:: VECTSP_1 semantic presentation
:: deftheorem Def1 VECTSP_1:def 1 :
canceled;
:: deftheorem Def2 VECTSP_1:def 2 :
canceled;
:: deftheorem Def3 VECTSP_1:def 3 :
canceled;
:: deftheorem Def4 VECTSP_1:def 4 :
canceled;
:: deftheorem Def5 VECTSP_1:def 5 :
canceled;
:: deftheorem Def6 defines G_Real VECTSP_1:def 6 :
theorem Th1: :: VECTSP_1:1
canceled;
theorem Th2: :: VECTSP_1:2
canceled;
theorem Th3: :: VECTSP_1:3
canceled;
theorem Th4: :: VECTSP_1:4
canceled;
theorem Th5: :: VECTSP_1:5
canceled;
theorem Th6: :: VECTSP_1:6
theorem Th7: :: VECTSP_1:7
:: deftheorem Def7 VECTSP_1:def 7 :
canceled;
:: deftheorem Def8 VECTSP_1:def 8 :
canceled;
:: deftheorem Def9 defines 1_ VECTSP_1:def 9 :
:: deftheorem Def10 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 Def14 VECTSP_1:def 14 :
canceled;
:: deftheorem Def15 defines F_Real VECTSP_1:def 15 :
:: deftheorem Def16 VECTSP_1:def 16 :
canceled;
:: deftheorem Def17 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 Field-like VECTSP_1:def 20 :
:: deftheorem Def21 defines degenerated VECTSP_1:def 21 :
set FR = F_Real ;
Lemma49:
1. F_Real = 1
Lemma50:
for L being non empty doubleLoopStr st L is distributive holds
( L is right-distributive & L is left-distributive )
theorem Th8: :: VECTSP_1:8
canceled;
theorem Th9: :: VECTSP_1:9
canceled;
theorem Th10: :: VECTSP_1:10
canceled;
theorem Th11: :: VECTSP_1:11
canceled;
theorem Th12: :: VECTSP_1:12
canceled;
theorem Th13: :: VECTSP_1:13
canceled;
theorem Th14: :: VECTSP_1:14
canceled;
theorem Th15: :: VECTSP_1:15
canceled;
theorem Th16: :: VECTSP_1:16
canceled;
theorem Th17: :: VECTSP_1:17
canceled;
theorem Th18: :: VECTSP_1:18
canceled;
theorem Th19: :: VECTSP_1:19
canceled;
theorem Th20: :: VECTSP_1:20
theorem Th21: :: VECTSP_1:21
theorem Th22: :: VECTSP_1:22
theorem Th23: :: VECTSP_1:23
canceled;
theorem Th24: :: VECTSP_1:24
canceled;
theorem Th25: :: VECTSP_1:25
canceled;
theorem Th26: :: VECTSP_1:26
canceled;
theorem Th27: :: VECTSP_1:27
canceled;
theorem Th28: :: VECTSP_1:28
canceled;
theorem Th29: :: VECTSP_1:29
canceled;
theorem Th30: :: VECTSP_1:30
canceled;
theorem Th31: :: VECTSP_1:31
canceled;
theorem Th32: :: VECTSP_1:32
canceled;
theorem Th33: :: VECTSP_1:33
:: deftheorem Def22 defines " VECTSP_1:def 22 :
:: deftheorem Def23 defines / VECTSP_1:def 23 :
theorem Th34: :: VECTSP_1:34
canceled;
theorem Th35: :: VECTSP_1:35
canceled;
theorem Th36: :: VECTSP_1:36
theorem Th37: :: VECTSP_1:37
canceled;
theorem Th38: :: 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 Th43: :: VECTSP_1:43
theorem Th44: :: VECTSP_1:44
theorem Th45: :: VECTSP_1:45
:: deftheorem Def24 defines * VECTSP_1:def 24 :
:: deftheorem Def25 defines comp VECTSP_1:def 25 :
E72:
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,the
Zero of
F,
MLT #);
for
x,
y,
z being
Element of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #) holds
(
x + y = y + x &
(x + y) + z = x + (y + z) &
x + (0. VectSpStr(# the carrier of F,the add of F,the Zero of F,MLT #)) = x & ex
x' being
Element of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #) st
x + x' = 0. VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #) )
proof
let x be
Element of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #),
y be
Element of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #),
z be
Element of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #);
reconsider x' =
x,
y' =
y,
z' =
z as
Element of
F ;
thus x + y =
the
add of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #)
. x,
y
by RLVECT_1:5
.=
y' + x'
by RLVECT_1:5
.=
the
add of
F . y',
x'
by RLVECT_1:5
.=
y + x
by RLVECT_1:5
;
thus (x + y) + z =
the
add of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #)
. (x + y),
z
by RLVECT_1:5
.=
the
add of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #)
. (the add of VectSpStr(# the carrier of F,the add of F,the Zero of F,MLT #) . x,y),
z
by RLVECT_1:5
.=
the
add of
F . (x' + y'),
z'
by RLVECT_1:5
.=
(x' + y') + z'
by RLVECT_1:5
.=
x' + (y' + z')
by RLVECT_1:def 6
.=
the
add of
F . x',
(y' + z')
by RLVECT_1:5
.=
the
add of
F . x',
(the add of F . y',z')
by RLVECT_1:5
.=
the
add of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #)
. x,
(y + z)
by RLVECT_1:5
.=
x + (y + z)
by RLVECT_1:5
;
thus x + (0. VectSpStr(# the carrier of F,the add of F,the Zero of F,MLT #)) =
the
add of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #)
. x,
(0. VectSpStr(# the carrier of F,the add of F,the Zero of F,MLT #))
by RLVECT_1:5
.=
x' + (0. F)
by RLVECT_1:5
.=
x
by RLVECT_1:10
;
consider t being
Element of
F such that E46:
x' + t = 0. F
by RLVECT_1:def 8;
reconsider t' =
t as
Element of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #) ;
take
t'
;
thus x + t' =
the
add of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #)
. x,
t'
by RLVECT_1:5
.=
0. VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #)
by RLVECT_1:5,
;
end;
hence
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #) is
AbGroup
by RLVECT_1:def 5, RLVECT_1:def 6, RLVECT_1:def 7, RLVECT_1:def 8;
end;
E77:
now
let F be non
empty 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;
assume E46:
MLT = the
mult of
F
;
set LS =
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #);
let x be
Element of
F,
y be
Element of
F;
let v be
Element of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #),
w be
Element of
VectSpStr(# the
carrier of
F,the
add of
F,the
Zero of
F,
MLT #);
reconsider v' =
v,
w' =
w as
Element of
F ;
thus x * (v + w) =
MLT . x,
(the add of F . v',w')
by RLVECT_1:5
.=
x * (v' + w')
by RLVECT_1:5,
.=
(x * v') + (x * w')
by
.=
the
add of
F . (MLT . x,v'),
(x * w')
by RLVECT_1:5,
.=
(x * v) + (x * w)
by , RLVECT_1:5
;
thus (x + y) * v =
(x + y) * v'
by
.=
(x * v') + (y * v')
by
.=
the
add of
F . (MLT . x,v'),
(y * v')
by RLVECT_1:5,
.=
(x * v) + (y * v)
by , RLVECT_1:5
;
thus (x * y) * v =
(x * y) * v'
by
.=
x * (y * v')
by GROUP_1:def 4
.=
x * (y * v)
by
;
thus (1. F) * v =
(1. F) * v'
by
.=
v
by
;
end;
:: deftheorem Def26 defines VectSp-like VECTSP_1:def 26 :
theorem Th46: :: VECTSP_1:46
canceled;
theorem Th47: :: VECTSP_1:47
canceled;
theorem Th48: :: VECTSP_1:48
canceled;
theorem Th49: :: VECTSP_1:49
canceled;
theorem Th50: :: VECTSP_1:50
canceled;
theorem Th51: :: VECTSP_1:51
canceled;
theorem Th52: :: VECTSP_1:52
canceled;
theorem Th53: :: VECTSP_1:53
canceled;
theorem Th54: :: VECTSP_1:54
canceled;
theorem Th55: :: VECTSP_1:55
canceled;
theorem Th56: :: VECTSP_1:56
canceled;
theorem Th57: :: VECTSP_1:57
canceled;
theorem Th58: :: VECTSP_1:58
canceled;
theorem Th59: :: VECTSP_1:59
theorem Th60: :: VECTSP_1:60
theorem Th61: :: VECTSP_1:61
canceled;
theorem Th62: :: VECTSP_1:62
canceled;
theorem Th63: :: VECTSP_1:63
Lemma84:
for V being non empty add-associative right_zeroed right_complementable LoopStr
for v, w being Element of V holds - (w + (- v)) = v - w
Lemma85:
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 Th64: :: VECTSP_1:64
theorem Th65: :: VECTSP_1:65
theorem Th66: :: VECTSP_1:66
theorem Th67: :: VECTSP_1:67
theorem Th68: :: VECTSP_1:68
theorem Th69: :: VECTSP_1:69
theorem Th70: :: VECTSP_1:70
theorem Th71: :: VECTSP_1:71
canceled;
theorem Th72: :: VECTSP_1:72
canceled;
theorem Th73: :: VECTSP_1:73
theorem Th74: :: VECTSP_1:74
theorem Th75: :: VECTSP_1:75
canceled;
theorem Th76: :: VECTSP_1:76
canceled;
theorem Th77: :: VECTSP_1:77
canceled;
theorem Th78: :: VECTSP_1:78
:: deftheorem Def27 VECTSP_1:def 27 :
canceled;
:: deftheorem Def28 defines Fanoian VECTSP_1:def 28 :
:: deftheorem Def29 defines Fanoian VECTSP_1:def 29 :
theorem Th79: :: VECTSP_1:79
canceled;
theorem Th80: :: VECTSP_1:80
canceled;
theorem Th81: :: VECTSP_1:81
theorem Th82: :: VECTSP_1:82
canceled;
theorem Th83: :: VECTSP_1:83
canceled;
theorem Th84: :: VECTSP_1:84
theorem Th85: :: VECTSP_1:85
canceled;
theorem Th86: :: VECTSP_1:86
theorem Th87: :: VECTSP_1:87
theorem Th88: :: VECTSP_1:88
theorem Th89: :: VECTSP_1:89
theorem Th90: :: VECTSP_1:90
theorem Th91: :: VECTSP_1:91
canceled;
theorem Th92: :: VECTSP_1:92
theorem Th93: :: VECTSP_1:93
theorem Th94: :: VECTSP_1:94
theorem Th95: :: VECTSP_1:95