:: Subspaces of Real Linear Space Generated by One, Two, or Three Vectorsand Their Cosets
:: by Wojciech A. Trybulec
::
:: Received February 24, 1993
:: Copyright (c) 1993 Association of Mizar Users

scheme :: RLVECT_4:sch 1
LambdaSep3{ F₁() -> non empty set , F₂() -> non empty set , F₃() -> Element of F₁(), F₄() -> Element of F₁(), F₅() -> Element of F₁(), F₆() -> Element of F₂(), F₇() -> Element of F₂(), F₈() -> Element of F₂(), F₉( set ) -> Element of F₂() } :

ex f being Function of F₁(),F₂() st
( f . F₃() = F₆() & f . F₄() = F₇() & f . F₅() = F₈() & ( for C being Element of F₁() st C <> F₃() & C <> F₄() & C <> F₅() holds
f . C = F₉(C) ) )

provided

A1: F₃() <> F₄() and
A2: F₃() <> F₅() and
A3: F₄() <> F₅()

proof end;

scheme :: RLVECT_4:sch 2
LinCEx1{ F₁() -> RealLinearSpace, F₂() -> VECTOR of F₁(), F₃() -> Real } :

ex l being Linear_Combination of {F₂()} st l . F₂() = F₃()

proof end;

scheme :: RLVECT_4:sch 3
LinCEx2{ F₁() -> RealLinearSpace, F₂() -> VECTOR of F₁(), F₃() -> VECTOR of F₁(), F₄() -> Real, F₅() -> Real } :

ex l being Linear_Combination of {F₂(),F₃()} st
( l . F₂() = F₄() & l . F₃() = F₅() )

provided

A1: F₂() <> F₃()

proof end;

scheme :: RLVECT_4:sch 4
LinCEx3{ F₁() -> RealLinearSpace, F₂() -> VECTOR of F₁(), F₃() -> VECTOR of F₁(), F₄() -> VECTOR of F₁(), F₅() -> Real, F₆() -> Real, F₇() -> Real } :

ex l being Linear_Combination of {F₂(),F₃(),F₄()} st
( l . F₂() = F₅() & l . F₃() = F₆() & l . F₄() = F₇() )

provided

A1: F₂() <> F₃() and
A2: F₂() <> F₄() and
A3: F₃() <> F₄()

proof end;

theorem :: RLVECT_4:1

for V being RealLinearSpace
for v, w being VECTOR of V holds
( (v + w) - v = w & (w + v) - v = w & (v - v) + w = w & (w - v) + v = w & v + (w - v) = w & w + (v - v) = w & v - (v - w) = w )

proof end;

theorem :: RLVECT_4:2

canceled;

theorem :: RLVECT_4:3

canceled;

theorem :: RLVECT_4:4

for V being RealLinearSpace
for v1, w, v2 being VECTOR of V st v1 - w = v2 - w holds
v1 = v2

proof end;

theorem :: RLVECT_4:5

canceled;

theorem Th6: :: RLVECT_4:6

for a being Real
for V being RealLinearSpace
for v being VECTOR of V holds - (a * v) = (- a) * v

proof end;

theorem :: RLVECT_4:7

for V being RealLinearSpace
for v being VECTOR of V
for W1, W2 being Subspace of V st W1 is Subspace of W2 holds
v + W1 c= v + W2

proof end;

theorem :: RLVECT_4:8

for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st u in v + W holds
v + W = u + W

proof end;

theorem Th9: :: RLVECT_4:9

for V being RealLinearSpace
for u, v, w being VECTOR of V
for l being Linear_Combination of {u,v,w} st u <> v & u <> w & v <> w holds
Sum l = (((l . u) * u) + ((l . v) * v)) + ((l . w) * w)

proof end;

theorem Th10: :: RLVECT_4:10

for V being RealLinearSpace
for u, v, w being VECTOR of V holds
( ( u <> v & u <> w & v <> w & {u,v,w} is linearly-independent ) iff for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) )

proof end;

theorem Th11: :: RLVECT_4:11

for x being set
for V being RealLinearSpace
for v being VECTOR of V holds
( x in Lin {v} iff ex a being Real st x = a * v )

proof end;

theorem :: RLVECT_4:12

for V being RealLinearSpace
for v being VECTOR of V holds v in Lin {v}

proof end;

theorem :: RLVECT_4:13

for x being set
for V being RealLinearSpace
for v, w being VECTOR of V holds
( x in v + (Lin {w}) iff ex a being Real st x = v + (a * w) )

proof end;

theorem Th14: :: RLVECT_4:14

for x being set
for V being RealLinearSpace
for w1, w2 being VECTOR of V holds
( x in Lin {w1,w2} iff ex a, b being Real st x = (a * w1) + (b * w2) )

proof end;

theorem :: RLVECT_4:15

for V being RealLinearSpace
for w1, w2 being VECTOR of V holds
( w1 in Lin {w1,w2} & w2 in Lin {w1,w2} )

proof end;

theorem :: RLVECT_4:16

for x being set
for V being RealLinearSpace
for v, w1, w2 being VECTOR of V holds
( x in v + (Lin {w1,w2}) iff ex a, b being Real st x = (v + (a * w1)) + (b * w2) )

proof end;

theorem Th17: :: RLVECT_4:17

for x being set
for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V holds
( x in Lin {v1,v2,v3} iff ex a, b, c being Real st x = ((a * v1) + (b * v2)) + (c * v3) )

proof end;

theorem :: RLVECT_4:18

for V being RealLinearSpace
for w1, w2, w3 being VECTOR of V holds
( w1 in Lin {w1,w2,w3} & w2 in Lin {w1,w2,w3} & w3 in Lin {w1,w2,w3} )

proof end;

theorem :: RLVECT_4:19

for x being set
for V being RealLinearSpace
for v, w1, w2, w3 being VECTOR of V holds
( x in v + (Lin {w1,w2,w3}) iff ex a, b, c being Real st x = ((v + (a * w1)) + (b * w2)) + (c * w3) )

proof end;

theorem :: RLVECT_4:20

for V being RealLinearSpace
for u, v being VECTOR of V st {u,v} is linearly-independent & u <> v holds
{u,(v - u)} is linearly-independent

proof end;

theorem :: RLVECT_4:21

for V being RealLinearSpace
for u, v being VECTOR of V st {u,v} is linearly-independent & u <> v holds
{u,(v + u)} is linearly-independent

proof end;

theorem Th22: :: RLVECT_4:22

for a being Real
for V being RealLinearSpace
for u, v being VECTOR of V st {u,v} is linearly-independent & u <> v & a <> 0 holds
{u,(a * v)} is linearly-independent

proof end;

theorem :: RLVECT_4:23

for V being RealLinearSpace
for u, v being VECTOR of V st {u,v} is linearly-independent & u <> v holds
{u,(- v)} is linearly-independent

proof end;

theorem Th24: :: RLVECT_4:24

for a, b being Real
for V being RealLinearSpace
for v being VECTOR of V st a <> b holds
not {(a * v),(b * v)} is linearly-independent

proof end;

theorem :: RLVECT_4:25

for a being Real
for V being RealLinearSpace
for v being VECTOR of V st a <> 1 holds
not {v,(a * v)} is linearly-independent

proof end;

theorem :: RLVECT_4:26

for V being RealLinearSpace
for u, w, v being VECTOR of V st {u,w,v} is linearly-independent & u <> v & u <> w & v <> w holds
{u,w,(v - u)} is linearly-independent

proof end;

theorem :: RLVECT_4:27

for V being RealLinearSpace
for u, w, v being VECTOR of V st {u,w,v} is linearly-independent & u <> v & u <> w & v <> w holds
{u,(w - u),(v - u)} is linearly-independent

proof end;

theorem :: RLVECT_4:28

for V being RealLinearSpace
for u, w, v being VECTOR of V st {u,w,v} is linearly-independent & u <> v & u <> w & v <> w holds
{u,w,(v + u)} is linearly-independent

proof end;

theorem :: RLVECT_4:29

for V being RealLinearSpace
for u, w, v being VECTOR of V st {u,w,v} is linearly-independent & u <> v & u <> w & v <> w holds
{u,(w + u),(v + u)} is linearly-independent

proof end;

theorem Th30: :: RLVECT_4:30

for a being Real
for V being RealLinearSpace
for u, w, v being VECTOR of V st {u,w,v} is linearly-independent & u <> v & u <> w & v <> w & a <> 0 holds
{u,w,(a * v)} is linearly-independent

proof end;

theorem Th31: :: RLVECT_4:31

for a, b being Real
for V being RealLinearSpace
for u, w, v being VECTOR of V st {u,w,v} is linearly-independent & u <> v & u <> w & v <> w & a <> 0 & b <> 0 holds
{u,(a * w),(b * v)} is linearly-independent

proof end;

theorem :: RLVECT_4:32

for V being RealLinearSpace
for u, w, v being VECTOR of V st {u,w,v} is linearly-independent & u <> v & u <> w & v <> w holds
{u,w,(- v)} is linearly-independent

proof end;

theorem :: RLVECT_4:33

for V being RealLinearSpace
for u, w, v being VECTOR of V st {u,w,v} is linearly-independent & u <> v & u <> w & v <> w holds
{u,(- w),(- v)} is linearly-independent

proof end;

theorem Th34: :: RLVECT_4:34

for a, b being Real
for V being RealLinearSpace
for v, w being VECTOR of V st a <> b holds
not {(a * v),(b * v),w} is linearly-independent

proof end;

theorem :: RLVECT_4:35

for a being Real
for V being RealLinearSpace
for v, w being VECTOR of V st a <> 1 holds
not {v,(a * v),w} is linearly-independent

proof end;

theorem :: RLVECT_4:36

for V being RealLinearSpace
for v, w being VECTOR of V st v in Lin {w} & v <> 0. V holds
Lin {v} = Lin {w}

proof end;

theorem :: RLVECT_4:37

for V being RealLinearSpace
for v1, v2, w1, w2 being VECTOR of V st v1 <> v2 & {v1,v2} is linearly-independent & v1 in Lin {w1,w2} & v2 in Lin {w1,w2} holds
( Lin {w1,w2} = Lin {v1,v2} & {w1,w2} is linearly-independent & w1 <> w2 )

proof end;

theorem :: RLVECT_4:38

for V being RealLinearSpace
for w, v being VECTOR of V st w <> 0. V & not {v,w} is linearly-independent holds
ex a being Real st v = a * w

proof end;

theorem :: RLVECT_4:39

for V being RealLinearSpace
for v, w, u being VECTOR of V st v <> w & {v,w} is linearly-independent & not {u,v,w} is linearly-independent holds
ex a, b being Real st u = (a * v) + (b * w)

proof end;