:: Mazur-Ulam Theorem
:: by Artur Korni{\l}owicz
::
:: Received December 21, 2010
:: Copyright (c) 2010-2019 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies FUNCT_1, NORMSP_1, VECTMETR, PRE_TOPC, ARYTM_1, RUSUB_4, XREAL_0,
CARD_1, XXREAL_0, RELAT_1, ARYTM_3, FUNCT_2, SEQ_2, MEMBERED, ORDINAL2,
MAZURULM, BINOP_1, VALUED_1, SUPINF_2, COMPLEX1, SUBSET_1, XXREAL_2,
XBOOLE_0, NUMBERS, MEMBER_1, SUPINF_1, TARSKI, RLVECT_1, CFCONT_1, NAT_1,
PARTFUN1, REAL_1, NEWTON, TOPMETR, BORSUK_1, TOPS_1, XXREAL_1, URYSOHN1,
METRIC_1, STRUCT_0, SEQ_1, FUNCOP_1, RCOMP_1, XCMPLX_0;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, FUNCT_1, RELSET_1, PARTFUN1,
FUNCT_2, FUNCOP_1, NUMBERS, XCMPLX_0, XXREAL_0, XXREAL_1, XXREAL_3,
XREAL_0, REAL_1, XXREAL_2, MEMBERED, VALUED_1, SEQ_1, SEQ_2, COMPLEX1,
MEMBER_1, SUPINF_1, NEWTON, MEASURE6, STRUCT_0, PRE_TOPC, TOPS_1, TOPS_2,
BORSUK_1, METRIC_1, TOPMETR, ALGSTR_0, RLVECT_1, NORMSP_0, NORMSP_1,
NDIFF_1, NFCONT_1, CONNSP_3, URYSOHN1;
constructors TOPS_2, REAL_1, SUPINF_2, INTEGRA2, EXTREAL1, BINOP_2, NFCONT_1,
NEWTON, CONNSP_3, URYSOHN1, TOPS_1, TOPMETR, PCOMPS_1, MEASURE6, NDIFF_1,
COMSEQ_2;
registrations RELSET_1, XREAL_0, ORDINAL1, STRUCT_0, MEMBERED, FUNCT_1,
FUNCT_2, XBOOLE_0, NORMSP_1, FUNCTOR1, NUMBERS, SEQ_4, XXREAL_2,
MEMBER_1, XXREAL_0, RLVECT_1, TOPS_1, TOPMETR, METRIC_1, XXREAL_1,
VALUED_1, VALUED_0, FUNCOP_1, BORSUK_1, INT_1, NEWTON, XXREAL_3, NAT_1,
SEQ_1, SEQ_2;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
begin
:: Mazur-Ulam Theorem
:: cluster bijective isometric -> Affine Function of E,F;
:: is a consequence of two registrations:
:: cluster bijective isometric -> midpoints-preserving Function of E,F;
:: cluster isometric midpoints-preserving -> Affine Function of E,F;
reserve E,F,G for RealNormSpace;
reserve f for Function of E,F;
reserve g for Function of F,G;
reserve a,b,c for Point of E;
reserve t for Real;
registration
cluster I[01] -> closed for SubSpace of R^1;
end;
theorem :: MAZURULM:1
DYADIC is dense Subset of I[01];
theorem :: MAZURULM:2
Cl DYADIC = [. 0,1 .];
theorem :: MAZURULM:3
a + a = 2*a;
theorem :: MAZURULM:4
a + b - b = a;
registration
let A be bounded_above real-membered set;
let r be non negative Real;
cluster r**A -> bounded_above;
end;
registration
let A be bounded_above real-membered set;
let r be non positive Real;
cluster r**A -> bounded_below;
end;
registration
let A be bounded_below real-membered set;
let r be non negative Real;
cluster r**A -> bounded_below;
end;
registration
let A be bounded_below non empty real-membered set;
let r be non positive Real;
cluster r**A -> bounded_above;
end;
theorem :: MAZURULM:5
for f being Real_Sequence holds f + (NAT --> t) = t + f;
theorem :: MAZURULM:6
for r being Element of REAL holds lim (NAT --> r) = r;
theorem :: MAZURULM:7
for f being convergent Real_Sequence holds
lim (t + f) = t + lim f;
registration
let f be convergent Real_Sequence;
let t;
cluster t + f -> convergent for Real_Sequence;
end;
theorem :: MAZURULM:8
for f be Real_Sequence holds f (#) (NAT --> a) = f * a;
theorem :: MAZURULM:9
lim (NAT --> a) = a;
theorem :: MAZURULM:10
for f being convergent Real_Sequence holds
lim (f * a) = (lim f) * a;
registration
let f be convergent Real_Sequence;
let E,a;
cluster f * a -> convergent;
end;
definition
let E,F be non empty NORMSTR, f be Function of E,F;
attr f is isometric means
:: MAZURULM:def 1
for a,b being Point of E holds ||. f.a - f.b .|| = ||. a - b .||;
end;
definition
let E,F be non empty RLSStruct, f be Function of E,F;
attr f is Affine means
:: MAZURULM:def 2
for a,b being Point of E, t being Real st 0 <= t & t <= 1 holds
f.((1-t)*a+t*b) = (1-t)*(f.a) + t*(f.b);
attr f is midpoints-preserving means
:: MAZURULM:def 3
for a,b being Point of E holds f.(1/2*(a+b)) = 1/2*(f.a+f.b);
end;
registration
let E be non empty NORMSTR;
cluster id E -> isometric;
end;
registration
let E be non empty RLSStruct;
cluster id E -> midpoints-preserving Affine;
end;
registration
let E be non empty NORMSTR;
cluster bijective isometric midpoints-preserving Affine for UnOp of E;
end;
theorem :: MAZURULM:11
f is isometric & g is isometric implies g*f is isometric;
registration
let E; let f,g be isometric UnOp of E;
cluster g*f -> isometric for UnOp of E;
end;
theorem :: MAZURULM:12
f is bijective isometric implies f/" is isometric;
registration
let E; let f be bijective isometric UnOp of E;
cluster f/" -> isometric;
end;
theorem :: MAZURULM:13
f is midpoints-preserving & g is midpoints-preserving implies
g*f is midpoints-preserving;
registration
let E; let f,g be midpoints-preserving UnOp of E;
cluster g*f -> midpoints-preserving for UnOp of E;
end;
theorem :: MAZURULM:14
f is bijective midpoints-preserving implies f/" is midpoints-preserving;
registration
let E; let f be bijective midpoints-preserving UnOp of E;
cluster f/" -> midpoints-preserving;
end;
theorem :: MAZURULM:15
f is Affine & g is Affine implies g*f is Affine;
registration
let E; let f,g be Affine UnOp of E;
cluster g*f -> Affine for UnOp of E;
end;
theorem :: MAZURULM:16
f is bijective Affine implies f/" is Affine;
registration
let E; let f be bijective Affine UnOp of E;
cluster f/" -> Affine;
end;
definition
let E be non empty RLSStruct, a be Point of E;
func a-reflection -> UnOp of E means
:: MAZURULM:def 4
for b being Point of E holds it.b = 2*a - b;
end;
theorem :: MAZURULM:17
(a-reflection) * (a-reflection) = id E;
registration
let E,a;
cluster a-reflection -> bijective;
end;
theorem :: MAZURULM:18 :: a is the only fixed point of R
(a-reflection).a = a & for b st (a-reflection).b = b holds a = b;
theorem :: MAZURULM:19
(a-reflection).b - a = a - b;
theorem :: MAZURULM:20
||. (a-reflection).b - a .|| = ||. b - a .||;
theorem :: MAZURULM:21
(a-reflection).b - b = 2 * (a - b);
theorem :: MAZURULM:22
||. (a-reflection).b - b .|| = 2 * ||. b - a .||;
theorem :: MAZURULM:23
(a-reflection)/" = a-reflection;
registration
let E,a;
cluster a-reflection -> isometric;
end;
theorem :: MAZURULM:24
f is isometric implies f is_continuous_on dom f;
registration
let E,F;
cluster bijective isometric -> midpoints-preserving for Function of E,F;
end;
registration
let E,F;
cluster isometric midpoints-preserving -> Affine for Function of E,F;
end;