:: Subspaces and Cosets of Subspaces in Real Linear Space
::  by Wojciech A. Trybulec
::
:: Received July 24, 1989
:: Copyright (c) 1990-2021 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 RLVECT_1, REAL_1, SUBSET_1, ARYTM_3, RELAT_1, XBOOLE_0, SUPINF_2,
      CARD_1, ARYTM_1, STRUCT_0, TARSKI, ALGSTR_0, REALSET1, ZFMISC_1, NUMBERS,
      FUNCT_1, BINOP_1, RLSUB_1;
 notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,
      XREAL_0, REAL_1, MCART_1, RELAT_1, FUNCT_1, FUNCT_2, BINOP_1, REALSET1,
      DOMAIN_1, STRUCT_0, ALGSTR_0, RLVECT_1;
 constructors PARTFUN1, BINOP_1, REAL_1, NAT_1, REALSET1, RLVECT_1, RELSET_1,
      NUMBERS;
 registrations XBOOLE_0, SUBSET_1, FUNCT_1, RELSET_1, NUMBERS, REALSET1,
      STRUCT_0, RLVECT_1, ORDINAL1, ALGSTR_0, XREAL_0;
 requirements NUMERALS, BOOLE, SUBSET, ARITHM;


begin

reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;

::
::  Introduction of predicate linearly closed subsets of the carrier.
::

definition
  let V;
  let V1;
  attr V1 is linearly-closed means
:: RLSUB_1:def 1

  (for v,u st v in V1 & u in V1 holds
  v + u in V1) & for a,v st v in V1 holds a * v in V1;
end;

theorem :: RLSUB_1:1
  V1 <> {} & V1 is linearly-closed implies 0.V in V1;

theorem :: RLSUB_1:2
  V1 is linearly-closed implies for v st v in V1 holds - v in V1;

theorem :: RLSUB_1:3
  V1 is linearly-closed implies for v,u st v in V1 & u in V1 holds v - u in V1;

theorem :: RLSUB_1:4
  {0.V} is linearly-closed;

theorem :: RLSUB_1:5
  the carrier of V = V1 implies V1 is linearly-closed;

theorem :: RLSUB_1:6
  V1 is linearly-closed & V2 is linearly-closed & V3 = {v + u : v in V1
  & u in V2} implies V3 is linearly-closed;

theorem :: RLSUB_1:7
  V1 is linearly-closed & V2 is linearly-closed implies V1 /\ V2 is
  linearly-closed;

definition
  let V;
  mode Subspace of V -> RealLinearSpace means
:: RLSUB_1:def 2

    the carrier of it c= the
carrier of V & 0.it = 0.V & the addF of it = (the addF of V)||the carrier of it
    & the Mult of it = (the Mult of V) | [:REAL, the carrier of it:];
end;

reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;

::
::  Axioms of the subspaces of real linear spaces.
::

theorem :: RLSUB_1:8
  x in W1 & W1 is Subspace of W2 implies x in W2;

theorem :: RLSUB_1:9
  x in W implies x in V;

theorem :: RLSUB_1:10
  w is VECTOR of V;

theorem :: RLSUB_1:11
  0.W = 0.V;

theorem :: RLSUB_1:12
  0.W1 = 0.W2;

theorem :: RLSUB_1:13
  w1 = v & w2 = u implies w1 + w2 = v + u;

theorem :: RLSUB_1:14
  w = v implies a * w = a * v;

theorem :: RLSUB_1:15
  w = v implies - v = - w;

theorem :: RLSUB_1:16
  w1 = v & w2 = u implies w1 - w2 = v - u;

theorem :: RLSUB_1:17
  0.V in W;

theorem :: RLSUB_1:18
  0.W1 in W2;

theorem :: RLSUB_1:19
  0.W in V;

theorem :: RLSUB_1:20
  u in W & v in W implies u + v in W;

theorem :: RLSUB_1:21
  v in W implies a * v in W;

theorem :: RLSUB_1:22
  v in W implies - v in W;

theorem :: RLSUB_1:23
  u in W & v in W implies u - v in W;

reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;

theorem :: RLSUB_1:24
  V1 = D & d1 = 0.V & A = (the addF of V)||V1 & M = (the Mult of V
  ) | [:REAL,V1:] implies RLSStruct (# D,d1,A,M #) is Subspace of V;

theorem :: RLSUB_1:25
  V is Subspace of V;

theorem :: RLSUB_1:26
  for V,X being strict RealLinearSpace holds V is Subspace of X &
  X is Subspace of V implies V = X;

theorem :: RLSUB_1:27
  V is Subspace of X & X is Subspace of Y implies V is Subspace of Y;

theorem :: RLSUB_1:28
  the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2;

theorem :: RLSUB_1:29
  (for v st v in W1 holds v in W2) implies W1 is Subspace of W2;

registration
  let V;
  cluster strict for Subspace of V;
end;

theorem :: RLSUB_1:30
  for W1,W2 being strict Subspace of V holds the carrier of W1 =
  the carrier of W2 implies W1 = W2;

theorem :: RLSUB_1:31
  for W1,W2 being strict Subspace of V holds (for v holds v in W1
  iff v in W2) implies W1 = W2;

theorem :: RLSUB_1:32
  for V being strict RealLinearSpace, W being strict Subspace of V holds
  the carrier of W = the carrier of V implies W = V;

theorem :: RLSUB_1:33
  for V being strict RealLinearSpace, W being strict Subspace of V holds
  (for v being VECTOR of V holds v in W iff v in V) implies W = V;

theorem :: RLSUB_1:34
  the carrier of W = V1 implies V1 is linearly-closed;

theorem :: RLSUB_1:35
  V1 <> {} & V1 is linearly-closed implies ex W being strict
  Subspace of V st V1 = the carrier of W;

::
::  Definition of zero subspace and improper subspace of real linear space.
::

definition
  let V;
  func (0).V -> strict Subspace of V means
:: RLSUB_1:def 3

  the carrier of it = {0.V};
end;

definition
  let V;
  func (Omega).V -> strict Subspace of V equals
:: RLSUB_1:def 4
  the RLSStruct of V;
end;

::
::  Definitional theorems of zero subspace and improper subspace.
::

theorem :: RLSUB_1:36
  (0).W = (0).V;

theorem :: RLSUB_1:37
  (0).W1 = (0).W2;

theorem :: RLSUB_1:38
  (0).W is Subspace of V;

theorem :: RLSUB_1:39
  (0).V is Subspace of W;

theorem :: RLSUB_1:40
  (0).W1 is Subspace of W2;

theorem :: RLSUB_1:41
  for V being strict RealLinearSpace holds V is Subspace of (Omega).V;

::
::  Introduction of the cosets of subspace.
::

definition
  let V;
  let v,W;
  func v + W -> Subset of V equals
:: RLSUB_1:def 5
  {v + u : u in W};
end;

definition
  let V;
  let W;
  mode Coset of W -> Subset of V means
:: RLSUB_1:def 6

    ex v st it = v + W;
end;

reserve B,C for Coset of W;

::
::  Definitional theorems of the cosets.
::

theorem :: RLSUB_1:42
  0.V in v + W iff v in W;

theorem :: RLSUB_1:43
  v in v + W;

theorem :: RLSUB_1:44
  0.V + W = the carrier of W;

theorem :: RLSUB_1:45
  v + (0).V = {v};

theorem :: RLSUB_1:46
  v + (Omega).V = the carrier of V;

theorem :: RLSUB_1:47
  0.V in v + W iff v + W = the carrier of W;

theorem :: RLSUB_1:48
  v in W iff v + W = the carrier of W;

theorem :: RLSUB_1:49
  v in W implies (a * v) + W = the carrier of W;

theorem :: RLSUB_1:50
  a <> 0 & (a * v) + W = the carrier of W implies v in W;

theorem :: RLSUB_1:51
  v in W iff - v + W = the carrier of W;

theorem :: RLSUB_1:52
  u in W iff v + W = (v + u) + W;

theorem :: RLSUB_1:53
  u in W iff v + W = (v - u) + W;

theorem :: RLSUB_1:54
  v in u + W iff u + W = v + W;

theorem :: RLSUB_1:55
  v + W = (- v) + W iff v in W;

theorem :: RLSUB_1:56
  u in v1 + W & u in v2 + W implies v1 + W = v2 + W;

theorem :: RLSUB_1:57
  u in v + W & u in (- v) + W implies v in W;

theorem :: RLSUB_1:58
  a <> 1 & a * v in v + W implies v in W;

theorem :: RLSUB_1:59
  v in W implies a * v in v + W;

theorem :: RLSUB_1:60
  - v in v + W iff v in W;

theorem :: RLSUB_1:61
  u + v in v + W iff u in W;

theorem :: RLSUB_1:62
  v - u in v + W iff u in W;

theorem :: RLSUB_1:63
  u in v + W iff ex v1 st v1 in W & u = v + v1;

theorem :: RLSUB_1:64
  u in v + W iff ex v1 st v1 in W & u = v - v1;

theorem :: RLSUB_1:65
  (ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W;

theorem :: RLSUB_1:66
  v + W = u + W implies ex v1 st v1 in W & v + v1 = u;

theorem :: RLSUB_1:67
  v + W = u + W implies ex v1 st v1 in W & v - v1 = u;

theorem :: RLSUB_1:68
  for W1,W2 being strict Subspace of V holds v + W1 = v + W2 iff W1 = W2;

theorem :: RLSUB_1:69
  for W1,W2 being strict Subspace of V holds v + W1 = u + W2 implies W1 = W2;

::
::  Theorems concerning cosets of subspace
::  regarded as subsets of the carrier.
::

theorem :: RLSUB_1:70
  C is linearly-closed iff C = the carrier of W;

theorem :: RLSUB_1:71
  for W1,W2 being strict Subspace of V, C1 being Coset of W1, C2 being
  Coset of W2 holds C1 = C2 implies W1 = W2;

theorem :: RLSUB_1:72
  {v} is Coset of (0).V;

theorem :: RLSUB_1:73
  V1 is Coset of (0).V implies ex v st V1 = {v};

theorem :: RLSUB_1:74
  the carrier of W is Coset of W;

theorem :: RLSUB_1:75
  the carrier of V is Coset of (Omega).V;

theorem :: RLSUB_1:76
  V1 is Coset of (Omega).V implies V1 = the carrier of V;

theorem :: RLSUB_1:77
  0.V in C iff C = the carrier of W;

theorem :: RLSUB_1:78
  u in C iff C = u + W;

theorem :: RLSUB_1:79
  u in C & v in C implies ex v1 st v1 in W & u + v1 = v;

theorem :: RLSUB_1:80
  u in C & v in C implies ex v1 st v1 in W & u - v1 = v;

theorem :: RLSUB_1:81
  (ex C st v1 in C & v2 in C) iff v1 - v2 in W;

theorem :: RLSUB_1:82
  u in B & u in C implies B = C;