:: RLSUB_2 semantic presentation

definition

let V be RealLinearSpace;
let W1, W2 be Subspace of V;
func W1 + W2 -> strict Subspace of V means :Def1: :: RLSUB_2:def 1
the carrier of it = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ;
existence

proof end;

uniqueness by RLSUB_1:38;

end;

:: deftheorem Def1 defines + RLSUB_2:def 1 :

definition

let V be RealLinearSpace;
let W1, W2 be Subspace of V;
func W1 /\ W2 -> strict Subspace of V means :Def2: :: RLSUB_2:def 2
the carrier of it = the carrier of W1 /\ the carrier of W2;
existence

proof end;

uniqueness by RLSUB_1:38;

end;

:: deftheorem Def2 defines /\ RLSUB_2:def 2 :

theorem :: RLSUB_2:1

canceled;

theorem :: RLSUB_2:2

canceled;

theorem :: RLSUB_2:3

canceled;

theorem :: RLSUB_2:4

canceled;

theorem Th5: :: RLSUB_2:5

for V being RealLinearSpace
for W1, W2 being Subspace of V
for x being set holds
( x in W1 + W2 iff ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

proof end;

theorem Th6: :: RLSUB_2:6

for V being RealLinearSpace
for W1, W2 being Subspace of V
for v being VECTOR of V st ( v in W1 or v in W2 ) holds
v in W1 + W2

proof end;

theorem Th7: :: RLSUB_2:7

for V being RealLinearSpace
for W1, W2 being Subspace of V
for x being set holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )

proof end;

Lm1: for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1

proof end;

Lm2: for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of W1 c= the carrier of (W1 + W2)

proof end;

Lm3: for V being RealLinearSpace
for W1 being Subspace of V
for W2 being strict Subspace of V st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2

proof end;

theorem :: RLSUB_2:8

for V being RealLinearSpace
for W being strict Subspace of V holds W + W = W by Lm3;

theorem :: RLSUB_2:9

for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1 by Lm1;

theorem Th10: :: RLSUB_2:10

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds W1 + (W2 + W3) = (W1 + W2) + W3

proof end;

theorem Th11: :: RLSUB_2:11

for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 )

proof end;

theorem Th12: :: RLSUB_2:12

for V being RealLinearSpace
for W1 being Subspace of V
for W2 being strict Subspace of V holds
( W1 is Subspace of W2 iff W1 + W2 = W2 )

proof end;

theorem Th13: :: RLSUB_2:13

for V being RealLinearSpace
for W being strict Subspace of V holds
( ((0). V) + W = W & W + ((0). V) = W )

proof end;

theorem Th14: :: RLSUB_2:14

for V being RealLinearSpace holds
( ((0). V) + ((Omega). V) = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) & ((Omega). V) + ((0). V) = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) ) by Th13;

theorem Th15: :: RLSUB_2:15

for V being RealLinearSpace
for W being Subspace of V holds
( ((Omega). V) + W = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) & W + ((Omega). V) = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) )

proof end;

theorem :: RLSUB_2:16

for V being strict RealLinearSpace holds ((Omega). V) + ((Omega). V) = V by Th15;

theorem :: RLSUB_2:17

for V being RealLinearSpace
for W being strict Subspace of V holds W /\ W = W

proof end;

theorem Th18: :: RLSUB_2:18

for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 /\ W2 = W2 /\ W1

proof end;

theorem Th19: :: RLSUB_2:19

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3

proof end;

Lm4: for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of W1

proof end;

theorem Th20: :: RLSUB_2:20

for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 /\ W2 is Subspace of W1 & W1 /\ W2 is Subspace of W2 )

proof end;

theorem Th21: :: RLSUB_2:21

for V being RealLinearSpace
for W2 being Subspace of V
for W1 being strict Subspace of V holds
( W1 is Subspace of W2 iff W1 /\ W2 = W1 )

proof end;

theorem Th22: :: RLSUB_2:22

for V being RealLinearSpace
for W being Subspace of V holds
( ((0). V) /\ W = (0). V & W /\ ((0). V) = (0). V )

proof end;

theorem :: RLSUB_2:23

for V being RealLinearSpace holds
( ((0). V) /\ ((Omega). V) = (0). V & ((Omega). V) /\ ((0). V) = (0). V ) by Th22;

theorem Th24: :: RLSUB_2:24

for V being RealLinearSpace
for W being strict Subspace of V holds
( ((Omega). V) /\ W = W & W /\ ((Omega). V) = W )

proof end;

theorem :: RLSUB_2:25

for V being strict RealLinearSpace holds ((Omega). V) /\ ((Omega). V) = V by Th24;

Lm5: for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)

proof end;

theorem :: RLSUB_2:26

for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 /\ W2 is Subspace of W1 + W2

proof end;

Lm6: for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2

proof end;

theorem Th27: :: RLSUB_2:27

for V being RealLinearSpace
for W1 being Subspace of V
for W2 being strict Subspace of V holds (W1 /\ W2) + W2 = W2

proof end;

Lm7: for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1

proof end;

theorem Th28: :: RLSUB_2:28

for V being RealLinearSpace
for W2 being Subspace of V
for W1 being strict Subspace of V holds W1 /\ (W1 + W2) = W1

proof end;

Lm8: for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))

proof end;

theorem :: RLSUB_2:29

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds (W1 /\ W2) + (W2 /\ W3) is Subspace of W2 /\ (W1 + W3)

proof end;

Lm9: for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))

proof end;

theorem Th30: :: RLSUB_2:30

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3)

proof end;

Lm10: for V being RealLinearSpace
for W2, W1, W3 being Subspace of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))

proof end;

theorem :: RLSUB_2:31

for V being RealLinearSpace
for W2, W1, W3 being Subspace of V holds W2 + (W1 /\ W3) is Subspace of (W1 + W2) /\ (W2 + W3)

proof end;

Lm11: for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))

proof end;

theorem :: RLSUB_2:32

for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3)

proof end;

theorem Th33: :: RLSUB_2:33

for V being RealLinearSpace
for W1, W3, W2 being Subspace of V st W1 is strict Subspace of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3

proof end;

theorem :: RLSUB_2:34

for V being RealLinearSpace
for W1, W2 being strict Subspace of V holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )

proof end;

theorem :: RLSUB_2:35

for V being RealLinearSpace
for W1 being Subspace of V
for W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds
W1 + W3 is Subspace of W2 + W3

proof end;

theorem :: RLSUB_2:36

for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( ( W1 is Subspace of W2 or W2 is Subspace of W1 ) iff ex W being Subspace of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )

proof end;

definition

let V be RealLinearSpace;
func Subspaces V -> set means :Def3: :: RLSUB_2:def 3
for x being set holds
( x in it iff x is strict Subspace of V );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines Subspaces RLSUB_2:def 3 :

registration

let V be RealLinearSpace;
cluster Subspaces V -> non empty ;
coherence

proof end;

end;

theorem :: RLSUB_2:37

canceled;

theorem :: RLSUB_2:38

canceled;

theorem :: RLSUB_2:39

for V being strict RealLinearSpace holds V in Subspaces V

proof end;

definition

let V be RealLinearSpace;
let W1, W2 be Subspace of V;
pred V is_the_direct_sum_of W1,W2 means :Def4: :: RLSUB_2:def 4
( RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) = W1 + W2 & W1 /\ W2 = (0). V );

end;

:: deftheorem Def4 defines is_the_direct_sum_of RLSUB_2:def 4 :

Lm12: for V being RealLinearSpace
for W being strict Subspace of V st ( for v being VECTOR of V holds v in W ) holds
W = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #)

proof end;

Lm13: for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 + W2 = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )

proof end;

Lm14: for V being non empty add-associative right_zeroed right_complementable LoopStr
for v, v1, v2 being Element of V holds
( v = v1 + v2 iff v1 = v - v2 )

proof end;

Lm15: for V being RealLinearSpace
for W being Subspace of V ex C being strict Subspace of V st V is_the_direct_sum_of C,W

proof end;

definition

let V be RealLinearSpace;
let W be Subspace of V;
mode Linear_Compl of W -> Subspace of V means :Def5: :: RLSUB_2:def 5
V is_the_direct_sum_of it,W;
existence

proof end;

end;

:: deftheorem Def5 defines Linear_Compl RLSUB_2:def 5 :

registration

let V be RealLinearSpace;
let W be Subspace of V;
cluster strict Linear_Compl of W;
existence

proof end;

end;

Lm16: for V being RealLinearSpace
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1

proof end;

theorem :: RLSUB_2:40

canceled;

theorem :: RLSUB_2:41

canceled;

theorem :: RLSUB_2:42

for V being RealLinearSpace
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1

proof end;

theorem Th43: :: RLSUB_2:43

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )

proof end;

theorem Th44: :: RLSUB_2:44

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W holds
( W + L = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) & L + W = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) )

proof end;

theorem Th45: :: RLSUB_2:45

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W holds
( W /\ L = (0). V & L /\ W = (0). V )

proof end;

theorem :: RLSUB_2:46

for V being RealLinearSpace
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1 by Lm16;

theorem Th47: :: RLSUB_2:47

for V being RealLinearSpace holds
( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V )

proof end;

theorem :: RLSUB_2:48

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W holds W is Linear_Compl of L

proof end;

theorem :: RLSUB_2:49

for V being RealLinearSpace holds
( (0). V is Linear_Compl of (Omega). V & (Omega). V is Linear_Compl of (0). V )

proof end;

theorem Th50: :: RLSUB_2:50

for V being RealLinearSpace
for W1, W2 being Subspace of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2

proof end;

Lm17: for V being RealLinearSpace
for W being Subspace of V
for v being VECTOR of V ex C being Coset of W st v in C

proof end;

theorem Th51: :: RLSUB_2:51

for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} )

proof end;

theorem :: RLSUB_2:52

for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 + W2 = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) by Lm13;

theorem Th53: :: RLSUB_2:53

for V being RealLinearSpace
for W1, W2 being Subspace of V
for v, v1, v2, u1, u2 being VECTOR of V st V is_the_direct_sum_of W1,W2 & v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 )

proof end;

Lm18: for X, Y being set st X c< Y holds
ex x being set st
( x in Y & not x in X )

proof end;

theorem :: RLSUB_2:54

for V being RealLinearSpace
for W1, W2 being Subspace of V st V = W1 + W2 & ex v being VECTOR of V st
for v1, v2, u1, u2 being VECTOR of V st v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) holds
V is_the_direct_sum_of W1,W2

proof end;

definition

let V be RealLinearSpace;
let v be VECTOR of V;
let W1, W2 be Subspace of V;
assume A1: V is_the_direct_sum_of W1,W2 ;
func v |-- W1,W2 -> Element of [:the carrier of V,the carrier of V:] means :Def6: :: RLSUB_2:def 6
( v = (it `1 ) + (it `2 ) & it `1 in W1 & it `2 in W2 );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines |-- RLSUB_2:def 6 :

theorem :: RLSUB_2:55

canceled;

theorem :: RLSUB_2:56

canceled;

theorem :: RLSUB_2:57

canceled;

theorem :: RLSUB_2:58

canceled;

theorem Th59: :: RLSUB_2:59

for V being RealLinearSpace
for W1, W2 being Subspace of V
for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `1 = (v |-- W2,W1) `2

proof end;

theorem Th60: :: RLSUB_2:60

for V being RealLinearSpace
for W1, W2 being Subspace of V
for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1

proof end;

theorem :: RLSUB_2:61

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V
for t being Element of [:the carrier of V,the carrier of V:] st (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L holds
t = v |-- W,L

proof end;

theorem :: RLSUB_2:62

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V holds ((v |-- W,L) `1 ) + ((v |-- W,L) `2 ) = v

proof end;

theorem :: RLSUB_2:63

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V holds
( (v |-- W,L) `1 in W & (v |-- W,L) `2 in L )

proof end;

theorem :: RLSUB_2:64

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- W,L) `1 = (v |-- L,W) `2

proof end;

theorem :: RLSUB_2:65

for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- W,L) `2 = (v |-- L,W) `1

proof end;

definition

let V be RealLinearSpace;
func SubJoin V -> BinOp of Subspaces V means :Def7: :: RLSUB_2:def 7
for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
it . A1,A2 = W1 + W2;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines SubJoin RLSUB_2:def 7 :

definition

let V be RealLinearSpace;
func SubMeet V -> BinOp of Subspaces V means :Def8: :: RLSUB_2:def 8
for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
it . A1,A2 = W1 /\ W2;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines SubMeet RLSUB_2:def 8 :

theorem :: RLSUB_2:66

canceled;

theorem :: RLSUB_2:67

canceled;

theorem :: RLSUB_2:68

canceled;

theorem :: RLSUB_2:69

canceled;

theorem Th70: :: RLSUB_2:70

for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is Lattice

proof end;

registration

let V be RealLinearSpace;
cluster LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) -> Lattice-like ;
coherence by Th70;

end;

theorem Th71: :: RLSUB_2:71

for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is lower-bounded

proof end;

theorem Th72: :: RLSUB_2:72

for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is upper-bounded

proof end;

theorem Th73: :: RLSUB_2:73

for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is 01_Lattice

proof end;

theorem Th74: :: RLSUB_2:74

for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is modular

proof end;

Lm19: for l being Lattice
for a, b being Element of l holds
( a is_a_complement_of b iff ( a "\/" b = Top l & a "/\" b = Bottom l ) )

proof end;

Lm20: for l being Lattice
for b being Element of l st ( for a being Element of l holds a "/\" b = b ) holds
b = Bottom l

proof end;

Lm21: for l being Lattice
for b being Element of l st ( for a being Element of l holds a "\/" b = b ) holds
b = Top l

proof end;

theorem Th75: :: RLSUB_2:75

for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is complemented

proof end;

registration

let V be RealLinearSpace;
cluster LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) -> Lattice-like modular lower-bounded upper-bounded complemented ;
coherence by Th71, Th72, Th74, Th75;

end;

theorem :: RLSUB_2:76

for V being RealLinearSpace
for W1, W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds
W1 /\ W3 is Subspace of W2 /\ W3

proof end;

theorem :: RLSUB_2:77

for X, Y being set st X c< Y holds
ex x being set st
( x in Y & not x in X ) by Lm18;

theorem :: RLSUB_2:78

for V being non empty add-associative right_zeroed right_complementable LoopStr
for v, v1, v2 being Element of V holds
( v = v1 + v2 iff v1 = v - v2 ) by Lm14;

theorem :: RLSUB_2:79

for V being RealLinearSpace
for W being strict Subspace of V st ( for v being VECTOR of V holds v in W ) holds
W = RLSStruct(# the carrier of V,the Zero of V,the add of V,the Mult of V #) by Lm12;

theorem :: RLSUB_2:80

for V being RealLinearSpace
for W being Subspace of V
for v being VECTOR of V ex C being Coset of W st v in C by Lm17;

theorem :: RLSUB_2:81

canceled;

theorem :: RLSUB_2:82

canceled;

theorem :: RLSUB_2:83

canceled;

theorem :: RLSUB_2:84

for l being Lattice
for b being Element of l st ( for a being Element of l holds a "/\" b = b ) holds
b = Bottom l by Lm20;

theorem :: RLSUB_2:85

for l being Lattice
for b being Element of l st ( for a being Element of l holds a "\/" b = b ) holds
b = Top l by Lm21;