JORDAN20: Behaviour of an Arc Crossing a Line

:: Behaviour of an Arc Crossing a Line
:: by Yatsuka Nakamura
::
:: Received January 26, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

theorem Th1: :: JORDAN20:1

for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & p in P holds
Segment (P,p1,p2,p,p) = {p}

proof end;

theorem Th2: :: JORDAN20:2

for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg (p1,p2) & p1 `1 <= a & p2 `1 <= a holds
p `1 <= a

proof end;

theorem Th3: :: JORDAN20:3

for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg (p1,p2) & p1 `1 >= a & p2 `1 >= a holds
p `1 >= a

proof end;

theorem :: JORDAN20:4

for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg (p1,p2) & p1 `1 < a & p2 `1 < a holds
p `1 < a

proof end;

theorem :: JORDAN20:5

for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg (p1,p2) & p1 `1 > a & p2 `1 > a holds
p `1 > a

proof end;

theorem Th6: :: JORDAN20:6

for j being Nat
for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 > (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 >= q `1

proof end;

theorem Th7: :: JORDAN20:7

for j being Nat
for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 < (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 <= q `1

proof end;

definition

let P be Subset of (TOP-REAL 2);
let p1, p2, p be Point of (TOP-REAL 2);
let e be Real;

pred p is_Lin P,p1,p2,e means :: JORDAN20:def 1
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e ) ) );

pred p is_Rin P,p1,p2,e means :: JORDAN20:def 2
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e ) ) );

pred p is_Lout P,p1,p2,e means :: JORDAN20:def 3
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 <= e ) ) );

pred p is_Rout P,p1,p2,e means :: JORDAN20:def 4
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 >= e ) ) );

pred p is_OSin P,p1,p2,e means :: JORDAN20:def 5
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p7,p,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p7,p8,P,p1,p2 & LE p8,p,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p4,p7,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p4,p5,P,p1,p2 & LE p5,p7,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p4,p6,P,p1,p2 & LE p6,p7,P,p1,p2 & p6 `1 < e ) ) ) ) );

pred p is_OSout P,p1,p2,e means :: JORDAN20:def 6
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p,p7,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p8,p7,P,p1,p2 & LE p,p8,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p7,p4,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p5,p4,P,p1,p2 & LE p7,p5,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p6,p4,P,p1,p2 & LE p7,p6,P,p1,p2 & p6 `1 < e ) ) ) ) );

correctness ;

end;

:: deftheorem defines is_Lin JORDAN20:def 1 :

:: deftheorem defines is_Rin JORDAN20:def 2 :

:: deftheorem defines is_Lout JORDAN20:def 3 :

:: deftheorem defines is_Rout JORDAN20:def 4 :

:: deftheorem defines is_OSin JORDAN20:def 5 :

:: deftheorem defines is_OSout JORDAN20:def 6 :

theorem :: JORDAN20:8

for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 <= e & p2 `1 >= e holds
ex p3 being Point of (TOP-REAL 2) st
( p3 in P & p3 `1 = e )

proof end;

theorem :: JORDAN20:9

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e & not p is_Rin P,p1,p2,e holds
p is_OSin P,p1,p2,e

proof end;

theorem :: JORDAN20:10

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e & not p is_Rout P,p1,p2,e holds
p is_OSout P,p1,p2,e

proof end;

theorem Th11: :: JORDAN20:11

for P being Subset of I[01]
for s being Real st P = [.0,s.[ holds
P is open

proof end;

theorem Th12: :: JORDAN20:12

for P being Subset of I[01]
for s being Real st P = ].s,1.] holds
P is open

proof end;

theorem Th13: :: JORDAN20:13

for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for Q being Subset of I[01]
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ holds
f .: Q = P1

proof end;

theorem Th14: :: JORDAN20:14

for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for Q being Subset of I[01]
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] holds
f .: Q = P1

proof end;

Lm1: [#] I[01] <> {}
;

theorem Th15: :: JORDAN20:15

for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } holds
P1 is open

proof end;

theorem Th16: :: JORDAN20:16

for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } holds
P1 is open

proof end;

theorem Th17: :: JORDAN20:17

for T being non empty TopStruct
for Q1, Q2 being Subset of T
for p1, p2 being Point of T st Q1 /\ Q2 = {} & Q1 \/ Q2 = the carrier of T & p1 in Q1 & p2 in Q2 & Q1 is open & Q2 is open holds
for P being Function of I[01],T holds
( not P is continuous or not P . 0 = p1 or not P . 1 = p2 )

proof end;

theorem Th18: :: JORDAN20:18

for P being non empty Subset of (TOP-REAL 2)
for Q being Subset of ((TOP-REAL 2) | P)
for p1, p2, q being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q in P & q <> p1 & q <> p2 & Q = P \ {q} holds
( not Q is connected & ( for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) )

proof end;

theorem Th19: :: JORDAN20:19

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & q2 in P & not LE q1,q2,P,p1,p2 holds
LE q2,q1,P,p1,p2

proof end;

theorem Th20: :: JORDAN20:20

for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for P, P1 being non empty Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 & P1 is_an_arc_of p1,p2 & P1 c= P holds
P1 = P

proof end;

theorem Th21: :: JORDAN20:21

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & p2 <> q1 holds
Segment (P,p1,p2,q1,p2) is_an_arc_of q1,p2

proof end;

theorem Th22: :: JORDAN20:22

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
(Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) = Segment (P,p1,p2,q1,q3)

proof end;

theorem :: JORDAN20:23

proof end;

theorem Th24: :: JORDAN20:24

for P being non empty Subset of (TOP-REAL 2)
for p1, p2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 holds
Segment (P,p1,p2,p1,p2) = P

proof end;

theorem Th25: :: JORDAN20:25

for P, Q1 being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & Q1 is_an_arc_of q1,q2 & LE q1,q2,P,p1,p2 & Q1 c= P holds
Q1 = Segment (P,p1,p2,q1,q2)

proof end;

theorem :: JORDAN20:26

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Lin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Lin P,p1,p2,e

proof end;

theorem :: JORDAN20:27

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Rin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Rin P,p1,p2,e

proof end;

theorem :: JORDAN20:28

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Lout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Lout P,p1,p2,e

proof end;

theorem :: JORDAN20:29

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Rout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Rout P,p1,p2,e

proof end;

theorem :: JORDAN20:30

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_S-P_arc_joining p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e holds
p is_Rin P,p1,p2,e

proof end;

theorem :: JORDAN20:31

for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_S-P_arc_joining p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e holds
p is_Rout P,p1,p2,e

proof end;