:: Rank of Submodule, Linear Transformations and Linearly Independent Subsets
:: of $\mathbb Z$-module
:: by Kazuhisa Nakasho , Yuichi Futa , Hiroyuki Okazaki and Yasunari Shidama
::
:: Received July 10, 2014
:: Copyright (c) 2014-2021 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 TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2,
FUNCOP_1, REALSET1, FUNCT_4, FINSET_1, CARD_1, PBOOLE, CARD_3, ARYTM_3,
ARYTM_1, NUMBERS, XXREAL_0, REAL_1, NAT_1, INT_1, VALUED_0, ORDINAL4,
GRCAT_1, MESFUNC1, MOD_3, CLASSES1, RFINSEQ, FINSEQ_1, VALUED_1,
SUPINF_2, MSSUBFAM, STRUCT_0, RLVECT_1, RLSUB_1, RLSUB_2, RLVECT_2,
RLVECT_3, RMOD_2, PRELAMB, UNIALG_1, MSAFREE2, RANKNULL, UPROOTS,
VECTSP10, ZMODUL01, ZMODUL03, ZMODUL05, SETWISEO, BINOP_2, VECTSP_1,
INT_3, FUNCSDOM, VECTSP_2;
notations TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1,
CLASSES1, RELSET_1, PARTFUN1, FUNCT_2, DOMAIN_1, FUNCOP_1, FUNCT_4,
FINSET_1, SETWOP_2, RFINSEQ, CARD_1, NUMBERS, XCMPLX_0, XXREAL_0,
XREAL_0, INT_1, VALUED_0, FINSEQ_1, FINSEQOP, RVSUM_1, FUNCT_7, STRUCT_0,
ALGSTR_0, GROUP_1, RLVECT_1, GR_CY_1, VECTSP_1, MOD_2, GRCAT_1, RLVECT_2,
VECTSP_2, VECTSP_4, VECTSP_5, VECTSP_6, INT_3, LOPBAN_1, VECTSP_7,
ZMODUL01, ZMODUL02, ZMODUL03, RANKNULL;
constructors RELSET_1, REALSET1, CLASSES1, FINSEQOP, FINSOP_1, FUNCT_7,
ALGSTR_1, GR_CY_1, LOPBAN_1, ZMODUL02, ZMODUL03, SETWISEO, BINOP_2,
VECTSP_2, VECTSP_4, VECTSP_6, VECTSP_7, INT_3, ALGSTR_0, RLVECT_2,
REAL_1, VFUNCT_1, RFINSEQ, GRCAT_1, MOD_2, VECTSP_5, MOD_4, ZMODUL01,
RANKNULL;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1, RELSET_1,
FUNCT_2, FUNCOP_1, FINSET_1, CARD_1, NUMBERS, XREAL_0, NAT_1, INT_1,
VALUED_0, FINSEQ_1, STRUCT_0, RLVECT_1, ZMODUL02, ZMODUL03, ZMODUL04,
BINOP_2, INT_3, MEMBERED;
requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL;
begin :: 1. Rank of submodule of $\mathbb Z$-module
reserve V,W for Z_Module;
registration
let V be LeftMod of INT.Ring;
let A be finite Subset of V;
cluster Lin(A) -> finitely-generated;
end;
theorem :: ZMODUL05:1
for V be finite-rank free Z_Module holds
rank V = 0 iff (Omega).V = (0).V;
registration
let V be finite-rank free Z_Module;
cluster finite for Basis of V;
end;
registration
let V be finite-rank free Z_Module;
cluster -> finite for Basis of V;
end;
theorem :: ZMODUL05:2
for V being finite-rank free Z_Module, W being Submodule of V holds
rank W <= rank V;
theorem :: ZMODUL05:3
for V being Z_Module, A being finite linearly-independent Subset of V holds
card A = rank Lin(A);
theorem :: ZMODUL05:4
for V being finite-rank free Z_Module holds
rank V = rank (Omega).V;
theorem :: ZMODUL05:5
for V being finite-rank free Z_Module holds
rank V = 1 iff ex v being VECTOR of V st v <> 0.V & (Omega).V = Lin{v};
theorem :: ZMODUL05:6
for V being finite-rank free Z_Module holds
rank V = 2 iff ex u, v being VECTOR of V st u <> v &
{u, v} is linearly-independent & (Omega).V = Lin{u, v};
theorem :: ZMODUL05:7
for V being finite-rank free Z_Module, W being Submodule of V,
n being Nat holds
n <= rank V iff ex W being strict free Submodule of V st rank W = n;
definition
let V be finite-rank free Z_Module, n be Nat;
func n Submodules_of V -> set means
:: ZMODUL05:def 1
for x being object holds
x in it iff ex W being strict free Submodule of V st W = x & rank W = n;
end;
theorem :: ZMODUL05:8
for V being finite-rank free Z_Module, n being Nat holds
n <= rank V implies n Submodules_of V is non empty;
theorem :: ZMODUL05:9
for V being finite-rank free Z_Module, n being Nat holds
rank V < n implies n Submodules_of V = {};
theorem :: ZMODUL05:10
for V being finite-rank free Z_Module, W being Submodule of V,
n being Nat holds
n Submodules_of W c= n Submodules_of V;
theorem :: ZMODUL05:11
for F,G being FinSequence of INT, v be Integer holds
len F = len G + 1 & G = F | (dom G) & v = F.(len F) implies
Sum(F) = Sum(G) + v;
theorem :: ZMODUL05:12 ::: should be proven for real-valued
for F,G being FinSequence of INT
st rng F = rng G & F is one-to-one
& G is one-to-one holds Sum(F) = Sum(G);
definition
let T be finite Subset of the carrier of INT.Ring;
::: what for? defined for reals, then redefine
func Sum T -> Element of INT.Ring means
:: ZMODUL05:def 2
ex F being FinSequence of INT st
rng F = T & F is one-to-one & it = Sum F;
end;
theorem :: ZMODUL05:13
for K being Ring
for R1,R2 be FinSequence of K st R1,R2 are_fiberwise_equipotent
holds Sum R1 = Sum R2;
::$CT 2
::theorem
::: for X be finite Subset of INT st X = {}
::: holds Sum X = 0 by Def2,GR_CY_1:3,RELAT_1:38;
::theorem
:: for v be Element of INT holds Sum {v} = v
:: proof
:: let v be Element of INT;
:: A1: Sum <*v*> = v by RVSUM_1:73;
:: rng <*v*> = {v} & <*v*> is one-to-one by FINSEQ_1:39;
:: hence Sum {v} = v by A1,Def2;
:: end;
::theorem
:: for S, T being finite Subset of INT
:: st T misses S holds
:: Sum (T \/ S) = (Sum T) + (Sum S)
:: proof
:: let S, T be finite Subset of INT;
:: consider F being FinSequence of INT such that
:: A1: rng F = T \/ S and
:: A2: F is one-to-one & Sum (T \/ S) = Sum F by Def2;
:: consider G being FinSequence of INT such that
:: A3: rng G = T and
:: A4: G is one-to-one and
:: A5: Sum T = Sum G by Def2;
:: consider H being FinSequence of INT such that
:: A6: rng H = S and
:: A7: H is one-to-one and
:: A8: Sum S = Sum H by Def2;
:: set I = G ^ H;
:: reconsider G0 = G, H0 = H as real-valued FinSequence;
::: A9: Sum (G0 ^ H0) = Sum (G) + Sum (H) by RVSUM_1:75;
:: assume T misses S; then
:: A10: G ^ H is one-to-one by A3,A4,A6,A7,FINSEQ_3:91;
:: rng (G ^ H) = rng F by A1,A3,A6,FINSEQ_1:31;
:: hence Sum (T \/ S) = (Sum T) + (Sum S) by A2,A5,A8,A9,A10,RLVECT142;
:: end;
definition
::$CD 3
end;
theorem :: ZMODUL05:16
for R being Ring
for V1, V2 being LeftMod of R
for f being Function of V1, V2
for p being FinSequence of V1 st f is additive homogeneous holds
f . (Sum p) = Sum (f * p);
theorem :: ZMODUL05:17
for V be free Z_Module st [#]V is finite holds
(Omega).V = (0).V;
begin :: Basic facts of linear transformations
reserve T for linear-transformation of V,W;
theorem :: ZMODUL05:18
for F being Ring, V,W being VectSp of F,
T being linear-transformation of V,W
for x,y being Element of V holds T.x - T.y = T.(x - y);
theorem :: ZMODUL05:19
T.(0.V) = 0.W;
theorem :: ZMODUL05:20
for x being Element of V holds x in ker T iff T.x = 0.W;
theorem :: ZMODUL05:21
0.V in ker T;
theorem :: ZMODUL05:22
for X being Subset of V holds T .: X is Subset of im T;
theorem :: ZMODUL05:23
for y being Element of W holds y in im T iff
ex x being Element of V st y = T.x;
theorem :: ZMODUL05:24
for x being Element of ker T holds T.x = 0.W;
theorem :: ZMODUL05:25
T is one-to-one implies ker T = (0).V;
theorem :: ZMODUL05:26
for V being finite-rank free Z_Module holds rank ((0).V) = 0;
theorem :: ZMODUL05:27
for x,y being Element of V st T.x = T.y holds x - y in ker T;
theorem :: ZMODUL05:28
for A being Subset of V, x,y being Element of V st x - y in Lin A holds
x in Lin (A \/ {y});
begin :: Some basic facts about linearly independent subsets and linear
:: combinations
theorem :: ZMODUL05:29
for X being Subset of V st V is Submodule of W holds X is Subset of W;
theorem :: ZMODUL05:30
for R being Ring
for V being LeftMod of R,
A being Subset of V holds A is Subset of Lin(A);
theorem :: ZMODUL05:31
for V being LeftMod of INT.Ring,
A being linearly-independent Subset of V
holds A is Basis of Lin(A);
:: Adjoining an element x to A that is already in its linear span
:: results in a linearly dependent set.
theorem :: ZMODUL05:32
for V be finite-rank free Z_Module,
A being Subset of V, x being Element of V st x in Lin A &
not x in A holds A \/ {x} is linearly-dependent;
registration
let V be finite-rank free Z_Module, W be Z_Module;
let T be linear-transformation of V,W;
cluster ker T -> finite-rank free;
end;
reserve T for linear-transformation of V,W;
theorem :: ZMODUL05:33
for V be finite-rank free Z_Module,A being Subset of V,
B being Basis of V,
T being linear-transformation of V,W
st A is Basis of ker T & A c= B holds
T | (B \ A) is one-to-one;
theorem :: ZMODUL05:34
for R being Ring, V being LeftMod of R
for A being Subset of V, l being Linear_Combination of A,
x being Element of V, a being Element of R holds
l +* (x,a) is Linear_Combination of A \/ {x};
reserve l for Linear_Combination of V;
registration
let R be non degenerated Ring, V be LeftMod of R;
cluster linearly-dependent for Subset of V;
end;
:: Restricting a linear combination to a given set
definition
let R be Ring;
let V be LeftMod of R, l be Linear_Combination of V, A be Subset of V;
func l!A -> Linear_Combination of A equals
:: ZMODUL05:def 6
(l|A) +* ((A`) --> 0.R);
end;
theorem :: ZMODUL05:35
for R be Ring
for V be LeftMod of R, l be Linear_Combination of V holds
l = l!Carrier l;
theorem :: ZMODUL05:36
for R being Ring for V being LeftMod of R
for l being Linear_Combination of V
for A being Subset of V, v being Element of V st v in A holds
(l !A).v = l.v;
theorem :: ZMODUL05:37
for R being Ring, V being LeftMod of R
for l being Linear_Combination of V
for A being Subset of V, v being Element of V st not v in A holds
(l!A).v = 0.R;
theorem :: ZMODUL05:38
for R being Ring, V being LeftMod of R
for A,B being Subset of V, l being Linear_Combination of B st A c= B holds
l = (l!A) + (l!(B\A));
registration
let R be Ring, V be LeftMod of R,
l be Linear_Combination of V, X be Subset of V;
cluster l .: X -> finite;
end;
theorem :: ZMODUL05:39
for R being Ring, V being LeftMod of R,
l be Linear_Combination of V
for X being Subset of V st X misses Carrier l holds l .: X c= {0.R};
definition
let K be Ring;
let V,W be VectSp of K,
l be Linear_Combination of V,
T be linear-transformation of V,W;
let w be Element of W;
func lCFST(l,T,w) -> (the carrier of K) -valued FinSequence equals
:: ZMODUL05:def 7
l * canFS((T"{w}) /\ (Carrier l));
end;
reserve V,W for Z_Module;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;
theorem :: ZMODUL05:40
for V,W be non empty set,
f be FinSequence,
l be Function of V,W
st rng f c= V
holds l*f is W-valued FinSequence-like;
registration
let V,W be non empty set,
f be V-valued FinSequence,
l be Function of V,W;
cluster l*f -> W -valued FinSequence-like;
end;
registration
let V,W be non empty set,
A be finite Subset of V,
l be Function of V,W;
cluster l*canFS(A) -> W -valued FinSequence-like;
end;
registration
let R be Ring;
let V be LeftMod of R,
A be finite Subset of V,
l be Linear_Combination of V;
cluster l*canFS(A) -> (the carrier of R) -valued FinSequence-like;
end;
theorem :: ZMODUL05:41
for V,W be non empty set,
f,g be V-valued FinSequence,
l be Function of V,W
holds l*(f^g) = (l*f)^(l*g);
theorem :: ZMODUL05:42
for R being Ring
for V be LeftMod of R,
A,B be finite Subset of V,
l be Linear_Combination of V,
l0,l1,l2 being FinSequence of R
st A /\ B = {} & l0 = l*canFS(A \/ B) & l1 = l*canFS(A) & l2 = l*canFS(B)
holds Sum l0 = Sum l1 + Sum l2;
theorem :: ZMODUL05:43
for R being Ring
for V be LeftMod of R,
A be finite Subset of V,
l,l0 be Linear_Combination of V
st l | (Carrier l0) = l0 | (Carrier l0)
& Carrier l0 c= Carrier l & A c= Carrier l0
holds Sum(l*canFS(A)) = Sum(l0*canFS(A));
definition
let R be Ring;
let V,W be LeftMod of R,
l be Linear_Combination of V, T be linear-transformation of V,W;
func T @* l -> Linear_Combination of W means
:: ZMODUL05:def 8
Carrier it c= T.:(Carrier l)
& for w being Element of W holds it.w = Sum(lCFST(l,T,w));
end;
theorem :: ZMODUL05:44
for R being Ring
for V,W be LeftMod of R,
l be Linear_Combination of V, T be linear-transformation of V,W holds
(T@*l) is Linear_Combination of T .: (Carrier l);
theorem :: ZMODUL05:45
for K being Ring
for V,W being LeftMod of K,
T being linear-transformation of V,W,
s being Vector of W holds
(for A being Subset of V, l being Linear_Combination of A
st (for v being Vector of V st v in Carrier l holds T.v = s)
holds T.Sum(l) = Sum(lCFST(l,T,s)) * s);
theorem :: ZMODUL05:46
for K being Ring
for V, W being LeftMod of K,
T being linear-transformation of V,W,
A being Subset of V,
l being Linear_Combination of A,
Tl being Linear_Combination of T.:(Carrier l)
st Tl = T@*l
holds T.Sum(l) = Sum (Tl);
::: same as ZMODUL04:25
theorem :: ZMODUL05:47
for R being Ring
for V being LeftMod of R
for l,m being Linear_Combination of V
st (Carrier l) misses (Carrier m) holds
Carrier(l + m) = (Carrier l) \/ (Carrier m);
theorem :: ZMODUL05:48
for R being Ring
for V being LeftMod of R
for l, m being Linear_Combination of V
st (Carrier l) misses (Carrier m) holds
Carrier (l - m) = (Carrier l) \/ (Carrier m);
theorem :: ZMODUL05:49
for R being Ring
for V being LeftMod of R, A being Subset of V,
l1, l2 being Linear_Combination of A
st Carrier(l1) misses Carrier(l2) holds
Carrier(l1 - l2) = Carrier(l1) \/ Carrier(l2);
theorem :: ZMODUL05:50
for V be free Z_Module,
A,B being Subset of V st A c= B & B is Basis of V holds
V is_the_direct_sum_of Lin A, Lin (B \ A);
theorem :: ZMODUL05:51
for R being Ring
for V,W being LeftMod of R
for A being Subset of V, l being Linear_Combination of A,
T being linear-transformation of V,W,
v being Element of V st T|A is one-to-one & v in A holds
ex X being Subset of V
st X misses A & T"{T.v} = {v} \/ X;
theorem :: ZMODUL05:52
for R being Ring
for V being LeftMod of R, A being Subset of V,
l being Linear_Combination of A
for X being Subset of V st X misses Carrier l & X <> {} holds
l .: X = {0.R};
theorem :: ZMODUL05:53
for R being Ring
for V,W being LeftMod of R
for l being Linear_Combination of V,
T being linear-transformation of V,W
for w being Element of W st w in Carrier (T@*l) holds T"{w} meets Carrier l;
theorem :: ZMODUL05:54
for R being Ring
for V,W being LeftMod of R
for l being Linear_Combination of V,
T being linear-transformation of V,W
for v being Element of V st T | (Carrier l) is one-to-one &
v in Carrier l holds (T@*l).(T.v) = l.v;
theorem :: ZMODUL05:55
for R being Ring
for V,W being LeftMod of R
for l being Linear_Combination of V,
T being linear-transformation of V,W
for G being FinSequence of V st rng G = Carrier l &
T | (Carrier l) is one-to-one holds T*(l (#) G) = (T@*l) (#) (T*G);
theorem :: ZMODUL05:56
for R being Ring
for V,W being LeftMod of R
for l being Linear_Combination of V,
T being linear-transformation of V,W holds
T | (Carrier l) is one-to-one implies T .: (Carrier l) = Carrier(T@*l);
theorem :: ZMODUL05:57
for V being finite-rank free Z_Module,
A being Subset of V,
B being Basis of V,
T being linear-transformation of V,W,
l being Linear_Combination of B \ A
st A is Basis of ker T & A c= B holds T.(Sum l) = Sum(T@*l);
theorem :: ZMODUL05:58
for X being Subset of V st X is linearly-dependent holds
ex l being Linear_Combination of X st Carrier l <> {} & Sum l = 0.V;
:: "Pulling back" a linear combination from the image space of a
:: linear transformation to the base space.
definition
let R be Ring;
let V,W be LeftMod of R, X be Subset of V,
T be linear-transformation of V,W, l be Linear_Combination of T .: X;
assume
T|X is one-to-one;
func T#l -> Linear_Combination of X equals
:: ZMODUL05:def 9
(l*T) +* ((X`) --> 0.R);
end;
theorem :: ZMODUL05:59
for R being Ring
for V,W be LeftMod of R, X be Subset of V,
T be linear-transformation of V,W
for X being Subset of V,
l be Linear_Combination of T.:X,
v being Element of V st v in X & T|X is one-to-one holds (T#l).v = l.(T.v);
:: # is a right inverse of @*
theorem :: ZMODUL05:60
for R being Ring
for V,W be LeftMod of R, X be Subset of V,
T be linear-transformation of V,W,
X being Subset of V,
l be Linear_Combination of T .: X st
T|X is one-to-one holds T@*(T#l) = l;