:: MOD_3 semantic presentation

Lemma1: for b1 being Ring
for b2 being Scalar of b1 holds
( - b2 = 0. b1 implies b2 = 0. b1 )
proof end;

theorem Th1: :: MOD_3:1
canceled;

theorem Th2: :: MOD_3:2
for b1 being non empty add-associative right_zeroed right_complementable non degenerated doubleLoopStr holds
0. b1 <> - (1. b1)
proof end;

theorem Th3: :: MOD_3:3
canceled;

theorem Th4: :: MOD_3:4
canceled;

theorem Th5: :: MOD_3:5
canceled;

theorem Th6: :: MOD_3:6
for b1 being Ring
for b2 being LeftMod of b1
for b3 being Linear_Combination of b2
for b4 being finite Subset of b2 holds
not ( Carrier b3 c= b4 & ( for b5 being FinSequence of the carrier of b2 holds
not ( b5 is one-to-one & rng b5 = b4 & Sum b3 = Sum (b3 (#) b5) ) ) )
proof end;

theorem Th7: :: MOD_3:7
for b1 being Ring
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 Ring;
let c2 be LeftMod of c1;
let c3 be Subset of c2;
func Lin c3 -> strict Subspace of a2 means :Def1: :: MOD_3:def 1
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 Def1 defines Lin MOD_3:def 1 :
for b1 being Ring
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 Th8: :: MOD_3:8
canceled;

theorem Th9: :: MOD_3:9
canceled;

theorem Th10: :: MOD_3:10
canceled;

theorem Th11: :: MOD_3:11
for b1 being set
for b2 being Ring
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 Th12: :: MOD_3:12
for b1 being set
for b2 being Ring
for b3 being LeftMod of b2
for b4 being Subset of b3 holds
( b1 in b4 implies b1 in Lin b4 )
proof end;

theorem Th13: :: MOD_3:13
for b1 being Ring
for b2 being LeftMod of b1 holds Lin ({} the carrier of b2) = (0). b2
proof end;

theorem Th14: :: MOD_3:14
for b1 being Ring
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 Th15: :: MOD_3:15
for b1 being Ring
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 Th16: :: MOD_3:16
for b1 being Ring
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 Th17: :: MOD_3:17
for b1 being Ring
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 Th18: :: MOD_3:18
for b1 being Ring
for b2 being LeftMod of b1
for b3, b4 being Subset of b2 holds
( Lin b3 = b2 & b3 c= b4 implies Lin b4 = b2 )
proof end;

theorem Th19: :: MOD_3:19
for b1 being Ring
for b2 being LeftMod of b1
for b3, b4 being Subset of b2 holds Lin (b3 \/ b4) = (Lin b3) + (Lin b4)
proof end;

theorem Th20: :: MOD_3:20
for b1 being Ring
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;

definition
let c1 be Ring;
let c2 be LeftMod of c1;
let c3 be Subset of c2;
attr a3 is base means :Def2: :: MOD_3:def 2
( a3 is linearly-independent & Lin a3 = VectSpStr(# the carrier of a2,the add of a2,the Zero of a2,the lmult of a2 #) );
end;

:: deftheorem Def2 defines base MOD_3:def 2 :
for b1 being Ring
for b2 being LeftMod of b1
for b3 being Subset of b2 holds
( b3 is base iff ( b3 is linearly-independent & Lin b3 = VectSpStr(# the carrier of b2,the add of b2,the Zero of b2,the lmult of b2 #) ) );

definition
let c1 be Ring;
let c2 be LeftMod of c1;
attr a2 is free means :Def3: :: MOD_3:def 3
ex b1 being Subset of a2 st b1 is base;
end;

:: deftheorem Def3 defines free MOD_3:def 3 :
for b1 being Ring
for b2 being LeftMod of b1 holds
( b2 is free iff ex b3 being Subset of b2 st b3 is base );

theorem Th21: :: MOD_3:21
for b1 being Ring
for b2 being LeftMod of b1 holds (0). b2 is free
proof end;

registration
let c1 be Ring;
cluster strict free VectSpStr of a1;
existence
ex b1 being LeftMod of c1 st
( b1 is strict & b1 is free )
proof end;
end;

Lemma14: for b1 being Skew-Field
for b2 being Scalar of b1
for b3 being LeftMod of b1
for b4 being Vector of b3 holds
( b2 <> 0. b1 implies ( (b2 " ) * (b2 * b4) = (1. b1) * b4 & ((b2 " ) * b2) * b4 = (1. b1) * b4 ) )
proof end;

theorem Th22: :: MOD_3:22
canceled;

theorem Th23: :: MOD_3:23
for b1 being Skew-Field
for b2 being LeftMod of b1
for b3 being Vector of b2 holds
( {b3} is linearly-independent iff b3 <> 0. b2 )
proof end;

theorem Th24: :: MOD_3:24
for b1 being Skew-Field
for b2 being LeftMod of b1
for b3, b4 being Vector of b2 holds
( ( b3 <> b4 & {b3,b4} is linearly-independent implies ( b4 <> 0. b2 & ( for b5 being Scalar of b1 holds
b3 <> b5 * b4 ) ) ) & ( b4 <> 0. b2 & ( for b5 being Scalar of b1 holds
b3 <> b5 * b4 ) implies ( b3 <> b4 & {b3,b4} is linearly-independent ) ) )
proof end;

theorem Th25: :: MOD_3:25
for b1 being Skew-Field
for b2 being LeftMod of b1
for b3, b4 being Vector of b2 holds
( ( b3 <> b4 & {b3,b4} is linearly-independent ) iff for b5, b6 being Scalar of b1 holds
( (b5 * b3) + (b6 * b4) = 0. b2 implies ( b5 = 0. b1 & b6 = 0. b1 ) ) )
proof end;

theorem Th26: :: MOD_3:26
for b1 being Skew-Field
for b2 being LeftMod 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 base ) ) )
proof end;

theorem Th27: :: MOD_3:27
for b1 being Skew-Field
for b2 being LeftMod 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 base ) ) )
proof end;

Lemma18: for b1 being Skew-Field
for b2 being LeftMod of b1 holds
ex b3 being Subset of b2 st b3 is base
proof end;

theorem Th28: :: MOD_3:28
for b1 being Skew-Field
for b2 being LeftMod of b1 holds b2 is free
proof end;

definition
let c1 be Skew-Field;
let c2 be LeftMod of c1;
canceled;
mode Basis of c2 -> Subset of a2 means :Def5: :: MOD_3:def 5
a3 is base;
existence
ex b1 being Subset of c2 st b1 is base
by Lemma18;
end;

:: deftheorem Def4 MOD_3:def 4 :
canceled;

:: deftheorem Def5 defines Basis MOD_3:def 5 :
for b1 being Skew-Field
for b2 being LeftMod of b1
for b3 being Subset of b2 holds
( b3 is Basis of b2 iff b3 is base );

theorem Th29: :: MOD_3:29
for b1 being Skew-Field
for b2 being LeftMod 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 Th30: :: MOD_3:30
for b1 being Skew-Field
for b2 being LeftMod 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;