:: RUSUB_3 semantic presentation

definition
let c1 be RealUnitarySpace;
let c2 be Subset of c1;
func Lin c2 -> strict Subspace of a1 means :Def1: :: RUSUB_3:def 1
the carrier of a3 = { (Sum b1) where B is Linear_Combination of a2 : verum } ;
existence
ex b1 being strict Subspace of c1 st the carrier of b1 = { (Sum b2) where B is Linear_Combination of c2 : verum }
proof end;
uniqueness
for b1, b2 being strict Subspace of c1 holds
( the carrier of b1 = { (Sum b3) where B is Linear_Combination of c2 : verum } & the carrier of b2 = { (Sum b3) where B is Linear_Combination of c2 : verum } implies b1 = b2 )
by RUSUB_1:24;
end;

:: deftheorem Def1 defines Lin RUSUB_3:def 1 :
for b1 being RealUnitarySpace
for b2 being Subset of b1
for b3 being strict Subspace of b1 holds
( b3 = Lin b2 iff the carrier of b3 = { (Sum b4) where B is Linear_Combination of b2 : verum } );

theorem Th1: :: RUSUB_3:1
for b1 being RealUnitarySpace
for b2 being Subset of b1
for b3 being set holds
( b3 in Lin b2 iff ex b4 being Linear_Combination of b2 st b3 = Sum b4 )
proof end;

theorem Th2: :: RUSUB_3:2
for b1 being RealUnitarySpace
for b2 being Subset of b1
for b3 being set holds
( b3 in b2 implies b3 in Lin b2 )
proof end;

Lemma4: for b1 being RealUnitarySpace
for b2 being set holds
( b2 in (0). b1 iff b2 = 0. b1 )
proof end;

theorem Th3: :: RUSUB_3:3
for b1 being RealUnitarySpace holds Lin ({} the carrier of b1) = (0). b1
proof end;

theorem Th4: :: RUSUB_3:4
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
not ( Lin b2 = (0). b1 & not b2 = {} & not b2 = {(0. b1)} )
proof end;

theorem Th5: :: RUSUB_3:5
for b1 being RealUnitarySpace
for b2 being strict Subspace of b1
for b3 being Subset of b1 holds
( b3 = the carrier of b2 implies Lin b3 = b2 )
proof end;

theorem Th6: :: RUSUB_3:6
for b1 being strict RealUnitarySpace
for b2 being Subset of b1 holds
( b2 = the carrier of b1 implies Lin b2 = b1 )
proof end;

Lemma6: for b1 being RealUnitarySpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b4 implies b2 /\ b3 is Subspace of b4 )
proof end;

Lemma7: for b1 being RealUnitarySpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b3 & b2 is Subspace of b4 implies b2 is Subspace of b3 /\ b4 )
proof end;

Lemma8: for b1 being RealUnitarySpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b3 implies b2 is Subspace of b3 + b4 )
proof end;

Lemma9: for b1 being RealUnitarySpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b4 & b3 is Subspace of b4 implies b2 + b3 is Subspace of b4 )
proof end;

theorem Th7: :: RUSUB_3:7
for b1 being RealUnitarySpace
for b2, b3 being Subset of b1 holds
( b2 c= b3 implies Lin b2 is Subspace of Lin b3 )
proof end;

theorem Th8: :: RUSUB_3:8
for b1 being strict RealUnitarySpace
for b2, b3 being Subset of b1 holds
( Lin b2 = b1 & b2 c= b3 implies Lin b3 = b1 )
proof end;

theorem Th9: :: RUSUB_3:9
for b1 being RealUnitarySpace
for b2, b3 being Subset of b1 holds Lin (b2 \/ b3) = (Lin b2) + (Lin b3)
proof end;

theorem Th10: :: RUSUB_3:10
for b1 being RealUnitarySpace
for b2, b3 being Subset of b1 holds
Lin (b2 /\ b3) is Subspace of (Lin b2) /\ (Lin b3)
proof end;

theorem Th11: :: RUSUB_3:11
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
not ( b2 is linearly-independent & ( for b3 being Subset of b1 holds
not ( b2 c= b3 & b3 is linearly-independent & Lin b3 = UNITSTR(# the carrier of b1,the Zero of b1,the add of b1,the Mult of b1,the scalar of b1 #) ) ) )
proof end;

theorem Th12: :: RUSUB_3:12
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
not ( Lin b2 = b1 & ( for b3 being Subset of b1 holds
not ( b3 c= b2 & b3 is linearly-independent & Lin b3 = b1 ) ) )
proof end;

definition
let c1 be RealUnitarySpace;
mode Basis of c1 -> Subset of a1 means :Def2: :: RUSUB_3:def 2
( a2 is linearly-independent & Lin a2 = UNITSTR(# the carrier of a1,the Zero of a1,the add of a1,the Mult of a1,the scalar of a1 #) );
existence
ex b1 being Subset of c1 st
( b1 is linearly-independent & Lin b1 = UNITSTR(# the carrier of c1,the Zero of c1,the add of c1,the Mult of c1,the scalar of c1 #) )
proof end;
end;

:: deftheorem Def2 defines Basis RUSUB_3:def 2 :
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
( b2 is Basis of b1 iff ( b2 is linearly-independent & Lin b2 = UNITSTR(# the carrier of b1,the Zero of b1,the add of b1,the Mult of b1,the scalar of b1 #) ) );

theorem Th13: :: RUSUB_3:13
for b1 being strict RealUnitarySpace
for b2 being Subset of b1 holds
not ( b2 is linearly-independent & ( for b3 being Basis of b1 holds
not b2 c= b3 ) )
proof end;

theorem Th14: :: RUSUB_3:14
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
not ( Lin b2 = b1 & ( for b3 being Basis of b1 holds
not b3 c= b2 ) )
proof end;

theorem Th15: :: RUSUB_3:15
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
not ( b2 is linearly-independent & ( for b3 being Basis of b1 holds
not b2 c= b3 ) )
proof end;

theorem Th16: :: RUSUB_3:16
for b1 being RealUnitarySpace
for b2 being Linear_Combination of b1
for b3 being Subset of b1
for b4 being FinSequence of the carrier of b1 holds
not ( rng b4 c= the carrier of (Lin b3) & ( for b5 being Linear_Combination of b3 holds
not Sum (b2 (#) b4) = Sum b5 ) )
proof end;

theorem Th17: :: RUSUB_3:17
for b1 being RealUnitarySpace
for b2 being Linear_Combination of b1
for b3 being Subset of b1 holds
not ( Carrier b2 c= the carrier of (Lin b3) & ( for b4 being Linear_Combination of b3 holds
not Sum b2 = Sum b4 ) )
proof end;

theorem Th18: :: RUSUB_3:18
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being Linear_Combination of b1 holds
( Carrier b3 c= the carrier of b2 implies for b4 being Linear_Combination of b2 holds
( b4 = b3 | the carrier of b2 implies ( Carrier b3 = Carrier b4 & Sum b3 = Sum b4 ) ) )
proof end;

theorem Th19: :: RUSUB_3:19
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being Linear_Combination of b2 holds
ex b4 being Linear_Combination of b1 st
( Carrier b3 = Carrier b4 & Sum b3 = Sum b4 )
proof end;

theorem Th20: :: RUSUB_3:20
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being Linear_Combination of b1 holds
not ( Carrier b3 c= the carrier of b2 & ( for b4 being Linear_Combination of b2 holds
not ( Carrier b4 = Carrier b3 & Sum b4 = Sum b3 ) ) )
proof end;

theorem Th21: :: RUSUB_3:21
for b1 being RealUnitarySpace
for b2 being Basis of b1
for b3 being VECTOR of b1 holds b3 in Lin b2
proof end;

theorem Th22: :: RUSUB_3:22
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being Subset of b2 holds
( b3 is linearly-independent implies b3 is linearly-independent Subset of b1 )
proof end;

theorem Th23: :: RUSUB_3:23
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being Subset of b1 holds
( b3 is linearly-independent & b3 c= the carrier of b2 implies b3 is linearly-independent Subset of b2 )
proof end;

theorem Th24: :: RUSUB_3:24
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being Basis of b2 holds
ex b4 being Basis of b1 st b3 c= b4
proof end;

theorem Th25: :: RUSUB_3:25
for b1 being RealUnitarySpace
for b2 being Subset of b1 holds
( b2 is linearly-independent implies for b3 being VECTOR of b1 holds
( b3 in b2 implies for b4 being Subset of b1 holds
not ( b4 = b2 \ {b3} & b3 in Lin b4 ) ) )
proof end;

Lemma22: for b1, b2 being set holds
( not b2 in b1 implies b1 \ {b2} = b1 )
proof end;

theorem Th26: :: RUSUB_3:26
for b1 being RealUnitarySpace
for b2 being Basis of b1
for b3 being non empty Subset of b1 holds
( b3 misses b2 implies for b4 being Subset of b1 holds
not ( b4 = b2 \/ b3 & b4 is linearly-independent ) )
proof end;

theorem Th27: :: RUSUB_3:27
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being Subset of b1 holds
( b3 c= the carrier of b2 implies Lin b3 is Subspace of b2 )
proof end;

theorem Th28: :: RUSUB_3:28
for b1 being RealUnitarySpace
for b2 being Subspace of b1
for b3 being Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 implies Lin b3 = Lin b4 )
proof end;

theorem Th29: :: RUSUB_3:29
for b1 being RealUnitarySpace
for b2 being VECTOR of b1
for b3 being set holds
( b3 in Lin {b2} iff ex b4 being Real st b3 = b4 * b2 )
proof end;

theorem Th30: :: RUSUB_3:30
for b1 being RealUnitarySpace
for b2 being VECTOR of b1 holds b2 in Lin {b2}
proof end;

theorem Th31: :: RUSUB_3:31
for b1 being RealUnitarySpace
for b2, b3 being VECTOR of b1
for b4 being set holds
( b4 in b2 + (Lin {b3}) iff ex b5 being Real st b4 = b2 + (b5 * b3) )
proof end;

theorem Th32: :: RUSUB_3:32
for b1 being RealUnitarySpace
for b2, b3 being VECTOR of b1
for b4 being set holds
( b4 in Lin {b2,b3} iff ex b5, b6 being Real st b4 = (b5 * b2) + (b6 * b3) )
proof end;

theorem Th33: :: RUSUB_3:33
for b1 being RealUnitarySpace
for b2, b3 being VECTOR of b1 holds
( b2 in Lin {b2,b3} & b3 in Lin {b2,b3} )
proof end;

theorem Th34: :: RUSUB_3:34
for b1 being RealUnitarySpace
for b2, b3, b4 being VECTOR of b1
for b5 being set holds
( b5 in b2 + (Lin {b3,b4}) iff ex b6, b7 being Real st b5 = (b2 + (b6 * b3)) + (b7 * b4) )
proof end;

theorem Th35: :: RUSUB_3:35
for b1 being RealUnitarySpace
for b2, b3, b4 being VECTOR of b1
for b5 being set holds
( b5 in Lin {b2,b3,b4} iff ex b6, b7, b8 being Real st b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4) )
proof end;

theorem Th36: :: RUSUB_3:36
for b1 being RealUnitarySpace
for b2, b3, b4 being VECTOR of b1 holds
( b2 in Lin {b2,b3,b4} & b3 in Lin {b2,b3,b4} & b4 in Lin {b2,b3,b4} )
proof end;

theorem Th37: :: RUSUB_3:37
for b1 being RealUnitarySpace
for b2, b3, b4, b5 being VECTOR of b1
for b6 being set holds
( b6 in b2 + (Lin {b3,b4,b5}) iff ex b7, b8, b9 being Real st b6 = ((b2 + (b7 * b3)) + (b8 * b4)) + (b9 * b5) )
proof end;

theorem Th38: :: RUSUB_3:38
for b1 being RealUnitarySpace
for b2, b3 being VECTOR of b1 holds
( b2 in Lin {b3} & b2 <> 0. b1 implies Lin {b2} = Lin {b3} )
proof end;

theorem Th39: :: RUSUB_3:39
for b1 being RealUnitarySpace
for b2, b3, b4, b5 being VECTOR of b1 holds
( b2 <> b3 & {b2,b3} is linearly-independent & b2 in Lin {b4,b5} & b3 in Lin {b4,b5} implies ( Lin {b4,b5} = Lin {b2,b3} & {b4,b5} is linearly-independent & b4 <> b5 ) )
proof end;

theorem Th40: :: RUSUB_3:40
for b1 being RealUnitarySpace
for b2 being set holds
( b2 in (0). b1 iff b2 = 0. b1 ) by Lemma4;

theorem Th41: :: RUSUB_3:41
for b1 being RealUnitarySpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b4 implies b2 /\ b3 is Subspace of b4 ) by Lemma6;

theorem Th42: :: RUSUB_3:42
for b1 being RealUnitarySpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b3 & b2 is Subspace of b4 implies b2 is Subspace of b3 /\ b4 ) by Lemma7;

theorem Th43: :: RUSUB_3:43
for b1 being RealUnitarySpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b4 & b3 is Subspace of b4 implies b2 + b3 is Subspace of b4 ) by Lemma9;

theorem Th44: :: RUSUB_3:44
for b1 being RealUnitarySpace
for b2, b3, b4 being Subspace of b1 holds
( b2 is Subspace of b3 implies b2 is Subspace of b3 + b4 ) by Lemma8;

theorem Th45: :: RUSUB_3:45
for b1 being RealUnitarySpace
for b2, b3 being Subspace of b1
for b4 being VECTOR of b1 holds
( b2 is Subspace of b3 implies b4 + b2 c= b4 + b3 )
proof end;