:: MIDSP_2 semantic presentation
:: deftheorem Def1 defines Double MIDSP_2:def 1 :
:: deftheorem Def2 defines are_associated_wrp MIDSP_2:def 2 :
theorem Th1: :: MIDSP_2:1
definition
let c
1 be non
empty set ;
let c
2 be non
empty LoopStr ;
let c
3 be
Function of
[:c1,c1:],the
carrier of c
2;
pred c
3 is_atlas_of c
1,c
2 means :
Def3:
:: MIDSP_2:def 3
( ( for b
1 being
Element of a
1for b
2 being
Element of a
2 holds
ex b
3 being
Element of a
1 st a
3 . b
1,b
3 = b
2 ) & ( for b
1, b
2, b
3 being
Element of a
1 holds
( a
3 . b
1,b
2 = a
3 . b
1,b
3 implies b
2 = b
3 ) ) & ( for b
1, b
2, b
3 being
Element of a
1 holds
(a3 . b1,b2) + (a3 . b2,b3) = a
3 . b
1,b
3 ) );
end;
:: deftheorem Def3 defines is_atlas_of MIDSP_2:def 3 :
for b
1 being non
empty set for b
2 being non
empty LoopStr for b
3 being
Function of
[:b1,b1:],the
carrier of b
2 holds
( b
3 is_atlas_of b
1,b
2 iff ( ( for b
4 being
Element of b
1for b
5 being
Element of b
2 holds
ex b
6 being
Element of b
1 st b
3 . b
4,b
6 = b
5 ) & ( for b
4, b
5, b
6 being
Element of b
1 holds
( b
3 . b
4,b
5 = b
3 . b
4,b
6 implies b
5 = b
6 ) ) & ( for b
4, b
5, b
6 being
Element of b
1 holds
(b3 . b4,b5) + (b3 . b5,b6) = b
3 . b
4,b
6 ) ) );
definition
let c
1 be non
empty set ;
let c
2 be non
empty LoopStr ;
let c
3 be
Function of
[:c1,c1:],the
carrier of c
2;
let c
4 be
Element of c
1;
let c
5 be
Element of c
2;
assume E4:
c
3 is_atlas_of c
1,c
2
;
func c
4,c
5 . c
3 -> Element of a
1 means :
Def4:
:: MIDSP_2:def 4
a
3 . a
4,a
6 = a
5;
existence
ex b1 being Element of c1 st c3 . c4,b1 = c5
by E4, Def3;
uniqueness
for b1, b2 being Element of c1 holds
( c3 . c4,b1 = c5 & c3 . c4,b2 = c5 implies b1 = b2 )
by E4, Def3;
end;
:: deftheorem Def4 defines . MIDSP_2:def 4 :
theorem Th2: :: MIDSP_2:2
theorem Th3: :: MIDSP_2:3
canceled;
theorem Th4: :: MIDSP_2:4
theorem Th5: :: MIDSP_2:5
theorem Th6: :: MIDSP_2:6
theorem Th7: :: MIDSP_2:7
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of b
1for b
6 being non
empty add-associative right_zeroed right_complementable LoopStr for b
7 being
Function of
[:b1,b1:],the
carrier of b
6 holds
( b
7 is_atlas_of b
1,b
6 & b
7 . b
2,b
3 = b
7 . b
4,b
5 implies b
7 . b
3,b
2 = b
7 . b
5,b
4 )
theorem Th8: :: MIDSP_2:8
theorem Th9: :: MIDSP_2:9
theorem Th10: :: MIDSP_2:10
theorem Th11: :: MIDSP_2:11
theorem Th12: :: MIDSP_2:12
canceled;
theorem Th13: :: MIDSP_2:13
theorem Th14: :: MIDSP_2:14
theorem Th15: :: MIDSP_2:15
theorem Th16: :: MIDSP_2:16
theorem Th17: :: MIDSP_2:17
for b
1 being non
empty set for b
2 being non
empty Abelian add-associative right_zeroed right_complementable LoopStr for b
3 being
Function of
[:b1,b1:],the
carrier of b
2 holds
( b
3 is_atlas_of b
1,b
2 implies for b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
3 . b
4,b
5 = b
3 . b
5,b
7 & b
3 . b
4,b
6 = b
3 . b
6,b
8 implies b
3 . b
7,b
8 = Double (b3 . b5,b6) ) )
theorem Th18: :: MIDSP_2:18
Lemma20:
for b1 being MidSp holds
( ( for b2 being Element of (vectgroup b1) holds
ex b3 being Element of (vectgroup b1) st Double b3 = b2 ) & ( for b2 being Element of (vectgroup b1) holds
( Double b2 = 0. (vectgroup b1) implies b2 = 0. (vectgroup b1) ) ) )
:: deftheorem Def5 defines midpoint_operator MIDSP_2:def 5 :
theorem Th19: :: MIDSP_2:19
theorem Th20: :: MIDSP_2:20
:: deftheorem Def6 defines Half MIDSP_2:def 6 :
theorem Th21: :: MIDSP_2:21
theorem Th22: :: MIDSP_2:22
theorem Th23: :: MIDSP_2:23
:: deftheorem Def7 defines vector MIDSP_2:def 7 :
definition
let c
1 be
MidSp;
func vect c
1 -> Function of
[:the carrier of a1,the carrier of a1:],the
carrier of
(vectgroup a1) means :
Def8:
:: MIDSP_2:def 8
for b
1, b
2 being
Point of a
1 holds a
2 . b
1,b
2 = vect b
1,b
2;
existence
ex b1 being Function of [:the carrier of c1,the carrier of c1:],the carrier of (vectgroup c1) st
for b2, b3 being Point of c1 holds b1 . b2,b3 = vect b2,b3
uniqueness
for b1, b2 being Function of [:the carrier of c1,the carrier of c1:],the carrier of (vectgroup c1) holds
( ( for b3, b4 being Point of c1 holds b1 . b3,b4 = vect b3,b4 ) & ( for b3, b4 being Point of c1 holds b2 . b3,b4 = vect b3,b4 ) implies b1 = b2 )
end;
:: deftheorem Def8 defines vect MIDSP_2:def 8 :
theorem Th24: :: MIDSP_2:24
theorem Th25: :: MIDSP_2:25
for b
1 being
MidSpfor b
2, b
3, b
4, b
5 being
Point of b
1 holds
(
vect b
2,b
3 = vect b
4,b
5 iff b
2 @ b
5 = b
3 @ b
4 )
theorem Th26: :: MIDSP_2:26
theorem Th27: :: MIDSP_2:27
definition
let c
1 be non
empty set ;
let c
2 be non
empty Abelian add-associative right_zeroed right_complementable midpoint_operator LoopStr ;
let c
3 be
Function of
[:c1,c1:],the
carrier of c
2;
assume E32:
c
3 is_atlas_of c
1,c
2
;
func @ c
3 -> BinOp of a
1 means :
Def9:
:: MIDSP_2:def 9
for b
1, b
2 being
Element of a
1 holds a
3 . b
1,
(a4 . b1,b2) = a
3 . (a4 . b1,b2),b
2;
existence
ex b1 being BinOp of c1 st
for b2, b3 being Element of c1 holds c3 . b2,(b1 . b2,b3) = c3 . (b1 . b2,b3),b3
uniqueness
for b1, b2 being BinOp of c1 holds
( ( for b3, b4 being Element of c1 holds c3 . b3,(b1 . b3,b4) = c3 . (b1 . b3,b4),b4 ) & ( for b3, b4 being Element of c1 holds c3 . b3,(b2 . b3,b4) = c3 . (b2 . b3,b4),b4 ) implies b1 = b2 )
end;
:: deftheorem Def9 defines @ MIDSP_2:def 9 :
theorem Th28: :: MIDSP_2:28
definition
let c
1 be non
empty set ;
let c
2 be non
empty Abelian add-associative right_zeroed right_complementable midpoint_operator LoopStr ;
let c
3 be
Function of
[:c1,c1:],the
carrier of c
2;
func Atlas c
3 -> Function of
[:the carrier of MidStr(# a1,(@ a3) #),the carrier of MidStr(# a1,(@ a3) #):],the
carrier of a
2 equals :: MIDSP_2:def 10
a
3;
coherence
c3 is Function of [:the carrier of MidStr(# c1,(@ c3) #),the carrier of MidStr(# c1,(@ c3) #):],the carrier of c2
;
end;
:: deftheorem Def10 defines Atlas MIDSP_2:def 10 :
Lemma34:
for b1 being non empty set
for b2 being non empty Abelian add-associative right_zeroed right_complementable midpoint_operator LoopStr
for b3 being Function of [:b1,b1:],the carrier of b2 holds
( b3 is_atlas_of b1,b2 implies for b4, b5, b6 being Point of MidStr(# b1,(@ b3) #) holds
( b4 @ b5 = b6 iff (Atlas b3) . b4,b6 = (Atlas b3) . b6,b5 ) )
theorem Th29: :: MIDSP_2:29
canceled;
theorem Th30: :: MIDSP_2:30
canceled;
theorem Th31: :: MIDSP_2:31
canceled;
theorem Th32: :: MIDSP_2:32
:: deftheorem Def11 defines MidSp. MIDSP_2:def 11 :
theorem Th33: :: MIDSP_2:33
:: deftheorem Def12 defines ATLAS-like MIDSP_2:def 12 :
:: deftheorem Def13 defines . MIDSP_2:def 13 :
:: deftheorem Def14 defines . MIDSP_2:def 14 :
:: deftheorem Def15 defines 0. MIDSP_2:def 15 :
theorem Th34: :: MIDSP_2:34
for b
1 being non
empty Abelian add-associative right_zeroed right_complementable midpoint_operator LoopStr for b
2 being non
empty MidStr for b
3 being
Function of
[:the carrier of b2,the carrier of b2:],the
carrier of b
1for b
4, b
5, b
6, b
7 being
Point of b
2 holds
( b
3 is_atlas_of the
carrier of b
2,b
1 & b
2,b
1 are_associated_wrp b
3 implies ( b
4 @ b
5 = b
6 @ b
7 iff b
3 . b
4,b
5 = (b3 . b4,b6) + (b3 . b4,b7) ) )
theorem Th35: :: MIDSP_2:35
theorem Th36: :: MIDSP_2:36
for b
1 being
MidSpfor b
2 being
ATLAS of b
1for b
3, b
4, b
5, b
6 being
Point of b
1 holds
( b
3 @ b
4 = b
5 @ b
6 iff b
2 . b
3,b
4 = (b2 . b3,b5) + (b2 . b3,b6) )
theorem Th37: :: MIDSP_2:37
for b
1 being
MidSpfor b
2 being
ATLAS of b
1for b
3, b
4, b
5 being
Point of b
1 holds
( b
3 @ b
4 = b
5 iff b
2 . b
3,b
4 = Double (b2 . b3,b5) )
theorem Th38: :: MIDSP_2:38
for b
1 being
MidSpfor b
2 being
ATLAS of b
1 holds
( ( for b
3 being
Point of b
1for b
4 being
Vector of b
2 holds
ex b
5 being
Point of b
1 st b
2 . b
3,b
5 = b
4 ) & ( for b
3, b
4, b
5 being
Point of b
1 holds
( b
2 . b
3,b
4 = b
2 . b
3,b
5 implies b
4 = b
5 ) ) & ( for b
3, b
4, b
5 being
Point of b
1 holds
(b2 . b3,b4) + (b2 . b4,b5) = b
2 . b
3,b
5 ) )
theorem Th39: :: MIDSP_2:39
for b
1 being
MidSpfor b
2 being
ATLAS of b
1for b
3, b
4, b
5, b
6 being
Point of b
1for b
7 being
Vector of b
2 holds
( b
2 . b
3,b
3 = 0. b
2 & ( b
2 . b
3,b
4 = 0. b
2 implies b
3 = b
4 ) & b
2 . b
3,b
4 = - (b2 . b4,b3) & ( b
2 . b
3,b
4 = b
2 . b
5,b
6 implies b
2 . b
4,b
3 = b
2 . b
6,b
5 ) & ( for b
8 being
Point of b
1for b
9 being
Vector of b
2 holds
ex b
10 being
Point of b
1 st b
2 . b
10,b
8 = b
9 ) & ( b
2 . b
4,b
3 = b
2 . b
5,b
3 implies b
4 = b
5 ) & ( b
3 @ b
4 = b
5 implies b
2 . b
3,b
5 = b
2 . b
5,b
4 ) & ( b
2 . b
3,b
5 = b
2 . b
5,b
4 implies b
3 @ b
4 = b
5 ) & ( b
3 @ b
4 = b
5 @ b
6 implies b
2 . b
3,b
6 = b
2 . b
5,b
4 ) & ( b
2 . b
3,b
6 = b
2 . b
5,b
4 implies b
3 @ b
4 = b
5 @ b
6 ) & ( b
2 . b
3,b
4 = b
7 implies b
3,b
7 . b
2 = b
4 ) & ( b
3,b
7 . b
2 = b
4 implies b
2 . b
3,b
4 = b
7 ) )
theorem Th40: :: MIDSP_2:40