:: The Brouwer Fixed Point Theorem for Intervals
:: by Toshihiko Watanabe
::
:: Received August 17, 1992
:: Copyright (c) 1992-2021 Association of Mizar Users

Lm1: for a, b being Real
for x being set st x in [.a,b.] holds
x is Element of REAL
;

Lm2: for a, b being Real
for x being Point of (Closed-Interval-TSpace (a,b)) st a <= b holds
x is Element of REAL

proof end;

theorem Th1: :: TREAL_1:1

for a, b being Real
for A being Subset of R^1 st A = [.a,b.] holds
A is closed

proof end;

theorem Th2: :: TREAL_1:2

for a, b being Real st a <= b holds
Closed-Interval-TSpace (a,b) is closed

proof end;

theorem :: TREAL_1:3

for a, b, c, d being Real st a <= c & d <= b & c <= d holds
Closed-Interval-TSpace (c,d) is closed SubSpace of Closed-Interval-TSpace (a,b)

proof end;

theorem :: TREAL_1:4

for a, b, c, d being Real st a <= c & b <= d & c <= b holds
( Closed-Interval-TSpace (a,d) = (Closed-Interval-TSpace (a,b)) union (Closed-Interval-TSpace (c,d)) & Closed-Interval-TSpace (c,b) = (Closed-Interval-TSpace (a,b)) meet (Closed-Interval-TSpace (c,d)) )

proof end;

definition

let a, b be Real;
assume A1: a <= b ;

func (#) (a,b) -> Point of (Closed-Interval-TSpace (a,b)) equals :Def1: :: TREAL_1:def 1
a;

coherence
a is Point of (Closed-Interval-TSpace (a,b))

proof end;

correctness
;

func (a,b) (#) -> Point of (Closed-Interval-TSpace (a,b)) equals :Def2: :: TREAL_1:def 2
b;

coherence
b is Point of (Closed-Interval-TSpace (a,b))

proof end;

correctness
;

end;

:: deftheorem Def1 defines (#) TREAL_1:def 1 :
for a, b being Real st a <= b holds
(#) (a,b) = a;

:: deftheorem Def2 defines (#) TREAL_1:def 2 :
for a, b being Real st a <= b holds
(a,b) (#) = b;

theorem :: TREAL_1:5

( 0[01] = (#) (0,1) & 1[01] = (0,1) (#) ) by Def1, Def2, BORSUK_1:def 14, BORSUK_1:def 15;

theorem :: TREAL_1:6

for a, b, c being Real st a <= b & b <= c holds
( (#) (a,b) = (#) (a,c) & (b,c) (#) = (a,c) (#) )

proof end;

:: 2. Continuous Mappings Between Topological Intervals.

definition

let a, b be Real;
assume A1: a <= b ;
let t1, t2 be Point of (Closed-Interval-TSpace (a,b));

func L[01] (t1,t2) -> Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace (a,b)) means :Def3: :: TREAL_1:def 3
for s being Point of (Closed-Interval-TSpace (0,1)) holds it . s = ((1 - s) * t1) + (s * t2);

existence
ex b₁ being Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace (a,b)) st
for s being Point of (Closed-Interval-TSpace (0,1)) holds b₁ . s = ((1 - s) * t1) + (s * t2)

proof end;

uniqueness
for b₁, b₂ being Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace (a,b)) st ( for s being Point of (Closed-Interval-TSpace (0,1)) holds b₁ . s = ((1 - s) * t1) + (s * t2) ) & ( for s being Point of (Closed-Interval-TSpace (0,1)) holds b₂ . s = ((1 - s) * t1) + (s * t2) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines L[01] TREAL_1:def 3 :
for a, b being Real st a <= b holds
for t1, t2 being Point of (Closed-Interval-TSpace (a,b))
for b₅ being Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace (a,b)) holds
( b₅ = L[01] (t1,t2) iff for s being Point of (Closed-Interval-TSpace (0,1)) holds b₅ . s = ((1 - s) * t1) + (s * t2) );

theorem Th7: :: TREAL_1:7

for a, b being Real st a <= b holds
for t1, t2 being Point of (Closed-Interval-TSpace (a,b))
for s being Point of (Closed-Interval-TSpace (0,1)) holds (L[01] (t1,t2)) . s = ((t2 - t1) * s) + t1

proof end;

theorem Th8: :: TREAL_1:8

for a, b being Real st a <= b holds
for t1, t2 being Point of (Closed-Interval-TSpace (a,b)) holds L[01] (t1,t2) is continuous

proof end;

theorem :: TREAL_1:9

for a, b being Real st a <= b holds
for t1, t2 being Point of (Closed-Interval-TSpace (a,b)) holds
( (L[01] (t1,t2)) . ((#) (0,1)) = t1 & (L[01] (t1,t2)) . ((0,1) (#)) = t2 )

proof end;

theorem :: TREAL_1:10

L[01] (((#) (0,1)),((0,1) (#))) = id (Closed-Interval-TSpace (0,1))

proof end;

definition

let a, b be Real;
assume A1: a < b ;
let t1, t2 be Point of (Closed-Interval-TSpace (0,1));

func P[01] (a,b,t1,t2) -> Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (0,1)) means :Def4: :: TREAL_1:def 4
for s being Point of (Closed-Interval-TSpace (a,b)) holds it . s = (((b - s) * t1) + ((s - a) * t2)) / (b - a);

existence
ex b₁ being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (0,1)) st
for s being Point of (Closed-Interval-TSpace (a,b)) holds b₁ . s = (((b - s) * t1) + ((s - a) * t2)) / (b - a)

proof end;

uniqueness
for b₁, b₂ being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (0,1)) st ( for s being Point of (Closed-Interval-TSpace (a,b)) holds b₁ . s = (((b - s) * t1) + ((s - a) * t2)) / (b - a) ) & ( for s being Point of (Closed-Interval-TSpace (a,b)) holds b₂ . s = (((b - s) * t1) + ((s - a) * t2)) / (b - a) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines P[01] TREAL_1:def 4 :
for a, b being Real st a < b holds
for t1, t2 being Point of (Closed-Interval-TSpace (0,1))
for b₅ being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (0,1)) holds
( b₅ = P[01] (a,b,t1,t2) iff for s being Point of (Closed-Interval-TSpace (a,b)) holds b₅ . s = (((b - s) * t1) + ((s - a) * t2)) / (b - a) );

theorem Th11: :: TREAL_1:11

for a, b being Real st a < b holds
for t1, t2 being Point of (Closed-Interval-TSpace (0,1))
for s being Point of (Closed-Interval-TSpace (a,b)) holds (P[01] (a,b,t1,t2)) . s = (((t2 - t1) / (b - a)) * s) + (((b * t1) - (a * t2)) / (b - a))

proof end;

theorem Th12: :: TREAL_1:12

for a, b being Real st a < b holds
for t1, t2 being Point of (Closed-Interval-TSpace (0,1)) holds P[01] (a,b,t1,t2) is continuous

proof end;

theorem :: TREAL_1:13

for a, b being Real st a < b holds
for t1, t2 being Point of (Closed-Interval-TSpace (0,1)) holds
( (P[01] (a,b,t1,t2)) . ((#) (a,b)) = t1 & (P[01] (a,b,t1,t2)) . ((a,b) (#)) = t2 )

proof end;

theorem :: TREAL_1:14

P[01] (0,1,((#) (0,1)),((0,1) (#))) = id (Closed-Interval-TSpace (0,1))

proof end;

theorem Th15: :: TREAL_1:15

for a, b being Real st a < b holds
( id (Closed-Interval-TSpace (a,b)) = (L[01] (((#) (a,b)),((a,b) (#)))) * (P[01] (a,b,((#) (0,1)),((0,1) (#)))) & id (Closed-Interval-TSpace (0,1)) = (P[01] (a,b,((#) (0,1)),((0,1) (#)))) * (L[01] (((#) (a,b)),((a,b) (#)))) )

proof end;

theorem Th16: :: TREAL_1:16

for a, b being Real st a < b holds
( id (Closed-Interval-TSpace (a,b)) = (L[01] (((a,b) (#)),((#) (a,b)))) * (P[01] (a,b,((0,1) (#)),((#) (0,1)))) & id (Closed-Interval-TSpace (0,1)) = (P[01] (a,b,((0,1) (#)),((#) (0,1)))) * (L[01] (((a,b) (#)),((#) (a,b)))) )

proof end;

theorem Th17: :: TREAL_1:17

for a, b being Real st a < b holds
( L[01] (((#) (a,b)),((a,b) (#))) is being_homeomorphism & (L[01] (((#) (a,b)),((a,b) (#)))) " = P[01] (a,b,((#) (0,1)),((0,1) (#))) & P[01] (a,b,((#) (0,1)),((0,1) (#))) is being_homeomorphism & (P[01] (a,b,((#) (0,1)),((0,1) (#)))) " = L[01] (((#) (a,b)),((a,b) (#))) )

proof end;

theorem :: TREAL_1:18

for a, b being Real st a < b holds
( L[01] (((a,b) (#)),((#) (a,b))) is being_homeomorphism & (L[01] (((a,b) (#)),((#) (a,b)))) " = P[01] (a,b,((0,1) (#)),((#) (0,1))) & P[01] (a,b,((0,1) (#)),((#) (0,1))) is being_homeomorphism & (P[01] (a,b,((0,1) (#)),((#) (0,1)))) " = L[01] (((a,b) (#)),((#) (a,b))) )

proof end;

:: 3. Connectedness of Intervals and Brouwer Fixed Point Theorem for Intervals.

theorem Th19: :: TREAL_1:19

I[01] is connected

proof end;

theorem :: TREAL_1:20

for a, b being Real st a <= b holds
Closed-Interval-TSpace (a,b) is connected

proof end;

theorem Th21: :: TREAL_1:21

for f being continuous Function of I[01],I[01] ex x being Point of I[01] st f . x = x

proof end;

theorem Th22: :: TREAL_1:22

for a, b being Real st a <= b holds
for f being continuous Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (a,b)) ex x being Point of (Closed-Interval-TSpace (a,b)) st f . x = x

proof end;

theorem Th23: :: TREAL_1:23

for X, Y being non empty SubSpace of R^1
for f being continuous Function of X,Y st ex a, b being Real st
( a <= b & [.a,b.] c= the carrier of X & [.a,b.] c= the carrier of Y & f .: [.a,b.] c= [.a,b.] ) holds
ex x being Point of X st f . x = x

proof end;

:: Brouwer Fixed Point Theorem for Intervals

theorem :: TREAL_1:24

for X, Y being non empty SubSpace of R^1
for f being continuous Function of X,Y st ex a, b being Real st
( a <= b & [.a,b.] c= the carrier of X & f .: [.a,b.] c= [.a,b.] ) holds
ex x being Point of X st f . x = x

proof end;