:: LMOD_5 semantic presentation

definition
let c1 be non empty doubleLoopStr ;
let c2 be non empty VectSpStr of c1;
let c3 be Subset of c2;
attr a3 is linearly-independent means :Def1: :: LMOD_5:def 1
for b1 being Linear_Combination of a3 holds
( Sum b1 = 0. a2 implies Carrier b1 = {} );
end;

:: deftheorem Def1 defines linearly-independent LMOD_5:def 1 :
for b1 being non empty doubleLoopStr
for b2 being non empty VectSpStr 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 non empty doubleLoopStr ;
let c2 be non empty VectSpStr of c1;
let c3 be Subset of c2;
antonym linearly-dependent c3 for linearly-independent c3;
end;

theorem Th1: :: LMOD_5:1
canceled;

theorem Th2: :: LMOD_5:2
for b1 being Ring
for b2 being LeftMod 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: :: LMOD_5:3
for b1 being Ring
for b2 being LeftMod of b1
for b3 being Subset of b2 holds
not ( 0. b1 <> 1. b1 & b3 is linearly-independent & 0. b2 in b3 )
proof end;

theorem Th4: :: LMOD_5:4
for b1 being Ring
for b2 being LeftMod of b1 holds {} the carrier of b2 is linearly-independent
proof end;

theorem Th5: :: LMOD_5:5
for b1 being Ring
for b2 being LeftMod of b1
for b3, b4 being Vector of b2 holds
( 0. b1 <> 1. b1 & {b3,b4} is linearly-independent implies ( b3 <> 0. b2 & b4 <> 0. b2 ) )
proof end;

theorem Th6: :: LMOD_5:6
for b1 being Ring
for b2 being LeftMod of b1
for b3 being Vector of b2 holds
( 0. b1 <> 1. b1 implies ( not {b3,(0. b2)} is linearly-independent & not {(0. b2),b3} is linearly-independent ) ) by Th5;

theorem Th7: :: LMOD_5:7
for b1 being domRing
for b2 being LeftMod of b1
for b3 being Linear_Combination of b2
for b4 being Scalar of b1 holds
( b4 <> 0. b1 implies Carrier (b4 * b3) = Carrier b3 )
proof end;

theorem Th8: :: LMOD_5:8
for b1 being domRing
for b2 being LeftMod of b1
for b3 being Linear_Combination of b2
for b4 being Scalar of b1 holds Sum (b4 * b3) = b4 * (Sum b3)
proof end;

definition
let c1 be domRing;
let c2 be LeftMod of c1;
let c3 be Subset of c2;
func Lin c3 -> strict Subspace of a2 means :Def2: :: LMOD_5: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 LMOD_5:def 2 :
for b1 being domRing
for b2 being LeftMod 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 Th9: :: LMOD_5:9
for b1 being set
for b2 being domRing
for b3 being LeftMod 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 Th10: :: LMOD_5:10
for b1 being set
for b2 being domRing
for b3 being LeftMod of b2
for b4 being Subset of b3 holds
( b1 in b4 implies b1 in Lin b4 )
proof end;

theorem Th11: :: LMOD_5:11
for b1 being domRing
for b2 being LeftMod of b1 holds Lin ({} the carrier of b2) = (0). b2
proof end;

theorem Th12: :: LMOD_5:12
for b1 being domRing
for b2 being LeftMod of b1
for b3 being Subset of b2 holds
not ( Lin b3 = (0). b2 & not b3 = {} & not b3 = {(0. b2)} )
proof end;

theorem Th13: :: LMOD_5:13
for b1 being domRing
for b2 being LeftMod of b1
for b3 being Subset of b2
for b4 being strict Subspace of b2 holds
( 0. b1 <> 1. b1 & b3 = the carrier of b4 implies Lin b3 = b4 )
proof end;

theorem Th14: :: LMOD_5:14
for b1 being domRing
for b2 being strict LeftMod of b1
for b3 being Subset of b2 holds
( 0. b1 <> 1. b1 & b3 = the carrier of b2 implies Lin b3 = b2 )
proof end;

theorem Th15: :: LMOD_5:15
for b1 being domRing
for b2 being LeftMod of b1
for b3, b4 being Subset of b2 holds
( b3 c= b4 implies Lin b3 is Subspace of Lin b4 )
proof end;

theorem Th16: :: LMOD_5:16
for b1 being domRing
for b2 being strict LeftMod of b1
for b3, b4 being Subset of b2 holds
( Lin b3 = b2 & b3 c= b4 implies Lin b4 = b2 )
proof end;

theorem Th17: :: LMOD_5:17
for b1 being domRing
for b2 being LeftMod of b1
for b3, b4 being Subset of b2 holds Lin (b3 \/ b4) = (Lin b3) + (Lin b4)
proof end;

theorem Th18: :: LMOD_5:18
for b1 being domRing
for b2 being LeftMod of b1
for b3, b4 being Subset of b2 holds
Lin (b3 /\ b4) is Subspace of (Lin b3) /\ (Lin b4)
proof end;