:: 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
E28:
now
take ZS =
{0};
reconsider O = 0 as
Element of
ZS by TARSKI:def 1;
take O =
O;
deffunc H1(
Element of
ZS,
Element of
ZS)
-> Element of
ZS =
O;
consider F being
BinOp of
ZS such that E30:
for
x,
y being
Element of
ZS holds
F . x,
y = H1(
x,
y)
from BINOP_1:sch 4();
deffunc H2(
Element of
REAL ,
Element of
ZS)
-> Element of
ZS =
O;
consider G being
Function of
[:REAL ,ZS:],
ZS such that E32:
for
a being
Element of
REAL for
x being
Element of
ZS holds
G . a,
x = H2(
a,
x)
from BINOP_1:sch 4();
take F =
F;
take G =
G;
set W =
RLSStruct(#
ZS,
O,
F,
G #);
thus
for
x,
y being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) holds
x + y = y + x
proof
let x be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #),
y be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #);
E34:
(
x + y = F . x,
y &
y + x = F . y,
x )
;
reconsider X =
x,
Y =
y as
Element of
ZS ;
(
x + y = H1(
X,
Y) &
y + x = H1(
Y,
X) )
by , ;
hence
x + y = y + x
;
end;
thus
for
x,
y,
z being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) holds
(x + y) + z = x + (y + z)
proof
let x be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #),
y be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #),
z be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #);
E38:
(
(x + y) + z = F . (x + y),
z &
x + (y + z) = F . x,
(y + z) )
;
reconsider X =
x,
Y =
y,
Z =
z as
Element of
ZS ;
(
(x + y) + z = H1(
H1(
X,
Y),
Z) &
x + (y + z) = H1(
X,
H1(
Y,
Z)) )
by , ;
hence
(x + y) + z = x + (y + z)
;
end;
thus
for
x being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) holds
x + (0. RLSStruct(# ZS,O,F,G #)) = x
proof
let x be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #);
reconsider X =
x as
Element of
ZS ;
x + (0. RLSStruct(# ZS,O,F,G #)) =
F . x,
(0. RLSStruct(# ZS,O,F,G #))
.=
H1(
X,
O)
by
;
hence
x + (0. RLSStruct(# ZS,O,F,G #)) = x
by TARSKI:def 1;
end;
thus
for
x being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) ex
y being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) st
x + y = 0. RLSStruct(#
ZS,
O,
F,
G #)
proof
let x be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #);
reconsider y =
O as
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) ;
take
y
;
thus x + y =
F . x,
y
.=
0. RLSStruct(#
ZS,
O,
F,
G #)
by
;
end;
thus
for
a being
Real for
x,
y being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) holds
a * (x + y) = (a * x) + (a * y)
proof
let a be
Real;
let x be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #),
y be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #);
reconsider X =
x,
Y =
y as
Element of
ZS ;
(a * x) + (a * y) =
F . (a * x),
(a * y)
.=
O
by
.=
G . a,
(F . X,Y)
by
;
hence
a * (x + y) = (a * x) + (a * y)
;
end;
thus
for
a,
b being
Real for
x being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) holds
(a + b) * x = (a * x) + (b * x)
proof
let a be
Real;
let b be
Real;
let x be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #);
set c =
a + b;
reconsider X =
x as
Element of
ZS ;
E41:
(a + b) * x =
G . (a + b),
x
.=
H2(
a + b,
X)
by
;
(a * x) + (b * x) =
F . (a * x),
(b * x)
.=
H1(
H2(
a,
X),
H2(
b,
X))
by
;
hence
(a + b) * x = (a * x) + (b * x)
by ;
end;
thus
for
a,
b being
Real for
x being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) holds
(a * b) * x = a * (b * x)
proof
let a be
Real;
let b be
Real;
let x be
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #);
set c =
a * b;
reconsider X =
x as
Element of
ZS ;
E42:
(a * b) * x =
G . (a * b),
x
.=
H2(
a * b,
X)
by
;
a * (b * x) =
G . a,
(b * x)
.=
H2(
a,
H2(
b,
X))
by
;
hence
(a * b) * x = a * (b * x)
by ;
end;
thus
for
x being
VECTOR of
RLSStruct(#
ZS,
O,
F,
G #) holds 1
* x = x
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
V being non
empty RLSStruct st ( for
v,
w being
VECTOR of
V holds
v + w = w + v ) & ( for
u,
v,
w being
VECTOR of
V holds
(u + v) + w = u + (v + w) ) & ( for
v being
VECTOR of
V holds
v + (0. V) = v ) & ( for
v being
VECTOR of
V ex
w being
VECTOR of
V st
v + w = 0. V ) & ( for
a being
Real for
v,
w being
VECTOR of
V holds
a * (v + w) = (a * v) + (a * w) ) & ( for
a,
b being
Real for
v being
VECTOR of
V holds
(a + b) * v = (a * v) + (b * v) ) & ( for
a,
b being
Real for
v being
VECTOR of
V holds
(a * b) * v = a * (b * v) ) & ( for
v being
VECTOR of
V holds 1
* v = v ) holds
V is
RealLinearSpace by , , , , ;
Lemma51:
for V being non empty add-associative right_zeroed right_complementable LoopStr
for v, w being Element of V st v + w = 0. V holds
w + v = 0. V
theorem Th8: :: RLVECT_1:8
canceled;
theorem Th9: :: RLVECT_1:9
canceled;
theorem Th10: :: RLVECT_1:10
:: deftheorem Def10 defines - RLVECT_1:def 10 :
Lemma56:
for V being non empty add-associative right_zeroed right_complementable LoopStr
for v, u being Element of V ex w being Element of V st v + w = u
:: 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
Lemma72:
for V being non empty add-associative right_zeroed right_complementable LoopStr
for u, w being Element of V holds - (u + w) = (- w) + (- u)
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 :
Lemma118:
for V being non empty LoopStr holds Sum (<*> the carrier of V) = 0. V
Lemma120:
for V being non empty LoopStr
for F being FinSequence of the carrier of V st len F = 0 holds
Sum F = 0. V
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
Lemma126:
for j being natural number st j < 1 holds
j = 0
by NAT_1:39;
theorem Th58: :: RLVECT_1:58
Lemma128:
for V being non empty add-associative right_zeroed right_complementable LoopStr
for v being Element of V holds Sum <*v*> = v
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 :