:: Some Properties of Circles on the Plane
:: by Artur Korni{\l}owicz and Yasunari Shidama
::
:: Received October 18, 2004
:: Copyright (c) 2004 Association of Mizar Users
set P2 = 2 * PI ;
set o = |[0 ,0 ]|;
set R = the carrier of R^1 ;
Lm1:
0 in INT
by INT_1:def 1;
reconsider p0 = - 1 as real negative number ;
reconsider p1 = 1 as real positive number ;
set CIT = Closed-Interval-TSpace (- 1),1;
set cCIT = the carrier of (Closed-Interval-TSpace (- 1),1);
Lm2:
the carrier of (Closed-Interval-TSpace (- 1),1) = [.(- 1),1.]
by TOPMETR:25;
Lm3:
1 - 0 <= 1
;
Lm4:
(3 / 2) - (1 / 2) <= 1
;
Lm5:
PI / 2 < PI / 1
by XREAL_1:78;
Lm6:
1 * PI < (3 / 2) * PI
by XREAL_1:70;
Lm7:
(3 / 2) * PI < 2 * PI
by XREAL_1:70;
Lm9:
for X being non empty TopSpace
for Y being non empty SubSpace of X
for Z being non empty SubSpace of Y
for p being Point of Z holds p is Point of X
Lm10:
for X being TopSpace
for Y being SubSpace of X
for Z being SubSpace of Y
for A being Subset of Z holds A is Subset of X
theorem Th1: :: TOPREALB:1
theorem Th2: :: TOPREALB:2
:: deftheorem defines IntIntervals TOPREALB:def 1 :
theorem :: TOPREALB:3
theorem :: TOPREALB:4
:: deftheorem defines R^1 TOPREALB:def 2 :
:: deftheorem defines R^1 TOPREALB:def 3 :
:: deftheorem defines R^1 TOPREALB:def 4 :
theorem Th5: :: TOPREALB:5
theorem Th6: :: TOPREALB:6
theorem Th7: :: TOPREALB:7
theorem Th8: :: TOPREALB:8
Lm11:
sin is Function of R^1 ,R^1
Lm12:
cos is Function of R^1 ,R^1
set A = R^1 ].0 ,1.[;
Lm13:
now
let a be non
zero real number ;
:: thesis: for b being real number holds
( R^1 = R^1 | (R^1 (dom (AffineMap a,b))) & R^1 = R^1 | (R^1 (rng (AffineMap a,b))) )let b be
real number ;
:: thesis: ( R^1 = R^1 | (R^1 (dom (AffineMap a,b))) & R^1 = R^1 | (R^1 (rng (AffineMap a,b))) )A1:
dom (AffineMap a,b) = REAL
by FUNCT_2:def 1;
A2:
rng (AffineMap a,b) = REAL
by JORDAN16:32;
[#] R^1 = REAL
by TOPMETR:24;
hence
(
R^1 = R^1 | (R^1 (dom (AffineMap a,b))) &
R^1 = R^1 | (R^1 (rng (AffineMap a,b))) )
by A1, A2, TSEP_1:3;
:: thesis: verum
end;
:: deftheorem Def5 defines being_simple_closed_curve TOPREALB:def 5 :
:: deftheorem defines Tcircle TOPREALB:def 6 :
theorem Th9: :: TOPREALB:9
theorem Th10: :: TOPREALB:10
theorem :: TOPREALB:11
:: deftheorem defines Tunit_circle TOPREALB:def 7 :
set TUC = Tunit_circle 2;
set cS1 = the carrier of (Tunit_circle 2);
Lm14:
the carrier of (Tunit_circle 2) = Sphere |[0 ,0 ]|,1
by Th9, EUCLID:58;
theorem Th12: :: TOPREALB:12
theorem Th13: :: TOPREALB:13
theorem Th14: :: TOPREALB:14
theorem Th15: :: TOPREALB:15
theorem :: TOPREALB:16
theorem :: TOPREALB:17
set TREC = Trectangle p0,p1,p0,p1;
theorem :: TOPREALB:18
theorem Th19: :: TOPREALB:19
Lm15:
for n being non empty Element of NAT
for r being real positive number
for x being Point of (TOP-REAL n) holds Tunit_circle n, Tcircle x,r are_homeomorphic
theorem :: TOPREALB:20
:: deftheorem defines c[10] TOPREALB:def 8 :
:: deftheorem defines c[-10] TOPREALB:def 9 :
Lm16:
c[10] <> c[-10]
by SPPOL_2:1;
:: deftheorem Def10 defines Topen_unit_circle TOPREALB:def 10 :
theorem :: TOPREALB:21
canceled;
theorem Th22: :: TOPREALB:22
theorem Th23: :: TOPREALB:23
theorem Th24: :: TOPREALB:24
theorem Th25: :: TOPREALB:25
theorem Th26: :: TOPREALB:26
set TOUC = Topen_unit_circle c[10] ;
set TOUCm = Topen_unit_circle c[-10] ;
set X = the carrier of (Topen_unit_circle c[10] );
set Xm = the carrier of (Topen_unit_circle c[-10] );
set Y = the carrier of (R^1 | (R^1 ].0 ,(0 + p1).[));
set Ym = the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[));
Lm17:
the carrier of (Topen_unit_circle c[10] ) = [#] (Topen_unit_circle c[10] )
;
Lm18:
the carrier of (Topen_unit_circle c[-10] ) = [#] (Topen_unit_circle c[-10] )
;
theorem Th27: :: TOPREALB:27
theorem Th28: :: TOPREALB:28
theorem Th29: :: TOPREALB:29
theorem :: TOPREALB:30
theorem :: TOPREALB:31
:: deftheorem Def11 defines CircleMap TOPREALB:def 11 :
Lm19:
dom CircleMap = REAL
by FUNCT_2:def 1, TOPMETR:24;
theorem Th32: :: TOPREALB:32
theorem Th33: :: TOPREALB:33
theorem Th34: :: TOPREALB:34
Lm20:
CircleMap . (1 / 2) = |[(- 1),0 ]|
theorem Th35: :: TOPREALB:35
theorem :: TOPREALB:36
theorem :: TOPREALB:37
Lm22:
for r being real number holds CircleMap . r = |[((cos * (AffineMap (2 * PI ),0 )) . r),((sin * (AffineMap (2 * PI ),0 )) . r)]|
theorem :: TOPREALB:38
Lm23:
for A being Subset of R^1 holds CircleMap | A is Function of (R^1 | A),(Tunit_circle 2)
Lm24:
for r being real number st - 1 <= r & r <= 1 holds
( 0 <= (arccos r) / (2 * PI ) & (arccos r) / (2 * PI ) <= 1 / 2 )
theorem Th39: :: TOPREALB:39
Lm25:
CircleMap | [.0 ,1.[ is one-to-one
theorem Th40: :: TOPREALB:40
theorem Th41: :: TOPREALB:41
:: deftheorem defines CircleMap TOPREALB:def 12 :
Lm26:
for a, r being real number holds rng ((AffineMap 1,(- a)) | ].(r + a),((r + a) + 1).[) = ].r,(r + 1).[
theorem Th42: :: TOPREALB:42
definition
func Circle2IntervalR -> Function of
(Topen_unit_circle c[10] ),
(R^1 | (R^1 ].0 ,1.[)) means :
Def13:
:: TOPREALB:def 13
for
p being
Point of
(Topen_unit_circle c[10] ) ex
x,
y being
real number st
(
p = |[x,y]| & (
y >= 0 implies
it . p = (arccos x) / (2 * PI ) ) & (
y <= 0 implies
it . p = 1
- ((arccos x) / (2 * PI )) ) );
existence
ex b1 being Function of (Topen_unit_circle c[10] ),(R^1 | (R^1 ].0 ,1.[)) st
for p being Point of (Topen_unit_circle c[10] ) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = (arccos x) / (2 * PI ) ) & ( y <= 0 implies b1 . p = 1 - ((arccos x) / (2 * PI )) ) )
uniqueness
for b1, b2 being Function of (Topen_unit_circle c[10] ),(R^1 | (R^1 ].0 ,1.[)) st ( for p being Point of (Topen_unit_circle c[10] ) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = (arccos x) / (2 * PI ) ) & ( y <= 0 implies b1 . p = 1 - ((arccos x) / (2 * PI )) ) ) ) & ( for p being Point of (Topen_unit_circle c[10] ) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b2 . p = (arccos x) / (2 * PI ) ) & ( y <= 0 implies b2 . p = 1 - ((arccos x) / (2 * PI )) ) ) ) holds
b1 = b2
end;
:: deftheorem Def13 defines Circle2IntervalR TOPREALB:def 13 :
set A1 = R^1 ].(1 / 2),((1 / 2) + p1).[;
definition
func Circle2IntervalL -> Function of
(Topen_unit_circle c[-10] ),
(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) means :
Def14:
:: TOPREALB:def 14
for
p being
Point of
(Topen_unit_circle c[-10] ) ex
x,
y being
real number st
(
p = |[x,y]| & (
y >= 0 implies
it . p = 1
+ ((arccos x) / (2 * PI )) ) & (
y <= 0 implies
it . p = 1
- ((arccos x) / (2 * PI )) ) );
existence
ex b1 being Function of (Topen_unit_circle c[-10] ),(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) st
for p being Point of (Topen_unit_circle c[-10] ) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = 1 + ((arccos x) / (2 * PI )) ) & ( y <= 0 implies b1 . p = 1 - ((arccos x) / (2 * PI )) ) )
uniqueness
for b1, b2 being Function of (Topen_unit_circle c[-10] ),(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) st ( for p being Point of (Topen_unit_circle c[-10] ) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = 1 + ((arccos x) / (2 * PI )) ) & ( y <= 0 implies b1 . p = 1 - ((arccos x) / (2 * PI )) ) ) ) & ( for p being Point of (Topen_unit_circle c[-10] ) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b2 . p = 1 + ((arccos x) / (2 * PI )) ) & ( y <= 0 implies b2 . p = 1 - ((arccos x) / (2 * PI )) ) ) ) holds
b1 = b2
end;
:: deftheorem Def14 defines Circle2IntervalL TOPREALB:def 14 :
set C = Circle2IntervalR ;
set Cm = Circle2IntervalL ;
theorem Th43: :: TOPREALB:43
theorem Th44: :: TOPREALB:44
set A = ].0 ,1.[;
set Q = [.(- 1),1.[;
set E = ].0 ,PI .];
set j = 1 / (2 * PI );
reconsider Q = [.(- 1),1.[, E = ].0 ,PI .] as non empty Subset of REAL ;
Lm27:
the carrier of (R^1 | (R^1 Q)) = R^1 Q
by PRE_TOPC:29;
Lm28:
the carrier of (R^1 | (R^1 E)) = R^1 E
by PRE_TOPC:29;
Lm29:
the carrier of (R^1 | (R^1 ].0 ,1.[)) = R^1 ].0 ,1.[
by PRE_TOPC:29;
set Af = (AffineMap (1 / (2 * PI )),0 ) | (R^1 E);
dom (AffineMap (1 / (2 * PI )),0 ) = the carrier of R^1
by FUNCT_2:def 1, TOPMETR:24;
then Lm30:
dom ((AffineMap (1 / (2 * PI )),0 ) | (R^1 E)) = R^1 E
by RELAT_1:91;
rng ((AffineMap (1 / (2 * PI )),0 ) | (R^1 E)) c= ].0 ,1.[
then reconsider Af = (AffineMap (1 / (2 * PI )),0 ) | (R^1 E) as Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].0 ,1.[)) by Lm28, Lm29, Lm30, FUNCT_2:4;
Lm31:
R^1 (AffineMap (1 / (2 * PI )),0 ) = AffineMap (1 / (2 * PI )),0
;
Lm32:
dom (AffineMap (1 / (2 * PI )),0 ) = REAL
by FUNCT_2:def 1;
Lm33:
rng (AffineMap (1 / (2 * PI )),0 ) = REAL
by JORDAN16:32;
R^1 | ([#] R^1 ) = R^1
by TSEP_1:3;
then reconsider Af = Af as continuous Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].0 ,1.[)) by Lm31, Lm32, Lm33, TOPMETR:24, TOPREALA:29;
set L = (R^1 (AffineMap (- 1),1)) | (R^1 ].0 ,1.[);
Lm34:
dom (AffineMap (- 1),1) = REAL
by FUNCT_2:def 1;
then Lm35:
dom ((R^1 (AffineMap (- 1),1)) | (R^1 ].0 ,1.[)) = ].0 ,1.[
by RELAT_1:91;
rng ((R^1 (AffineMap (- 1),1)) | (R^1 ].0 ,1.[)) c= ].0 ,1.[
then reconsider L = (R^1 (AffineMap (- 1),1)) | (R^1 ].0 ,1.[) as Function of (R^1 | (R^1 ].0 ,1.[)),(R^1 | (R^1 ].0 ,1.[)) by Lm29, Lm35, FUNCT_2:4;
Lm36:
rng (AffineMap (- 1),1) = REAL
by JORDAN16:32;
Lm37:
R^1 | ([#] R^1 ) = R^1
by TSEP_1:3;
then reconsider L = L as continuous Function of (R^1 | (R^1 ].0 ,1.[)),(R^1 | (R^1 ].0 ,1.[)) by Lm34, Lm36, TOPMETR:24, TOPREALA:29;
reconsider ac = R^1 arccos as continuous Function of (R^1 | (R^1 [.(- 1),1.])),(R^1 | (R^1 [.0 ,PI .])) by SIN_COS6:87, SIN_COS6:88;
set c = ac | (R^1 Q);
Lm38:
dom (ac | (R^1 Q)) = Q
by RELAT_1:91, SIN_COS6:88, XXREAL_1:35;
Lm39:
rng (ac | (R^1 Q)) c= E
then reconsider c = ac | (R^1 Q) as Function of (R^1 | (R^1 Q)),(R^1 | (R^1 E)) by Lm27, Lm28, Lm38, FUNCT_2:4;
the carrier of (R^1 | (R^1 [.(- 1),1.])) = [.(- 1),1.]
by PRE_TOPC:29;
then reconsider QQ = R^1 Q as Subset of (R^1 | (R^1 [.(- 1),1.])) by XXREAL_1:35;
the carrier of (R^1 | (R^1 [.0 ,PI .])) = [.0 ,PI .]
by PRE_TOPC:29;
then reconsider EE = R^1 E as Subset of (R^1 | (R^1 [.0 ,PI .])) by XXREAL_1:36;
( (R^1 | (R^1 [.(- 1),1.])) | QQ = R^1 | (R^1 Q) & (R^1 | (R^1 [.0 ,PI .])) | EE = R^1 | (R^1 E) )
by GOBOARD9:4;
then Lm40:
c is continuous
by TOPREALA:29;
reconsider p = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:24;
Lm41:
dom p = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;
Lm42:
p is continuous
by TOPREAL6:83;
Lm43:
for aX1 being Subset of (Topen_unit_circle c[10] ) st aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10] ) & 0 <= q `2 ) } holds
Circle2IntervalR | ((Topen_unit_circle c[10] ) | aX1) is continuous
Lm44:
for aX1 being Subset of (Topen_unit_circle c[10] ) st aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10] ) & 0 >= q `2 ) } holds
Circle2IntervalR | ((Topen_unit_circle c[10] ) | aX1) is continuous
Lm45:
for p being Point of (Topen_unit_circle c[10] ) st p = c[-10] holds
Circle2IntervalR is_continuous_at p
set h1 = REAL --> 1;
reconsider h1 = REAL --> 1 as PartFunc of REAL , REAL ;
Lm46:
Circle2IntervalR is continuous
set A = ].(1 / 2),((1 / 2) + p1).[;
set Q = ].(- 1),1.];
set E = [.0 ,PI .[;
reconsider Q = ].(- 1),1.], E = [.0 ,PI .[ as non empty Subset of REAL ;
Lm47:
the carrier of (R^1 | (R^1 Q)) = R^1 Q
by PRE_TOPC:29;
Lm48:
the carrier of (R^1 | (R^1 E)) = R^1 E
by PRE_TOPC:29;
Lm49:
the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) = R^1 ].(1 / 2),((1 / 2) + p1).[
by PRE_TOPC:29;
set Af = (AffineMap (- (1 / (2 * PI ))),1) | (R^1 E);
dom (AffineMap (- (1 / (2 * PI ))),1) = the carrier of R^1
by FUNCT_2:def 1, TOPMETR:24;
then Lm50:
dom ((AffineMap (- (1 / (2 * PI ))),1) | (R^1 E)) = R^1 E
by RELAT_1:91;
rng ((AffineMap (- (1 / (2 * PI ))),1) | (R^1 E)) c= ].(1 / 2),((1 / 2) + p1).[
then reconsider Af = (AffineMap (- (1 / (2 * PI ))),1) | (R^1 E) as Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by Lm48, Lm49, Lm50, FUNCT_2:4;
Lm51:
R^1 (AffineMap (- (1 / (2 * PI ))),1) = AffineMap (- (1 / (2 * PI ))),1
;
Lm52:
dom (AffineMap (- (1 / (2 * PI ))),1) = REAL
by FUNCT_2:def 1;
rng (AffineMap (- (1 / (2 * PI ))),1) = REAL
by JORDAN16:32;
then reconsider Af = Af as continuous Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by Lm37, Lm51, Lm52, TOPMETR:24, TOPREALA:29;
set Af1 = (AffineMap (1 / (2 * PI )),1) | (R^1 E);
dom (AffineMap (1 / (2 * PI )),1) = the carrier of R^1
by FUNCT_2:def 1, TOPMETR:24;
then Lm53:
dom ((AffineMap (1 / (2 * PI )),1) | (R^1 E)) = R^1 E
by RELAT_1:91;
rng ((AffineMap (1 / (2 * PI )),1) | (R^1 E)) c= ].(1 / 2),((1 / 2) + p1).[
then reconsider Af1 = (AffineMap (1 / (2 * PI )),1) | (R^1 E) as Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by Lm48, Lm49, Lm53, FUNCT_2:4;
Lm54:
R^1 (AffineMap (1 / (2 * PI )),1) = AffineMap (1 / (2 * PI )),1
;
Lm55:
dom (AffineMap (1 / (2 * PI )),1) = REAL
by FUNCT_2:def 1;
rng (AffineMap (1 / (2 * PI )),1) = REAL
by JORDAN16:32;
then reconsider Af1 = Af1 as continuous Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by Lm37, Lm54, Lm55, TOPMETR:24, TOPREALA:29;
set c = ac | (R^1 Q);
Lm56:
dom (ac | (R^1 Q)) = Q
by RELAT_1:91, SIN_COS6:88, XXREAL_1:36;
Lm57:
rng (ac | (R^1 Q)) c= E
then reconsider c = ac | (R^1 Q) as Function of (R^1 | (R^1 Q)),(R^1 | (R^1 E)) by Lm47, Lm48, Lm56, FUNCT_2:4;
the carrier of (R^1 | (R^1 [.(- 1),1.])) = [.(- 1),1.]
by PRE_TOPC:29;
then reconsider QQ = R^1 Q as Subset of (R^1 | (R^1 [.(- 1),1.])) by XXREAL_1:36;
the carrier of (R^1 | (R^1 [.0 ,PI .])) = [.0 ,PI .]
by PRE_TOPC:29;
then reconsider EE = R^1 E as Subset of (R^1 | (R^1 [.0 ,PI .])) by XXREAL_1:35;
( (R^1 | (R^1 [.(- 1),1.])) | QQ = R^1 | (R^1 Q) & (R^1 | (R^1 [.0 ,PI .])) | EE = R^1 | (R^1 E) )
by GOBOARD9:4;
then Lm58:
c is continuous
by TOPREALA:29;
Lm59:
for aX1 being Subset of (Topen_unit_circle c[-10] ) st aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10] ) & 0 <= q `2 ) } holds
Circle2IntervalL | ((Topen_unit_circle c[-10] ) | aX1) is continuous
Lm60:
for aX1 being Subset of (Topen_unit_circle c[-10] ) st aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10] ) & 0 >= q `2 ) } holds
Circle2IntervalL | ((Topen_unit_circle c[-10] ) | aX1) is continuous
Lm61:
for p being Point of (Topen_unit_circle c[-10] ) st p = c[10] holds
Circle2IntervalL is_continuous_at p
Lm62:
Circle2IntervalL is continuous
Lm63:
CircleMap (R^1 0 ) is open
Lm64:
CircleMap (R^1 (1 / 2)) is open
by Lm20, Th44, TOPREALA:35;
theorem :: TOPREALB:45
canceled;
theorem :: TOPREALB:46
theorem :: TOPREALB:47
canceled;
theorem :: TOPREALB:48
canceled;
theorem :: TOPREALB:49
ex
F being
Subset-Family of
(Tunit_circle 2) st
(
F = {(CircleMap .: ].0 ,1.[),(CircleMap .: ].(1 / 2),(3 / 2).[)} &
F is
Cover of
(Tunit_circle 2) &
F is
open & ( for
U being
Subset of
(Tunit_circle 2) holds
( (
U = CircleMap .: ].0 ,1.[ implies (
union (IntIntervals 0 ,1) = CircleMap " U & ( for
d being
Subset of
R^1 st
d in IntIntervals 0 ,1 holds
for
f being
Function of
(R^1 | d),
((Tunit_circle 2) | U) st
f = CircleMap | d holds
f is
being_homeomorphism ) ) ) & (
U = CircleMap .: ].(1 / 2),(3 / 2).[ implies (
union (IntIntervals (1 / 2),(3 / 2)) = CircleMap " U & ( for
d being
Subset of
R^1 st
d in IntIntervals (1 / 2),
(3 / 2) holds
for
f being
Function of
(R^1 | d),
((Tunit_circle 2) | U) st
f = CircleMap | d holds
f is
being_homeomorphism ) ) ) ) ) )