:: Subspaces and Cosets of Subspace of Real Unitary Space
:: by Noboru Endou , Takashi Mitsuishi and Yasunari Shidama
::
:: Received October 9, 2002
:: Copyright (c) 2002-2021 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 BHSP_1, RLSUB_1, STRUCT_0, TARSKI, SUPINF_2, ALGSTR_0, REALSET1,
RLVECT_1, RELAT_1, ZFMISC_1, NUMBERS, ARYTM_3, FUNCT_1, REAL_1, PROB_2,
ARYTM_1, SUBSET_1, XBOOLE_0, BINOP_1, CARD_1, XXREAL_0;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, STRUCT_0, ALGSTR_0, ORDINAL1,
NUMBERS, XCMPLX_0, XREAL_0, MCART_1, RELAT_1, FUNCT_1, REAL_1, FUNCT_2,
DOMAIN_1, BINOP_1, REALSET1, RLVECT_1, RLSUB_1, BHSP_1, XXREAL_0;
constructors PARTFUN1, BINOP_1, XXREAL_0, REAL_1, REALSET1, RLSUB_1, BHSP_1,
VALUED_1, RELSET_1, NUMBERS;
registrations XBOOLE_0, SUBSET_1, FUNCT_1, RELSET_1, NUMBERS, MEMBERED,
REALSET1, STRUCT_0, BHSP_1, VALUED_0, ALGSTR_0, XREAL_0;
requirements NUMERALS, SUBSET, BOOLE, ARITHM;
begin :: Definition and Axioms of the Subspace of Real Unitary Space
definition
let V be RealUnitarySpace;
mode Subspace of V -> RealUnitarySpace means
:: RUSUB_1:def 1
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:]) & the scalar
of it = (the scalar of V)||the carrier of it;
end;
theorem :: RUSUB_1:1
for V being RealUnitarySpace, W1,W2 being Subspace of V, x being object
st x in W1 & W1 is Subspace of W2 holds x in W2;
theorem :: RUSUB_1:2
for V being RealUnitarySpace, W being Subspace of V, x being object
st x in W holds x in V;
theorem :: RUSUB_1:3
for V being RealUnitarySpace, W being Subspace of V, w being
VECTOR of W holds w is VECTOR of V;
theorem :: RUSUB_1:4
for V being RealUnitarySpace, W being Subspace of V holds 0.W = 0.V;
theorem :: RUSUB_1:5
for V being RealUnitarySpace, W1,W2 being Subspace of V holds 0.W1 = 0.W2;
theorem :: RUSUB_1:6
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V, w1,w2 being VECTOR of W st w1 = v & w2 = u holds w1 + w2 = v + u
;
theorem :: RUSUB_1:7
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V, w being VECTOR of W, a being Real st w = v
holds a * w = a * v;
theorem :: RUSUB_1:8
for V being RealUnitarySpace, W being Subspace of V, v1,v2 being
VECTOR of V, w1,w2 being VECTOR of W st w1 = v1 & w2 = v2 holds w1 .|. w2 = v1
.|. v2;
theorem :: RUSUB_1:9
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V, w being VECTOR of W st w = v holds - v = - w;
theorem :: RUSUB_1:10
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V, w1,w2 being VECTOR of W st w1 = v & w2 = u holds w1 - w2 = v - u
;
theorem :: RUSUB_1:11
for V being RealUnitarySpace, W being Subspace of V holds 0.V in W;
theorem :: RUSUB_1:12
for V being RealUnitarySpace, W1,W2 being Subspace of V holds 0.W1 in W2;
theorem :: RUSUB_1:13
for V being RealUnitarySpace, W being Subspace of V holds 0.W in V;
theorem :: RUSUB_1:14
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V st u in W & v in W holds u + v in W;
theorem :: RUSUB_1:15
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V, a being Real st v in W holds a * v in W;
theorem :: RUSUB_1:16
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V st v in W holds - v in W;
theorem :: RUSUB_1:17
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V st u in W & v in W holds u - v in W;
theorem :: RUSUB_1:18
for V being RealUnitarySpace, V1 being Subset of V, D being non
empty set, d1 being Element of D, A being BinOp of D, M being Function of [:
REAL, D:], D, S being Function of [:D,D:],REAL st V1 = D & d1 = 0.V & A = (the
addF of V)||V1 & M = (the Mult of V) | [:REAL,V1:] & S = (the scalar of V)||V1
holds UNITSTR (# D,d1,A,M,S #) is Subspace of V;
theorem :: RUSUB_1:19
for V being RealUnitarySpace holds V is Subspace of V;
theorem :: RUSUB_1:20
for V,X being strict RealUnitarySpace holds V is Subspace of X &
X is Subspace of V implies V = X;
theorem :: RUSUB_1:21
for V,X,Y being RealUnitarySpace st V is Subspace of X & X is
Subspace of Y holds V is Subspace of Y;
theorem :: RUSUB_1:22
for V being RealUnitarySpace, W1,W2 being Subspace of V st the
carrier of W1 c= the carrier of W2 holds W1 is Subspace of W2;
theorem :: RUSUB_1:23
for V being RealUnitarySpace, W1,W2 being Subspace of V st (for v
being VECTOR of V st v in W1 holds v in W2) holds W1 is Subspace of W2;
registration
let V be RealUnitarySpace;
cluster strict for Subspace of V;
end;
theorem :: RUSUB_1:24
for V being RealUnitarySpace, W1,W2 being strict Subspace of V
st the carrier of W1 = the carrier of W2 holds W1 = W2;
theorem :: RUSUB_1:25
for V being RealUnitarySpace, W1,W2 being strict Subspace of V
st (for v being VECTOR of V holds v in W1 iff v in W2) holds W1 = W2;
theorem :: RUSUB_1:26
for V being strict RealUnitarySpace, W being strict Subspace of V st
the carrier of W = the carrier of V holds W = V;
theorem :: RUSUB_1:27
for V being strict RealUnitarySpace, W being strict Subspace of V st (
for v being VECTOR of V holds v in W iff v in V) holds W = V;
theorem :: RUSUB_1:28
for V being RealUnitarySpace, W being Subspace of V, V1 being Subset
of V st the carrier of W = V1 holds V1 is linearly-closed;
theorem :: RUSUB_1:29
for V being RealUnitarySpace, V1 being Subset of V st V1 <> {} &
V1 is linearly-closed holds ex W being strict Subspace of V st V1 = the carrier
of W;
begin
:: Definition of Zero Subspace and Improper Subspace of Real Unitary Space
definition
let V being RealUnitarySpace;
func (0).V -> strict Subspace of V means
:: RUSUB_1:def 2
the carrier of it = {0.V};
end;
definition
let V being RealUnitarySpace;
func (Omega).V -> strict Subspace of V equals
:: RUSUB_1:def 3
the UNITSTR of V;
end;
begin :: Theorems of Zero Subspace and Improper Subspace
theorem :: RUSUB_1:30
for V being RealUnitarySpace, W being Subspace of V holds (0).W = (0).V;
theorem :: RUSUB_1:31
for V being RealUnitarySpace, W1,W2 being Subspace of V holds (0).W1 = (0).W2
;
theorem :: RUSUB_1:32
for V being RealUnitarySpace, W being Subspace of V holds (0).W is
Subspace of V;
theorem :: RUSUB_1:33
for V being RealUnitarySpace, W being Subspace of V holds (0).V is
Subspace of W;
theorem :: RUSUB_1:34
for V being RealUnitarySpace, W1,W2 being Subspace of V holds (0).W1
is Subspace of W2;
theorem :: RUSUB_1:35
for V being strict RealUnitarySpace holds V is Subspace of (Omega).V;
begin :: The Cosets of Subspace of Real Unitary Space
definition
let V be RealUnitarySpace, v be VECTOR of V, W be Subspace of V;
func v + W -> Subset of V equals
:: RUSUB_1:def 4
{v + u where u is VECTOR of V : u in W};
end;
definition
let V be RealUnitarySpace;
let W be Subspace of V;
mode Coset of W -> Subset of V means
:: RUSUB_1:def 5
ex v be VECTOR of V st it = v + W;
end;
begin :: Theorems of the Cosets
theorem :: RUSUB_1:36
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V holds 0.V in v + W iff v in W;
theorem :: RUSUB_1:37
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V holds v in v + W;
theorem :: RUSUB_1:38
for V being RealUnitarySpace, W being Subspace of V holds 0.V + W =
the carrier of W;
theorem :: RUSUB_1:39
for V being RealUnitarySpace, v being VECTOR of V holds v + (0). V = {v};
theorem :: RUSUB_1:40
for V being RealUnitarySpace, v being VECTOR of V holds v +
(Omega).V = the carrier of V;
theorem :: RUSUB_1:41
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V holds 0.V in v + W iff v + W = the carrier of W;
theorem :: RUSUB_1:42
for V being RealUnitarySpace, W being Subspace of V, v being VECTOR of
V holds v in W iff v + W = the carrier of W;
theorem :: RUSUB_1:43
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V, a being Real st v in W holds (a * v) + W = the carrier of W;
theorem :: RUSUB_1:44
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V, a being Real st a <> 0 & (a * v) + W = the carrier of W holds v in
W;
theorem :: RUSUB_1:45
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V holds v in W iff - v + W = the carrier of W;
theorem :: RUSUB_1:46
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V holds u in W iff v + W = (v + u) + W;
theorem :: RUSUB_1:47
for V being RealUnitarySpace, W being Subspace of V, u,v being VECTOR
of V holds u in W iff v + W = (v - u) + W;
theorem :: RUSUB_1:48
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V holds v in u + W iff u + W = v + W;
theorem :: RUSUB_1:49
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V holds v + W = (- v) + W iff v in W;
theorem :: RUSUB_1:50
for V being RealUnitarySpace, W being Subspace of V, u,v1,v2
being VECTOR of V st u in v1 + W & u in v2 + W holds v1 + W = v2 + W;
theorem :: RUSUB_1:51
for V being RealUnitarySpace, W being Subspace of V, u,v being VECTOR
of V st u in v + W & u in (- v) + W holds v in W;
theorem :: RUSUB_1:52
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V, a being Real st a <> 1 & a * v in v + W holds v in W;
theorem :: RUSUB_1:53
for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V, a being Real st v in W holds a * v in v + W;
theorem :: RUSUB_1:54
for V being RealUnitarySpace, W being Subspace of V, v being VECTOR of
V holds - v in v + W iff v in W;
theorem :: RUSUB_1:55
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V holds u + v in v + W iff u in W;
theorem :: RUSUB_1:56
for V being RealUnitarySpace, W being Subspace of V, u,v being VECTOR
of V holds v - u in v + W iff u in W;
theorem :: RUSUB_1:57
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V holds u in v + W iff ex v1 being VECTOR of V st v1 in W & u = v +
v1;
theorem :: RUSUB_1:58
for V being RealUnitarySpace, W being Subspace of V, u,v being VECTOR
of V holds u in v + W iff ex v1 being VECTOR of V st v1 in W & u = v - v1;
theorem :: RUSUB_1:59
for V being RealUnitarySpace, W being Subspace of V, v1,v2 being
VECTOR of V holds (ex v being VECTOR of V st v1 in v + W & v2 in v + W) iff v1
- v2 in W;
theorem :: RUSUB_1:60
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V st v + W = u + W holds ex v1 being VECTOR of V st v1 in W & v + v1
= u;
theorem :: RUSUB_1:61
for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V st v + W = u + W holds ex v1 being VECTOR of V st v1 in W & v - v1
= u;
theorem :: RUSUB_1:62
for V being RealUnitarySpace, W1,W2 being strict Subspace of V,
v being VECTOR of V holds v + W1 = v + W2 iff W1 = W2;
theorem :: RUSUB_1:63
for V being RealUnitarySpace, W1,W2 being strict Subspace of V,
u,v being VECTOR of V st v + W1 = u + W2 holds W1 = W2;
theorem :: RUSUB_1:64
for V being RealUnitarySpace, W being Subspace of V, C being Coset of
W holds C is linearly-closed iff C = the carrier of W;
theorem :: RUSUB_1:65
for V being RealUnitarySpace, W1,W2 being strict Subspace of V, C1
being Coset of W1, C2 being Coset of W2 holds C1 = C2 implies W1 = W2;
theorem :: RUSUB_1:66
for V being RealUnitarySpace, v being VECTOR of V holds {v} is Coset of (0).V
;
theorem :: RUSUB_1:67
for V being RealUnitarySpace, V1 being Subset of V holds V1 is Coset
of (0).V implies ex v being VECTOR of V st V1 = {v};
theorem :: RUSUB_1:68
for V being RealUnitarySpace, W being Subspace of V holds the carrier
of W is Coset of W;
theorem :: RUSUB_1:69
for V being RealUnitarySpace holds the carrier of V is Coset of (Omega).V;
theorem :: RUSUB_1:70
for V being RealUnitarySpace, V1 being Subset of V st V1 is Coset of
(Omega).V holds V1 = the carrier of V;
theorem :: RUSUB_1:71
for V being RealUnitarySpace, W being Subspace of V, C being Coset of
W holds 0.V in C iff C = the carrier of W;
theorem :: RUSUB_1:72
for V being RealUnitarySpace, W being Subspace of V, C being
Coset of W, u being VECTOR of V holds u in C iff C = u + W;
theorem :: RUSUB_1:73
for V being RealUnitarySpace, W being Subspace of V, C being Coset of
W, u,v being VECTOR of V st u in C & v in C holds ex v1 being VECTOR of V st v1
in W & u + v1 = v;
theorem :: RUSUB_1:74
for V being RealUnitarySpace, W being Subspace of V, C being Coset of
W, u,v being VECTOR of V st u in C & v in C holds ex v1 being VECTOR of V st v1
in W & u - v1 = v;
theorem :: RUSUB_1:75
for V being RealUnitarySpace, W being Subspace of V, v1,v2 being
VECTOR of V holds (ex C being Coset of W st v1 in C & v2 in C) iff v1 - v2 in W
;
theorem :: RUSUB_1:76
for V being RealUnitarySpace, W being Subspace of V, u being VECTOR of
V, B,C being Coset of W st u in B & u in C holds B = C;