:: VECTSP10 semantic presentation
:: deftheorem Def1 defines StructVectSp VECTSP10:def 1 :
theorem Th1: :: VECTSP10:1
canceled;
theorem Th2: :: VECTSP10:2
theorem Th3: :: VECTSP10:3
theorem Th4: :: VECTSP10:4
theorem Th5: :: VECTSP10:5
theorem Th6: :: VECTSP10:6
theorem Th7: :: VECTSP10:7
theorem Th8: :: VECTSP10:8
theorem Th9: :: VECTSP10:9
theorem Th10: :: VECTSP10:10
theorem Th11: :: VECTSP10:11
theorem Th12: :: VECTSP10:12
theorem Th13: :: VECTSP10:13
theorem Th14: :: VECTSP10:14
theorem Th15: :: VECTSP10:15
theorem Th16: :: VECTSP10:16
theorem Th17: :: VECTSP10:17
theorem Th18: :: VECTSP10:18
theorem Th19: :: VECTSP10:19
theorem Th20: :: VECTSP10:20
theorem Th21: :: VECTSP10:21
:: deftheorem Def2 defines CosetSet VECTSP10:def 2 :
definition
let c
1 be non
empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let c
2 be
VectSp of c
1;
let c
3 be
Subspace of c
2;
func addCoset c
2,c
3 -> BinOp of
CosetSet a
2,a
3 means :
Def3:
:: VECTSP10:def 3
for b
1, b
2 being
Element of
CosetSet a
2,a
3for b
3, b
4 being
Vector of a
2 holds
( b
1 = b
3 + a
3 & b
2 = b
4 + a
3 implies a
4 . b
1,b
2 = (b3 + b4) + a
3 );
existence
ex b1 being BinOp of CosetSet c2,c3 st
for b2, b3 being Element of CosetSet c2,c3
for b4, b5 being Vector of c2 holds
( b2 = b4 + c3 & b3 = b5 + c3 implies b1 . b2,b3 = (b4 + b5) + c3 )
uniqueness
for b1, b2 being BinOp of CosetSet c2,c3 holds
( ( for b3, b4 being Element of CosetSet c2,c3
for b5, b6 being Vector of c2 holds
( b3 = b5 + c3 & b4 = b6 + c3 implies b1 . b3,b4 = (b5 + b6) + c3 ) ) & ( for b3, b4 being Element of CosetSet c2,c3
for b5, b6 being Vector of c2 holds
( b3 = b5 + c3 & b4 = b6 + c3 implies b2 . b3,b4 = (b5 + b6) + c3 ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines addCoset VECTSP10:def 3 :
:: deftheorem Def4 defines zeroCoset VECTSP10:def 4 :
definition
let c
1 be non
empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let c
2 be
VectSp of c
1;
let c
3 be
Subspace of c
2;
func lmultCoset c
2,c
3 -> Function of
[:the carrier of a1,(CosetSet a2,a3):],
CosetSet a
2,a
3 means :
Def5:
:: VECTSP10:def 5
for b
1 being
Element of a
1for b
2 being
Element of
CosetSet a
2,a
3for b
3 being
Vector of a
2 holds
( b
2 = b
3 + a
3 implies a
4 . b
1,b
2 = (b1 * b3) + a
3 );
existence
ex b1 being Function of [:the carrier of c1,(CosetSet c2,c3):], CosetSet c2,c3 st
for b2 being Element of c1
for b3 being Element of CosetSet c2,c3
for b4 being Vector of c2 holds
( b3 = b4 + c3 implies b1 . b2,b3 = (b2 * b4) + c3 )
uniqueness
for b1, b2 being Function of [:the carrier of c1,(CosetSet c2,c3):], CosetSet c2,c3 holds
( ( for b3 being Element of c1
for b4 being Element of CosetSet c2,c3
for b5 being Vector of c2 holds
( b4 = b5 + c3 implies b1 . b3,b4 = (b3 * b5) + c3 ) ) & ( for b3 being Element of c1
for b4 being Element of CosetSet c2,c3
for b5 being Vector of c2 holds
( b4 = b5 + c3 implies b2 . b3,b4 = (b3 * b5) + c3 ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines lmultCoset VECTSP10:def 5 :
definition
let c
1 be non
empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let c
2 be
VectSp of c
1;
let c
3 be
Subspace of c
2;
func VectQuot c
2,c
3 -> non
empty Abelian add-associative right_zeroed right_complementable strict VectSp-like VectSpStr of a
1 means :
Def6:
:: VECTSP10:def 6
( the
carrier of a
4 = CosetSet a
2,a
3 & the
add of a
4 = addCoset a
2,a
3 & the
Zero of a
4 = zeroCoset a
2,a
3 & the
lmult of a
4 = lmultCoset a
2,a
3 );
existence
ex b1 being non empty Abelian add-associative right_zeroed right_complementable strict VectSp-like VectSpStr of c1 st
( the carrier of b1 = CosetSet c2,c3 & the add of b1 = addCoset c2,c3 & the Zero of b1 = zeroCoset c2,c3 & the lmult of b1 = lmultCoset c2,c3 )
uniqueness
for b1, b2 being non empty Abelian add-associative right_zeroed right_complementable strict VectSp-like VectSpStr of c1 holds
( the carrier of b1 = CosetSet c2,c3 & the add of b1 = addCoset c2,c3 & the Zero of b1 = zeroCoset c2,c3 & the lmult of b1 = lmultCoset c2,c3 & the carrier of b2 = CosetSet c2,c3 & the add of b2 = addCoset c2,c3 & the Zero of b2 = zeroCoset c2,c3 & the lmult of b2 = lmultCoset c2,c3 implies b1 = b2 )
;
end;
:: deftheorem Def6 defines VectQuot VECTSP10:def 6 :
theorem Th22: :: VECTSP10:22
theorem Th23: :: VECTSP10:23
theorem Th24: :: VECTSP10:24
theorem Th25: :: VECTSP10:25
theorem Th26: :: VECTSP10:26
theorem Th27: :: VECTSP10:27
theorem Th28: :: VECTSP10:28
:: deftheorem Def7 defines constant VECTSP10:def 7 :
:: deftheorem Def8 defines coeffFunctional VECTSP10:def 8 :
theorem Th29: :: VECTSP10:29
theorem Th30: :: VECTSP10:30
theorem Th31: :: VECTSP10:31
theorem Th32: :: VECTSP10:32
theorem Th33: :: VECTSP10:33
:: deftheorem Def9 defines ker VECTSP10:def 9 :
theorem Th34: :: VECTSP10:34
:: deftheorem Def10 defines degenerated VECTSP10:def 10 :
:: deftheorem Def11 defines Ker VECTSP10:def 11 :
:: deftheorem Def12 defines QFunctional VECTSP10:def 12 :
theorem Th35: :: VECTSP10:35
:: deftheorem Def13 defines CQFunctional VECTSP10:def 13 :
theorem Th36: :: VECTSP10:36