:: Directed Geometrical Bundles and Their Analytical Representation
:: by Grzegorz Lewandowski, Krzysztof Pra\.zmowski and Bo\.zena Lewandowska
::
:: Received September 24, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let IT be non empty AffinStruct ;

attr IT is WeakAffVect-like means :Def1: :: AFVECT0:def 1
( ( for a, b, c being Element of IT st a,b // c,c holds
a = b ) & ( for a, b, c, d, p, q being Element of IT st a,b // p,q & c,d // p,q holds
a,b // c,d ) & ( for a, b, c being Element of IT ex d being Element of IT st a,b // c,d ) & ( for a, b, c, a9, b9, c9 being Element of IT st a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9 ) & ( for a, c being Element of IT ex b being Element of IT st a,b // b,c ) & ( for a, b, c, d being Element of IT st a,b // c,d holds
a,c // b,d ) );

end;

:: deftheorem Def1 defines WeakAffVect-like AFVECT0:def 1 :
for IT being non empty AffinStruct holds
( IT is WeakAffVect-like iff ( ( for a, b, c being Element of IT st a,b // c,c holds
a = b ) & ( for a, b, c, d, p, q being Element of IT st a,b // p,q & c,d // p,q holds
a,b // c,d ) & ( for a, b, c being Element of IT ex d being Element of IT st a,b // c,d ) & ( for a, b, c, a9, b9, c9 being Element of IT st a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9 ) & ( for a, c being Element of IT ex b being Element of IT st a,b // b,c ) & ( for a, b, c, d being Element of IT st a,b // c,d holds
a,c // b,d ) ) );

registration

cluster non empty non trivial strict WeakAffVect-like for AffinStruct ;

existence
ex b₁ being non trivial AffinStruct st
( b₁ is strict & b₁ is WeakAffVect-like )

proof end;

end;

definition

mode WeakAffVect is non trivial WeakAffVect-like AffinStruct ;

end;

registration

cluster non empty AffVect-like -> non empty WeakAffVect-like for AffinStruct ;

coherence
for b₁ being non empty AffinStruct st b₁ is AffVect-like holds
b₁ is WeakAffVect-like by TDGROUP:def 5;

end;

::
:: Properties of Relation of Congruence of Vectors
::

theorem Th1: :: AFVECT0:1

for AFV being WeakAffVect
for a, b being Element of AFV holds a,b // a,b

proof end;

theorem :: AFVECT0:2

for AFV being WeakAffVect
for a being Element of AFV holds a,a // a,a by Th1;

theorem Th3: :: AFVECT0:3

for AFV being WeakAffVect
for a, b, c, d being Element of AFV st a,b // c,d holds
c,d // a,b

proof end;

theorem Th4: :: AFVECT0:4

for AFV being WeakAffVect
for a, b, c being Element of AFV st a,b // a,c holds
b = c

proof end;

theorem Th5: :: AFVECT0:5

for AFV being WeakAffVect
for a, b, c, d, d9 being Element of AFV st a,b // c,d & a,b // c,d9 holds
d = d9

proof end;

theorem Th6: :: AFVECT0:6

for AFV being WeakAffVect
for a, b being Element of AFV holds a,a // b,b

proof end;

theorem Th7: :: AFVECT0:7

for AFV being WeakAffVect
for a, b, c, d being Element of AFV st a,b // c,d holds
b,a // d,c

proof end;

theorem :: AFVECT0:8

for AFV being WeakAffVect
for a, b, c, d, b9 being Element of AFV st a,b // c,d & a,c // b9,d holds
b = b9

proof end;

theorem :: AFVECT0:9

for AFV being WeakAffVect
for a, b, c, d, b9, c9, d9 being Element of AFV st b,c // b9,c9 & a,d // b,c & a,d9 // b9,c9 holds
d = d9

proof end;

theorem :: AFVECT0:10

for AFV being WeakAffVect
for a, b, c, d, a9, b9, d9 being Element of AFV st a,b // a9,b9 & c,d // b,a & c,d9 // b9,a9 holds
d = d9

proof end;

theorem :: AFVECT0:11

for AFV being WeakAffVect
for a, b, c, d, f, a9, b9, c9, d9, f9 being Element of AFV st a,b // a9,b9 & c,d // c9,d9 & b,f // c,d & b9,f9 // c9,d9 holds
a,f // a9,f9

proof end;

theorem Th12: :: AFVECT0:12

for AFV being WeakAffVect
for a, b, c, a9, b9, c9 being Element of AFV st a,b // a9,b9 & a,c // c9,b9 holds
b,c // c9,a9

proof end;

::
:: Relation of Maximal Distance
::

definition

let AFV be WeakAffVect;
let a, b be Element of AFV;

pred MDist a,b means :: AFVECT0:def 2
( a,b // b,a & a <> b );

irreflexivity
for a being Element of AFV holds
( not a,a // a,a or not a <> a ) ;
symmetry
for a, b being Element of AFV st a,b // b,a & a <> b holds
( b,a // a,b & b <> a ) by Th3;

end;

:: deftheorem defines MDist AFVECT0:def 2 :
for AFV being WeakAffVect
for a, b being Element of AFV holds
( MDist a,b iff ( a,b // b,a & a <> b ) );

theorem :: AFVECT0:13

for AFV being WeakAffVect ex a, b being Element of AFV st
( a <> b & not MDist a,b )

proof end;

theorem :: AFVECT0:14

for AFV being WeakAffVect
for a, b, c being Element of AFV st MDist a,b & MDist a,c & not b = c holds
MDist b,c

proof end;

theorem :: AFVECT0:15

for AFV being WeakAffVect
for a, b, c, d being Element of AFV st MDist a,b & a,b // c,d holds
MDist c,d

proof end;

::
:: Midpoint Relation
::

definition

let AFV be WeakAffVect;
let a, b, c be Element of AFV;

pred Mid a,b,c means :Def3: :: AFVECT0:def 3
a,b // b,c;

end;

:: deftheorem Def3 defines Mid AFVECT0:def 3 :
for AFV being WeakAffVect
for a, b, c being Element of AFV holds
( Mid a,b,c iff a,b // b,c );

theorem Th16: :: AFVECT0:16

for AFV being WeakAffVect
for a, b, c being Element of AFV st Mid a,b,c holds
Mid c,b,a

proof end;

theorem :: AFVECT0:17

for AFV being WeakAffVect
for a, b being Element of AFV holds
( Mid a,b,b iff a = b ) by Def1, Th6;

theorem Th18: :: AFVECT0:18

for AFV being WeakAffVect
for a, b being Element of AFV holds
( Mid a,b,a iff ( a = b or MDist a,b ) ) by Th6;

theorem Th19: :: AFVECT0:19

for AFV being WeakAffVect
for a, c being Element of AFV ex b being Element of AFV st Mid a,b,c

proof end;

theorem Th20: :: AFVECT0:20

for AFV being WeakAffVect
for a, b, c, b9 being Element of AFV st Mid a,b,c & Mid a,b9,c & not b = b9 holds
MDist b,b9

proof end;

theorem Th21: :: AFVECT0:21

for AFV being WeakAffVect
for a, b being Element of AFV ex c being Element of AFV st Mid a,b,c

proof end;

theorem Th22: :: AFVECT0:22

for AFV being WeakAffVect
for a, b, c, c9 being Element of AFV st Mid a,b,c & Mid a,b,c9 holds
c = c9

proof end;

theorem Th23: :: AFVECT0:23

for AFV being WeakAffVect
for a, b, c, b9 being Element of AFV st Mid a,b,c & MDist b,b9 holds
Mid a,b9,c

proof end;

theorem Th24: :: AFVECT0:24

for AFV being WeakAffVect
for a, b, c, b9, c9 being Element of AFV st Mid a,b,c & Mid a,b9,c9 & MDist b,b9 holds
c = c9

proof end;

theorem Th25: :: AFVECT0:25

for AFV being WeakAffVect
for a, b, a9, b9, p being Element of AFV st Mid a,p,a9 & Mid b,p,b9 holds
a,b // b9,a9

proof end;

theorem :: AFVECT0:26

for AFV being WeakAffVect
for a, b, a9, b9, p, q being Element of AFV st Mid a,p,a9 & Mid b,q,b9 & MDist p,q holds
a,b // b9,a9

proof end;

::
:: Point Symmetry
::

definition

let AFV be WeakAffVect;
let a, b be Element of AFV;

func PSym (a,b) -> Element of AFV means :Def4: :: AFVECT0:def 4
Mid b,a,it;

correctness
existence
ex b₁ being Element of AFV st Mid b,a,b₁;
uniqueness
for b₁, b₂ being Element of AFV st Mid b,a,b₁ & Mid b,a,b₂ holds
b₁ = b₂;
by Th21, Th22;

end;

:: deftheorem Def4 defines PSym AFVECT0:def 4 :
for AFV being WeakAffVect
for a, b, b₄ being Element of AFV holds
( b₄ = PSym (a,b) iff Mid b,a,b₄ );

theorem :: AFVECT0:27

for AFV being WeakAffVect
for a, b, p being Element of AFV holds
( PSym (p,a) = b iff a,p // p,b ) by Def3, Def4;

theorem Th28: :: AFVECT0:28

for AFV being WeakAffVect
for a, p being Element of AFV holds
( PSym (p,a) = a iff ( a = p or MDist a,p ) )

proof end;

theorem Th29: :: AFVECT0:29

for AFV being WeakAffVect
for a, p being Element of AFV holds PSym (p,(PSym (p,a))) = a

proof end;

theorem Th30: :: AFVECT0:30

for AFV being WeakAffVect
for a, b, p being Element of AFV st PSym (p,a) = PSym (p,b) holds
a = b

proof end;

theorem :: AFVECT0:31

for AFV being WeakAffVect
for b, p being Element of AFV ex a being Element of AFV st PSym (p,a) = b

proof end;

theorem Th32: :: AFVECT0:32

for AFV being WeakAffVect
for a, b, p being Element of AFV holds a,b // PSym (p,b), PSym (p,a)

proof end;

theorem Th33: :: AFVECT0:33

for AFV being WeakAffVect
for a, b, c, d, p being Element of AFV holds
( a,b // c,d iff PSym (p,a), PSym (p,b) // PSym (p,c), PSym (p,d) )

proof end;

theorem :: AFVECT0:34

for AFV being WeakAffVect
for a, b, p being Element of AFV holds
( MDist a,b iff MDist PSym (p,a), PSym (p,b) ) by Th30, Th33;

theorem Th35: :: AFVECT0:35

for AFV being WeakAffVect
for a, b, c, p being Element of AFV holds
( Mid a,b,c iff Mid PSym (p,a), PSym (p,b), PSym (p,c) ) by Th33;

theorem Th36: :: AFVECT0:36

for AFV being WeakAffVect
for a, p, q being Element of AFV holds
( PSym (p,a) = PSym (q,a) iff ( p = q or MDist p,q ) )

proof end;

theorem Th37: :: AFVECT0:37

for AFV being WeakAffVect
for a, p, q being Element of AFV holds PSym (q,(PSym (p,(PSym (q,a))))) = PSym ((PSym (q,p)),a)

proof end;

theorem :: AFVECT0:38

for AFV being WeakAffVect
for a, p, q being Element of AFV holds
( PSym (p,(PSym (q,a))) = PSym (q,(PSym (p,a))) iff ( p = q or MDist p,q or MDist q, PSym (p,q) ) )

proof end;

theorem Th39: :: AFVECT0:39

for AFV being WeakAffVect
for a, p, q, r being Element of AFV holds PSym (p,(PSym (q,(PSym (r,a))))) = PSym (r,(PSym (q,(PSym (p,a)))))

proof end;

theorem :: AFVECT0:40

for AFV being WeakAffVect
for a, b, c, p being Element of AFV ex d being Element of AFV st PSym (a,(PSym (b,(PSym (c,p))))) = PSym (d,p)

proof end;

theorem :: AFVECT0:41

for AFV being WeakAffVect
for a, b, p being Element of AFV ex c being Element of AFV st PSym (a,(PSym (c,p))) = PSym (c,(PSym (b,p)))

proof end;

::
:: Addition on the carrier
::

definition

let AFV be WeakAffVect;
let o, a, b be Element of AFV;

func Padd (o,a,b) -> Element of AFV means :Def5: :: AFVECT0:def 5
o,a // b,it;

correctness
existence
ex b₁ being Element of AFV st o,a // b,b₁;
uniqueness
for b₁, b₂ being Element of AFV st o,a // b,b₁ & o,a // b,b₂ holds
b₁ = b₂;
by Def1, Th5;

end;

:: deftheorem Def5 defines Padd AFVECT0:def 5 :
for AFV being WeakAffVect
for o, a, b, b₅ being Element of AFV holds
( b₅ = Padd (o,a,b) iff o,a // b,b₅ );

notation

let AFV be WeakAffVect;
let o, a be Element of AFV;
synonym Pcom (o,a) for PSym (o,a);

end;

Lm1: for AFV being WeakAffVect
for a, b, o being Element of AFV holds
( Pcom (o,a) = b iff a,o // o,b )
by Def4, Def3;

definition

let AFV be WeakAffVect;
let o be Element of AFV;

func Padd o -> BinOp of the carrier of AFV means :Def6: :: AFVECT0:def 6
for a, b being Element of AFV holds it . (a,b) = Padd (o,a,b);

existence
ex b₁ being BinOp of the carrier of AFV st
for a, b being Element of AFV holds b₁ . (a,b) = Padd (o,a,b)

proof end;

uniqueness
for b₁, b₂ being BinOp of the carrier of AFV st ( for a, b being Element of AFV holds b₁ . (a,b) = Padd (o,a,b) ) & ( for a, b being Element of AFV holds b₂ . (a,b) = Padd (o,a,b) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines Padd AFVECT0:def 6 :
for AFV being WeakAffVect
for o being Element of AFV
for b₃ being BinOp of the carrier of AFV holds
( b₃ = Padd o iff for a, b being Element of AFV holds b₃ . (a,b) = Padd (o,a,b) );

definition

let AFV be WeakAffVect;
let o be Element of AFV;

func Pcom o -> UnOp of the carrier of AFV means :Def7: :: AFVECT0:def 7
for a being Element of AFV holds it . a = Pcom (o,a);

existence
ex b₁ being UnOp of the carrier of AFV st
for a being Element of AFV holds b₁ . a = Pcom (o,a)

proof end;

uniqueness
for b₁, b₂ being UnOp of the carrier of AFV st ( for a being Element of AFV holds b₁ . a = Pcom (o,a) ) & ( for a being Element of AFV holds b₂ . a = Pcom (o,a) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines Pcom AFVECT0:def 7 :
for AFV being WeakAffVect
for o being Element of AFV
for b₃ being UnOp of the carrier of AFV holds
( b₃ = Pcom o iff for a being Element of AFV holds b₃ . a = Pcom (o,a) );

definition

let AFV be WeakAffVect;
let o be Element of AFV;

func GroupVect (AFV,o) -> strict addLoopStr equals :: AFVECT0:def 8
addLoopStr(# the carrier of AFV,(Padd o),o #);

correctness
coherence
addLoopStr(# the carrier of AFV,(Padd o),o #) is strict addLoopStr ;
;

end;

:: deftheorem defines GroupVect AFVECT0:def 8 :
for AFV being WeakAffVect
for o being Element of AFV holds GroupVect (AFV,o) = addLoopStr(# the carrier of AFV,(Padd o),o #);

registration

let AFV be WeakAffVect;
let o be Element of AFV;

cluster GroupVect (AFV,o) -> non empty strict ;

coherence
not GroupVect (AFV,o) is empty ;

end;

theorem :: AFVECT0:42

for AFV being WeakAffVect
for o being Element of AFV holds
( the carrier of (GroupVect (AFV,o)) = the carrier of AFV & the addF of (GroupVect (AFV,o)) = Padd o & 0. (GroupVect (AFV,o)) = o ) ;

theorem :: AFVECT0:43

for AFV being WeakAffVect
for o being Element of AFV
for a, b being Element of (GroupVect (AFV,o))
for a9, b9 being Element of AFV st a = a9 & b = b9 holds
a + b = (Padd o) . (a9,b9) ;

Lm2: for AFV being WeakAffVect
for o being Element of AFV
for a, b being Element of (GroupVect (AFV,o)) holds a + b = b + a

proof end;

Lm3: for AFV being WeakAffVect
for o being Element of AFV
for a, b, c being Element of (GroupVect (AFV,o)) holds (a + b) + c = a + (b + c)

proof end;

Lm4: for AFV being WeakAffVect
for o being Element of AFV
for a being Element of (GroupVect (AFV,o)) holds a + (0. (GroupVect (AFV,o))) = a

proof end;

Lm5: for AFV being WeakAffVect
for o being Element of AFV holds
( GroupVect (AFV,o) is Abelian & GroupVect (AFV,o) is add-associative & GroupVect (AFV,o) is right_zeroed )

proof end;

Lm6: for AFV being WeakAffVect
for o being Element of AFV holds GroupVect (AFV,o) is right_complementable

proof end;

registration

let AFV be WeakAffVect;
let o be Element of AFV;

cluster GroupVect (AFV,o) -> strict right_complementable Abelian add-associative right_zeroed ;

coherence
( GroupVect (AFV,o) is Abelian & GroupVect (AFV,o) is add-associative & GroupVect (AFV,o) is right_zeroed & GroupVect (AFV,o) is right_complementable ) by Lm5, Lm6;

end;

theorem Th44: :: AFVECT0:44

for AFV being WeakAffVect
for o being Element of AFV
for a being Element of (GroupVect (AFV,o))
for a9 being Element of AFV st a = a9 holds
- a = (Pcom o) . a9

proof end;

theorem :: AFVECT0:45

for AFV being WeakAffVect
for o being Element of AFV holds 0. (GroupVect (AFV,o)) = o ;

theorem Th46: :: AFVECT0:46

for AFV being WeakAffVect
for o being Element of AFV
for a being Element of (GroupVect (AFV,o)) ex b being Element of (GroupVect (AFV,o)) st b + b = a

proof end;

registration

let AFV be WeakAffVect;
let o be Element of AFV;

cluster GroupVect (AFV,o) -> strict Two_Divisible ;

coherence
GroupVect (AFV,o) is Two_Divisible

proof end;

end;

theorem Th47: :: AFVECT0:47

for AFV being AffVect
for o being Element of AFV
for a being Element of (GroupVect (AFV,o)) st a + a = 0. (GroupVect (AFV,o)) holds
a = 0. (GroupVect (AFV,o))

proof end;

registration

let AFV be AffVect;
let o be Element of AFV;

cluster GroupVect (AFV,o) -> strict Fanoian ;

coherence
GroupVect (AFV,o) is Fanoian

proof end;

end;

registration

cluster non empty non trivial strict right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible for addLoopStr ;

existence
ex b₁ being Uniquely_Two_Divisible_Group st
( b₁ is strict & not b₁ is trivial )

proof end;

end;

definition

mode Proper_Uniquely_Two_Divisible_Group is non trivial Uniquely_Two_Divisible_Group;

end;

theorem :: AFVECT0:48

for AFV being AffVect
for o being Element of AFV holds GroupVect (AFV,o) is Proper_Uniquely_Two_Divisible_Group ;

registration

let AFV be AffVect;
let o be Element of AFV;

cluster GroupVect (AFV,o) -> non trivial strict ;

coherence
not GroupVect (AFV,o) is trivial ;

end;

theorem Th49: :: AFVECT0:49

for ADG being Proper_Uniquely_Two_Divisible_Group holds AV ADG is AffVect

proof end;

registration

let ADG be Proper_Uniquely_Two_Divisible_Group;

cluster AV ADG -> non trivial AffVect-like ;

coherence
( AV ADG is AffVect-like & not AV ADG is trivial ) by Th49;

end;

theorem Th50: :: AFVECT0:50

for AFV being strict AffVect
for o being Element of AFV holds AFV = AV (GroupVect (AFV,o))

proof end;

theorem :: AFVECT0:51

for AS being strict AffinStruct holds
( AS is AffVect iff ex ADG being Proper_Uniquely_Two_Divisible_Group st AS = AV ADG )

proof end;

definition

let X, Y be non empty addLoopStr ;
let f be Function of the carrier of X, the carrier of Y;

pred f is_Iso_of X,Y means :: AFVECT0:def 9
( f is one-to-one & rng f = the carrier of Y & ( for a, b being Element of X holds
( f . (a + b) = (f . a) + (f . b) & f . (0. X) = 0. Y & f . (- a) = - (f . a) ) ) );

end;

:: deftheorem defines is_Iso_of AFVECT0:def 9 :
for X, Y being non empty addLoopStr
for f being Function of the carrier of X, the carrier of Y holds
( f is_Iso_of X,Y iff ( f is one-to-one & rng f = the carrier of Y & ( for a, b being Element of X holds
( f . (a + b) = (f . a) + (f . b) & f . (0. X) = 0. Y & f . (- a) = - (f . a) ) ) ) );

definition

let X, Y be non empty addLoopStr ;

pred X,Y are_Iso means :: AFVECT0:def 10
ex f being Function of the carrier of X, the carrier of Y st f is_Iso_of X,Y;

end;

:: deftheorem defines are_Iso AFVECT0:def 10 :
for X, Y being non empty addLoopStr holds
( X,Y are_Iso iff ex f being Function of the carrier of X, the carrier of Y st f is_Iso_of X,Y );

theorem Th52: :: AFVECT0:52

for ADG being Proper_Uniquely_Two_Divisible_Group
for f being Function of the carrier of ADG, the carrier of ADG
for o9 being Element of ADG
for o being Element of (AV ADG) st ( for x being Element of ADG holds f . x = o9 + x ) & o = o9 holds
for a, b being Element of ADG holds
( f . (a + b) = (Padd o) . ((f . a),(f . b)) & f . (0. ADG) = 0. (GroupVect ((AV ADG),o)) & f . (- a) = (Pcom o) . (f . a) )

proof end;

theorem Th53: :: AFVECT0:53

for ADG being Proper_Uniquely_Two_Divisible_Group
for f being Function of the carrier of ADG, the carrier of ADG
for o9 being Element of ADG st ( for b being Element of ADG holds f . b = o9 + b ) holds
f is one-to-one

proof end;

theorem Th54: :: AFVECT0:54

for ADG being Proper_Uniquely_Two_Divisible_Group
for f being Function of the carrier of ADG, the carrier of ADG
for o9 being Element of ADG
for o being Element of (AV ADG) st ( for b being Element of ADG holds f . b = o9 + b ) holds
rng f = the carrier of (GroupVect ((AV ADG),o))

proof end;

theorem :: AFVECT0:55

for ADG being Proper_Uniquely_Two_Divisible_Group
for o9 being Element of ADG
for o being Element of (AV ADG) st o = o9 holds
ADG, GroupVect ((AV ADG),o) are_Iso

proof end;