem :: VECTSP_1:35
canceled;
theorem Th36: :: VECTSP_1:36
theorem :: VECTSP_1:37
canceled;
theorem :: VECTSP_1:38
canceled;
theorem Th39: :: VECTSP_1:39
theorem Th40: :: VECTSP_1:40
theorem Th41: :: VECTSP_1:41
theorem Th42: :: VECTSP_1:42
theorem :: VECTSP_1:43
theorem Th44: :: VECTSP_1:44
theorem :: VECTSP_1:45
:: deftheorem defines * VECTSP_1:def 24 :
:: deftheorem defines comp VECTSP_1:def 25 :
Lm3:
now
let F be non
empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let MLT be
Function of
[:the carrier of F,the carrier of F:],the
carrier of
F;
set GF =
VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #);
for
x,
y,
z being
Element of
VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #) holds
(
x + y = y + x &
(x + y) + z = x + (y + z) &
x + (0. VectSpStr(# the carrier of F,the add of F,(0. F),MLT #)) = x & ex
x' being
Element of
VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #) st
x + x' = 0. VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #) )
proof
let x,
y,
z be
Element of
VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #);
reconsider x' =
x,
y' =
y,
z' =
z as
Element of
F ;
thus x + y =
y' + x'
by RLVECT_1:5
.=
y + x
;
thus (x + y) + z =
(x' + y') + z'
.=
x' + (y' + z')
by RLVECT_1:def 6
.=
x + (y + z)
;
thus x + (0. VectSpStr(# the carrier of F,the add of F,(0. F),MLT #)) =
x' + (0. F)
.=
x
by RLVECT_1:10
;
consider t being
Element of
F such that A1:
x' + t = 0. F
by RLVECT_1:def 8;
reconsider t' =
t as
Element of
VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #) ;
take
t'
;
thus
x + t' = 0. VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #)
by A1;
end;
hence
VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #) is
AbGroup
by RLVECT_1:def 5, RLVECT_1:def 6, RLVECT_1:def 7, RLVECT_1:def 8;
end;
Lm4:
now
let F be non
empty add-associative right_zeroed right_complementable associative well-unital distributive doubleLoopStr ;
let MLT be
Function of
[:the carrier of F,the carrier of F:],the
carrier of
F;
assume A1:
MLT = the
mult of
F
;
set LS =
VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #);
let x,
y be
Element of
F;
let v,
w be
Element of
VectSpStr(# the
carrier of
F,the
add of
F,
(0. F),
MLT #);
reconsider v' =
v,
w' =
w as
Element of
F ;
thus x * (v + w) =
x * (v' + w')
by A1
.=
(x * v') + (x * w')
by Def18
.=
(x * v) + (x * w)
by A1
;
thus (x + y) * v =
(x + y) * v'
by A1
.=
(x * v') + (y * v')
by Def18
.=
(x * v) + (y * v)
by A1
;
thus (x * y) * v =
(x * y) * v'
by A1
.=
x * (y * v')
by GROUP_1:def 4
.=
x * (y * v)
by A1
;
thus (1_ F) * v =
(1_ F) * v'
by A1
.=
v
by Def19
;
end;
:: deftheorem Def26 defines VectSp-like VECTSP_1:def 26 :
theorem :: VECTSP_1:46
canceled;
theorem :: VECTSP_1:47
canceled;
theorem :: VECTSP_1:48
canceled;
theorem :: VECTSP_1:49
canceled;
theorem :: VECTSP_1:50
canceled;
theorem :: VECTSP_1:51
canceled;
theorem :: VECTSP_1:52
canceled;
theorem :: VECTSP_1:53
canceled;
theorem :: VECTSP_1:54
canceled;
theorem :: VECTSP_1:55
canceled;
theorem :: VECTSP_1:56
canceled;
theorem :: VECTSP_1:57
canceled;
theorem :: VECTSP_1:58
canceled;
theorem Th59: :: VECTSP_1:59
theorem :: VECTSP_1:60
theorem :: VECTSP_1:61
canceled;
theorem :: VECTSP_1:62
canceled;
theorem :: VECTSP_1:63
Lm5:
for V being non empty add-associative right_zeroed right_complementable LoopStr
for v, w being Element of V holds - (w + (- v)) = v - w
Lm6:
for V being non empty add-associative right_zeroed right_complementable LoopStr
for v, w being Element of V holds - ((- v) - w) = w + v
theorem :: VECTSP_1:64
theorem :: VECTSP_1:65
theorem Th66: :: VECTSP_1:66
theorem :: VECTSP_1:67
theorem Th68: :: VECTSP_1:68
theorem Th69: :: VECTSP_1:69
theorem :: VECTSP_1:70
theorem :: VECTSP_1:71
canceled;
theorem :: VECTSP_1:72
canceled;
theorem :: VECTSP_1:73
theorem :: VECTSP_1:74
theorem :: VECTSP_1:75
canceled;
theorem :: VECTSP_1:76
canceled;
theorem :: VECTSP_1:77
canceled;
theorem Th78: :: VECTSP_1:78
:: deftheorem VECTSP_1:def 27 :
canceled;
:: deftheorem Def28 defines Fanoian VECTSP_1:def 28 :
:: deftheorem Def29 defines Fanoian VECTSP_1:def 29 :
Lm7:
for F being non empty add-associative right_zeroed right_complementable LoopStr
for a, b being Element of F holds - (a - b) = b - a
by RLVECT_1:47;
theorem :: VECTSP_1:79
canceled;
theorem :: VECTSP_1:80
canceled;
theorem :: VECTSP_1:81
canceled;
theorem :: VECTSP_1:82
canceled;
theorem :: VECTSP_1:83
canceled;
theorem :: VECTSP_1:84
theorem :: VECTSP_1:85
canceled;
theorem Th86: :: VECTSP_1:86
theorem :: VECTSP_1:87
theorem :: VECTSP_1:88
theorem :: VECTSP_1:89
theorem :: VECTSP_1:90
theorem :: VECTSP_1:91
canceled;
theorem :: VECTSP_1:92
theorem :: VECTSP_1:93
theorem :: VECTSP_1:94
theorem :: VECTSP_1:95
theorem :: VECTSP_1:96