:: VECTSP_7 semantic presentation

definition
let c1 be Field;
let c2 be VectSp of c1;
let c3 be Subset of c2;
attr a3 is linearly-independent means :Def1: :: VECTSP_7:def 1
for b1 being Linear_Combination of a3 holds
( Sum b1 = 0. a2 implies Carrier b1 = {} );
end;

:: deftheorem Def1 defines linearly-independent VECTSP_7:def 1 :
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2 holds
( b3 is linearly-independent iff for b4 being Linear_Combination of b3 holds
( Sum b4 = 0. b2 implies Carrier b4 = {} ) );

notation
let c1 be Field;
let c2 be VectSp of c1;
let c3 be Subset of c2;
antonym linearly-dependent c3 for linearly-independent c3;
end;

theorem Th1: :: VECTSP_7:1
canceled;

theorem Th2: :: VECTSP_7:2
for b1 being Field
for b2 being VectSp of b1
for b3, b4 being Subset of b2 holds
( b3 c= b4 & b4 is linearly-independent implies b3 is linearly-independent )
proof end;

theorem Th3: :: VECTSP_7:3
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2 holds
not ( b3 is linearly-independent & 0. b2 in b3 )
proof end;

theorem Th4: :: VECTSP_7:4
for b1 being Field
for b2 being VectSp of b1 holds {} the carrier of b2 is linearly-independent
proof end;

registration
let c1 be Field;
let c2 be VectSp of c1;
cluster linearly-independent Element of bool the carrier of a2;
existence
ex b1 being Subset of c2 st b1 is linearly-independent
proof end;
end;

theorem Th5: :: VECTSP_7:5
for b1 being Field
for b2 being VectSp of b1
for b3 being Vector of b2 holds
( {b3} is linearly-independent iff b3 <> 0. b2 )
proof end;

theorem Th6: :: VECTSP_7:6
for b1 being Field
for b2 being VectSp of b1
for b3, b4 being Vector of b2 holds
( {b3,b4} is linearly-independent implies ( b3 <> 0. b2 & b4 <> 0. b2 ) )
proof end;

theorem Th7: :: VECTSP_7:7
for b1 being Field
for b2 being VectSp of b1
for b3 being Vector of b2 holds
( not {b3,(0. b2)} is linearly-independent & not {(0. b2),b3} is linearly-independent ) by Th6;

theorem Th8: :: VECTSP_7:8
for b1 being Field
for b2 being VectSp of b1
for b3, b4 being Vector of b2 holds
( ( b3 <> b4 & {b3,b4} is linearly-independent implies ( b4 <> 0. b2 & ( for b5 being Element of b1 holds
b3 <> b5 * b4 ) ) ) & ( b4 <> 0. b2 & ( for b5 being Element of b1 holds
b3 <> b5 * b4 ) implies ( b3 <> b4 & {b3,b4} is linearly-independent ) ) )
proof end;

theorem Th9: :: VECTSP_7:9
for b1 being Field
for b2 being VectSp of b1
for b3, b4 being Vector of b2 holds
( ( b3 <> b4 & {b3,b4} is linearly-independent ) iff for b5, b6 being Element of b1 holds
( (b5 * b3) + (b6 * b4) = 0. b2 implies ( b5 = 0. b1 & b6 = 0. b1 ) ) )
proof end;

definition
let c1 be Field;
let c2 be VectSp of c1;
let c3 be Subset of c2;
func Lin c3 -> strict Subspace of a2 means :Def2: :: VECTSP_7:def 2
the carrier of a4 = { (Sum b1) where B is Linear_Combination of a3 : verum } ;
existence
ex b1 being strict Subspace of c2 st the carrier of b1 = { (Sum b2) where B is Linear_Combination of c3 : verum }
proof end;
uniqueness
for b1, b2 being strict Subspace of c2 holds
( the carrier of b1 = { (Sum b3) where B is Linear_Combination of c3 : verum } & the carrier of b2 = { (Sum b3) where B is Linear_Combination of c3 : verum } implies b1 = b2 )
by VECTSP_4:37;
end;

:: deftheorem Def2 defines Lin VECTSP_7:def 2 :
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2
for b4 being strict Subspace of b2 holds
( b4 = Lin b3 iff the carrier of b4 = { (Sum b5) where B is Linear_Combination of b3 : verum } );

theorem Th10: :: VECTSP_7:10
canceled;

theorem Th11: :: VECTSP_7:11
canceled;

theorem Th12: :: VECTSP_7:12
for b1 being set
for b2 being Field
for b3 being VectSp of b2
for b4 being Subset of b3 holds
( b1 in Lin b4 iff ex b5 being Linear_Combination of b4 st b1 = Sum b5 )
proof end;

theorem Th13: :: VECTSP_7:13
for b1 being set
for b2 being Field
for b3 being VectSp of b2
for b4 being Subset of b3 holds
( b1 in b4 implies b1 in Lin b4 )
proof end;

theorem Th14: :: VECTSP_7:14
for b1 being Field
for b2 being VectSp of b1 holds Lin ({} the carrier of b2) = (0). b2
proof end;

theorem Th15: :: VECTSP_7:15
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2 holds
not ( Lin b3 = (0). b2 & not b3 = {} & not b3 = {(0. b2)} )
proof end;

theorem Th16: :: VECTSP_7:16
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2
for b4 being strict Subspace of b2 holds
( b3 = the carrier of b4 implies Lin b3 = b4 )
proof end;

theorem Th17: :: VECTSP_7:17
for b1 being Field
for b2 being strict VectSp of b1
for b3 being Subset of b2 holds
( b3 = the carrier of b2 implies Lin b3 = b2 )
proof end;

theorem Th18: :: VECTSP_7:18
for b1 being Field
for b2 being VectSp of b1
for b3, b4 being Subset of b2 holds
( b3 c= b4 implies Lin b3 is Subspace of Lin b4 )
proof end;

theorem Th19: :: VECTSP_7:19
for b1 being Field
for b2 being strict VectSp of b1
for b3, b4 being Subset of b2 holds
( Lin b3 = b2 & b3 c= b4 implies Lin b4 = b2 )
proof end;

theorem Th20: :: VECTSP_7:20
for b1 being Field
for b2 being VectSp of b1
for b3, b4 being Subset of b2 holds Lin (b3 \/ b4) = (Lin b3) + (Lin b4)
proof end;

theorem Th21: :: VECTSP_7:21
for b1 being Field
for b2 being VectSp of b1
for b3, b4 being Subset of b2 holds
Lin (b3 /\ b4) is Subspace of (Lin b3) /\ (Lin b4)
proof end;

theorem Th22: :: VECTSP_7:22
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2 holds
not ( b3 is linearly-independent & ( for b4 being Subset of b2 holds
not ( b3 c= b4 & b4 is linearly-independent & Lin b4 = VectSpStr(# the carrier of b2,the add of b2,the Zero of b2,the lmult of b2 #) ) ) )
proof end;

theorem Th23: :: VECTSP_7:23
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2 holds
not ( Lin b3 = b2 & ( for b4 being Subset of b2 holds
not ( b4 c= b3 & b4 is linearly-independent & Lin b4 = b2 ) ) )
proof end;

definition
let c1 be Field;
let c2 be VectSp of c1;
mode Basis of c2 -> Subset of a2 means :Def3: :: VECTSP_7:def 3
( a3 is linearly-independent & Lin a3 = VectSpStr(# the carrier of a2,the add of a2,the Zero of a2,the lmult of a2 #) );
existence
ex b1 being Subset of c2 st
( b1 is linearly-independent & Lin b1 = VectSpStr(# the carrier of c2,the add of c2,the Zero of c2,the lmult of c2 #) )
proof end;
end;

:: deftheorem Def3 defines Basis VECTSP_7:def 3 :
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2 holds
( b3 is Basis of b2 iff ( b3 is linearly-independent & Lin b3 = VectSpStr(# the carrier of b2,the add of b2,the Zero of b2,the lmult of b2 #) ) );

theorem Th24: :: VECTSP_7:24
canceled;

theorem Th25: :: VECTSP_7:25
canceled;

theorem Th26: :: VECTSP_7:26
canceled;

theorem Th27: :: VECTSP_7:27
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2 holds
not ( b3 is linearly-independent & ( for b4 being Basis of b2 holds
not b3 c= b4 ) )
proof end;

theorem Th28: :: VECTSP_7:28
for b1 being Field
for b2 being VectSp of b1
for b3 being Subset of b2 holds
not ( Lin b3 = b2 & ( for b4 being Basis of b2 holds
not b4 c= b3 ) )
proof end;