:: VECTMETR semantic presentation
:: deftheorem Def1 defines convex VECTMETR:def 1 :
:: deftheorem Def2 defines internal VECTMETR:def 2 :
theorem Th1: :: VECTMETR:1
:: deftheorem Def3 defines isometric VECTMETR:def 3 :
:: deftheorem Def4 defines ISOM VECTMETR:def 4 :
theorem Th2: :: VECTMETR:2
theorem Th3: :: VECTMETR:3
theorem Th4: :: VECTMETR:4
theorem Th5: :: VECTMETR:5
registration
let c
1 be non
empty set ;
let c
2 be
Function of
[:c1,c1:],
REAL ;
let c
3 be
Element of c
1;
let c
4 be
BinOp of c
1;
let c
5 be
Function of
[:REAL ,c1:],c
1;
cluster RLSMetrStruct(# a
1,a
2,a
3,a
4,a
5 #)
-> non
empty ;
coherence
not RLSMetrStruct(# c1,c2,c3,c4,c5 #) is empty
by STRUCT_0:def 1;
end;
:: deftheorem Def5 defines homogeneous VECTMETR:def 5 :
:: deftheorem Def6 defines translatible VECTMETR:def 6 :
:: deftheorem Def7 defines Norm VECTMETR:def 7 :
theorem Th6: :: VECTMETR:6
theorem Th7: :: VECTMETR:7
theorem Th8: :: VECTMETR:8
definition
let c
1 be
Nat;
func RLMSpace c
1 -> strict RealLinearMetrSpace means :
Def8:
:: VECTMETR:def 8
( the
carrier of a
2 = REAL a
1 & the
distance of a
2 = Pitag_dist a
1 & the
Zero of a
2 = 0* a
1 & ( for b
1, b
2 being
Element of
REAL a
1 holds the
add of a
2 . b
1,b
2 = b
1 + b
2 ) & ( for b
1 being
Element of
REAL a
1for b
2 being
Element of
REAL holds the
Mult of a
2 . b
2,b
1 = b
2 * b
1 ) );
existence
ex b1 being strict RealLinearMetrSpace st
( the carrier of b1 = REAL c1 & the distance of b1 = Pitag_dist c1 & the Zero of b1 = 0* c1 & ( for b2, b3 being Element of REAL c1 holds the add of b1 . b2,b3 = b2 + b3 ) & ( for b2 being Element of REAL c1
for b3 being Element of REAL holds the Mult of b1 . b3,b2 = b3 * b2 ) )
uniqueness
for b1, b2 being strict RealLinearMetrSpace holds
( the carrier of b1 = REAL c1 & the distance of b1 = Pitag_dist c1 & the Zero of b1 = 0* c1 & ( for b3, b4 being Element of REAL c1 holds the add of b1 . b3,b4 = b3 + b4 ) & ( for b3 being Element of REAL c1
for b4 being Element of REAL holds the Mult of b1 . b4,b3 = b4 * b3 ) & the carrier of b2 = REAL c1 & the distance of b2 = Pitag_dist c1 & the Zero of b2 = 0* c1 & ( for b3, b4 being Element of REAL c1 holds the add of b2 . b3,b4 = b3 + b4 ) & ( for b3 being Element of REAL c1
for b4 being Element of REAL holds the Mult of b2 . b4,b3 = b4 * b3 ) implies b1 = b2 )
end;
:: deftheorem Def8 defines RLMSpace VECTMETR:def 8 :
theorem Th9: :: VECTMETR:9
:: deftheorem Def9 defines IsomGroup VECTMETR:def 9 :
theorem Th10: :: VECTMETR:10
theorem Th11: :: VECTMETR:11
:: deftheorem Def10 defines SubIsomGroupRel VECTMETR:def 10 :