:: BHSP_5 semantic presentation
theorem Th1: :: BHSP_5:1
:: deftheorem Def1 defines ++ BHSP_5:def 1 :
:: deftheorem Def2 defines setop_SUM BHSP_5:def 2 :
:: deftheorem Def3 defines PO BHSP_5:def 3 :
:: deftheorem Def4 defines Func_Seq BHSP_5:def 4 :
definition
let c
1, c
2 be non
empty set ;
let c
3 be
BinOp of c
1;
assume E2:
( c
3 is
commutative & c
3 is
associative & c
3 has_a_unity )
;
let c
4 be
finite Subset of c
2;
let c
5 be
Function of c
2,c
1;
assume E3:
c
4 c= dom c
5
;
func setopfunc c
4,c
2,c
1,c
5,c
3 -> Element of a
1 means :
Def5:
:: BHSP_5:def 5
ex b
1 being
FinSequence of a
2 st
( b
1 is
one-to-one &
rng b
1 = a
4 & a
6 = a
3 "**" (Func_Seq a5,b1) );
existence
ex b1 being Element of c1ex b2 being FinSequence of c2 st
( b2 is one-to-one & rng b2 = c4 & b1 = c3 "**" (Func_Seq c5,b2) )
uniqueness
for b1, b2 being Element of c1 holds
( ex b3 being FinSequence of c2 st
( b3 is one-to-one & rng b3 = c4 & b1 = c3 "**" (Func_Seq c5,b3) ) & ex b3 being FinSequence of c2 st
( b3 is one-to-one & rng b3 = c4 & b2 = c3 "**" (Func_Seq c5,b3) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines setopfunc BHSP_5:def 5 :
definition
let c
1 be
RealUnitarySpace;
let c
2 be
Point of c
1;
let c
3 be
finite Subset of c
1;
func setop_xPre_PROD c
2,c
3,c
1 -> Real means :: BHSP_5:def 6
ex b
1 being
FinSequence of the
carrier of a
1 st
( b
1 is
one-to-one &
rng b
1 = a
3 & ex b
2 being
FinSequence of
REAL st
(
dom b
2 = dom b
1 & ( for b
3 being
Nat holds
( b
3 in dom b
2 implies b
2 . b
3 = PO b
3,b
1,a
2 ) ) & a
4 = addreal "**" b
2 ) );
existence
ex b1 being Realex b2 being FinSequence of the carrier of c1 st
( b2 is one-to-one & rng b2 = c3 & ex b3 being FinSequence of REAL st
( dom b3 = dom b2 & ( for b4 being Nat holds
( b4 in dom b3 implies b3 . b4 = PO b4,b2,c2 ) ) & b1 = addreal "**" b3 ) )
uniqueness
for b1, b2 being Real holds
( ex b3 being FinSequence of the carrier of c1 st
( b3 is one-to-one & rng b3 = c3 & ex b4 being FinSequence of REAL st
( dom b4 = dom b3 & ( for b5 being Nat holds
( b5 in dom b4 implies b4 . b5 = PO b5,b3,c2 ) ) & b1 = addreal "**" b4 ) ) & ex b3 being FinSequence of the carrier of c1 st
( b3 is one-to-one & rng b3 = c3 & ex b4 being FinSequence of REAL st
( dom b4 = dom b3 & ( for b5 being Nat holds
( b5 in dom b4 implies b4 . b5 = PO b5,b3,c2 ) ) & b2 = addreal "**" b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines setop_xPre_PROD BHSP_5:def 6 :
:: deftheorem Def7 defines setop_xPROD BHSP_5:def 7 :
:: deftheorem Def8 defines OrthogonalFamily BHSP_5:def 8 :
theorem Th2: :: BHSP_5:2
:: deftheorem Def9 defines OrthonormalFamily BHSP_5:def 9 :
theorem Th3: :: BHSP_5:3
theorem Th4: :: BHSP_5:4
theorem Th5: :: BHSP_5:5
theorem Th6: :: BHSP_5:6
theorem Th7: :: BHSP_5:7
theorem Th8: :: BHSP_5:8
theorem Th9: :: BHSP_5:9
theorem Th10: :: BHSP_5:10
theorem Th11: :: BHSP_5:11
for b
1 being
RealUnitarySpace holds
( the
add of b
1 is
commutative & the
add of b
1 is
associative & the
add of b
1 has_a_unity implies for b
2 being
Point of b
1for b
3 being
finite OrthonormalFamily of b
1 holds
( not b
3 is
empty implies for b
4 being
Function of the
carrier of b
1,
REAL holds
( b
3 c= dom b
4 & ( for b
5 being
Point of b
1 holds
( b
5 in b
3 implies b
4 . b
5 = (b2 .|. b5) ^2 ) ) implies for b
5 being
Function of the
carrier of b
1,the
carrier of b
1 holds
( b
3 c= dom b
5 & ( for b
6 being
Point of b
1 holds
( b
6 in b
3 implies b
5 . b
6 = (b2 .|. b6) * b
6 ) ) implies b
2 .|. (setopfunc b3,the carrier of b1,the carrier of b1,b5,the add of b1) = setopfunc b
3,the
carrier of b
1,
REAL ,b
4,
addreal ) ) ) )
theorem Th12: :: BHSP_5:12
for b
1 being
RealUnitarySpace holds
( the
add of b
1 is
commutative & the
add of b
1 is
associative & the
add of b
1 has_a_unity implies for b
2 being
Point of b
1for b
3 being
finite OrthonormalFamily of b
1 holds
( not b
3 is
empty implies for b
4 being
Function of the
carrier of b
1,the
carrier of b
1 holds
( b
3 c= dom b
4 & ( for b
5 being
Point of b
1 holds
( b
5 in b
3 implies b
4 . b
5 = (b2 .|. b5) * b
5 ) ) implies for b
5 being
Function of the
carrier of b
1,
REAL holds
( b
3 c= dom b
5 & ( for b
6 being
Point of b
1 holds
( b
6 in b
3 implies b
5 . b
6 = (b2 .|. b6) ^2 ) ) implies
(setopfunc b3,the carrier of b1,the carrier of b1,b4,the add of b1) .|. (setopfunc b3,the carrier of b1,the carrier of b1,b4,the add of b1) = setopfunc b
3,the
carrier of b
1,
REAL ,b
5,
addreal ) ) ) )
theorem Th13: :: BHSP_5:13
theorem Th14: :: BHSP_5:14
for b
1, b
2 being non
empty set for b
3 being
BinOp of b
1 holds
( b
3 is
commutative & b
3 is
associative & b
3 has_a_unity implies for b
4, b
5 being
finite Subset of b
2 holds
( b
4 misses b
5 implies for b
6 being
Function of b
2,b
1 holds
( b
4 c= dom b
6 & b
5 c= dom b
6 implies for b
7 being
finite Subset of b
2 holds
( b
7 = b
4 \/ b
5 implies
setopfunc b
7,b
2,b
1,b
6,b
3 = b
3 . (setopfunc b4,b2,b1,b6,b3),
(setopfunc b5,b2,b1,b6,b3) ) ) ) )