:: Submodules
:: by Micha{\l} Muzalewski
::
:: Received June 19, 1992
:: Copyright (c) 1992-2021 Association of Mizar Users

notation

let K be Ring;
let V be LeftMod of K;
synonym Submodules V for Subspaces V;

end;

theorem :: LMOD_6:1

for x being set
for K being Ring
for M1, M2 being LeftMod of K st M1 = ModuleStr(# the carrier of M2, the addF of M2, the ZeroF of M2, the lmult of M2 #) holds
( x in M1 iff x in M2 ) ;

theorem :: LMOD_6:2

for K being Ring
for r being Scalar of K
for V being LeftMod of K
for a being Vector of V
for v being Vector of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) st a = v holds
r * a = r * v

proof end;

theorem Th3: :: LMOD_6:3

for K being Ring
for V being LeftMod of K holds ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is strict Subspace of V

proof end;

theorem Th4: :: LMOD_6:4

for K being Ring
for V being LeftMod of K holds V is Subspace of (Omega). V

proof end;

definition

let K be Ring;

redefine attr K is trivial means :: LMOD_6:def 1
0. K = 1_ K;

compatibility
( K is trivial iff 0. K = 1_ K )

proof end;

end;

:: deftheorem defines trivial LMOD_6:def 1 :
for K being Ring holds
( K is trivial iff 0. K = 1_ K );

theorem Th5: :: LMOD_6:5

for K being Ring
for V being LeftMod of K st K is trivial holds
( ( for r being Scalar of K holds r = 0. K ) & ( for a being Vector of V holds a = 0. V ) )

proof end;

theorem :: LMOD_6:6

for K being Ring
for V being LeftMod of K st K is trivial holds
V is trivial by Th5;

theorem :: LMOD_6:7

for K being Ring
for V being LeftMod of K holds
( V is trivial iff ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) = (0). V )

proof end;

definition

let K be Ring;
let V be LeftMod of K;
let W be strict Subspace of V;

func @ W -> Element of Submodules V equals :: LMOD_6:def 2
W;

coherence
W is Element of Submodules V by VECTSP_5:def 3;

end;

:: deftheorem defines @ LMOD_6:def 2 :
for K being Ring
for V being LeftMod of K
for W being strict Subspace of V holds @ W = W;

theorem Th8: :: LMOD_6:8

for K being Ring
for V being LeftMod of K
for W being Subspace of V
for A being Subset of W holds A is Subset of V

proof end;

definition

let K be Ring;
let V be LeftMod of K;
let W be Subspace of V;
let A be Subset of W;

func @ A -> Subset of V equals :: LMOD_6:def 3
A;

coherence
A is Subset of V by Th8;

end;

:: deftheorem defines @ LMOD_6:def 3 :
for K being Ring
for V being LeftMod of K
for W being Subspace of V
for A being Subset of W holds @ A = A;

registration

let K be Ring;
let V be LeftMod of K;
let W be Subspace of V;
let A be non empty Subset of W;

cluster @ A -> non empty ;

coherence
not @ A is empty ;

end;

theorem :: LMOD_6:9

for x being set
for K being Ring
for V being LeftMod of K holds
( x in [#] V iff x in V ) ;

theorem :: LMOD_6:10

for x being set
for K being Ring
for V being LeftMod of K
for W being Subspace of V holds
( x in @ ([#] W) iff x in W ) ;

theorem Th11: :: LMOD_6:11

for K being Ring
for V being LeftMod of K
for A being Subset of V holds A c= [#] (Lin A)

proof end;

theorem Th12: :: LMOD_6:12

for K being Ring
for V being LeftMod of K
for A being Subset of V
for l being Linear_Combination of A st A <> {} & A is linearly-closed holds
Sum l in A

proof end;

theorem :: LMOD_6:13

for K being Ring
for V being LeftMod of K
for A being Subset of V st 0. V in A & A is linearly-closed holds
A = [#] (Lin A)

proof end;

definition

let K be Ring;
let V be LeftMod of K;
let a be Vector of V;

func <:a:> -> strict Subspace of V equals :: LMOD_6:def 4
Lin {a};

correctness
coherence
Lin {a} is strict Subspace of V;
;

end;

:: deftheorem defines <: LMOD_6:def 4 :
for K being Ring
for V being LeftMod of K
for a being Vector of V holds <:a:> = Lin {a};

definition

let K be Ring;
let M, N be LeftMod of K;

pred M c= N means :: LMOD_6:def 5
M is Subspace of N;

reflexivity
for M being LeftMod of K holds M is Subspace of M by VECTSP_4:24;

end;

:: deftheorem defines c= LMOD_6:def 5 :
for K being Ring
for M, N being LeftMod of K holds
( M c= N iff M is Subspace of N );

theorem Th14: :: LMOD_6:14

for K being Ring
for M, N being LeftMod of K
for x being object st M c= N holds
( ( x in M implies x in N ) & ( x is Vector of M implies x is Vector of N ) ) by VECTSP_4:9, VECTSP_4:10;

theorem :: LMOD_6:15

for K being Ring
for r being Scalar of K
for M, N being LeftMod of K
for m, m1, m2 being Vector of M
for n, n1, n2 being Vector of N st M c= N holds
( 0. M = 0. N & ( m1 = n1 & m2 = n2 implies m1 + m2 = n1 + n2 ) & ( m = n implies r * m = r * n ) & ( m = n implies - n = - m ) & ( m1 = n1 & m2 = n2 implies m1 - m2 = n1 - n2 ) & 0. N in M & 0. M in N & ( n1 in M & n2 in M implies n1 + n2 in M ) & ( n in M implies r * n in M ) & ( n in M implies - n in M ) & ( n1 in M & n2 in M implies n1 - n2 in M ) ) by VECTSP_4:11, VECTSP_4:13, VECTSP_4:14, VECTSP_4:15, VECTSP_4:16, VECTSP_4:17, VECTSP_4:19, VECTSP_4:20, VECTSP_4:21, VECTSP_4:22, VECTSP_4:23;

theorem :: LMOD_6:16

for K being Ring
for M1, M2, N being LeftMod of K st M1 c= N & M2 c= N holds
( 0. M1 = 0. M2 & 0. M1 in M2 & ( the carrier of M1 c= the carrier of M2 implies M1 c= M2 ) & ( ( for n being Vector of N st n in M1 holds
n in M2 ) implies M1 c= M2 ) & ( the carrier of M1 = the carrier of M2 & M1 is strict & M2 is strict implies M1 = M2 ) & (0). M1 c= M2 ) by VECTSP_4:12, VECTSP_4:18, VECTSP_4:27, VECTSP_4:28, VECTSP_4:29, VECTSP_4:40;

theorem :: LMOD_6:17

for K being Ring
for V, M being strict LeftMod of K st V c= M & M c= V holds
V = M by VECTSP_4:25;

theorem :: LMOD_6:18

for K being Ring
for V, M, N being LeftMod of K st V c= M & M c= N holds
V c= N by VECTSP_4:26;

theorem :: LMOD_6:19

for K being Ring
for M, N being LeftMod of K st M c= N holds
(0). M c= N by VECTSP_4:38;

theorem :: LMOD_6:20

for K being Ring
for M, N being LeftMod of K st M c= N holds
(0). N c= M by VECTSP_4:39;

theorem :: LMOD_6:21

for K being Ring
for M, N being LeftMod of K st M c= N holds
M c= (Omega). N

proof end;

theorem :: LMOD_6:22

for K being Ring
for V being LeftMod of K
for W1, W2 being Subspace of V holds
( W1 c= W1 + W2 & W2 c= W1 + W2 ) by VECTSP_5:7;

theorem :: LMOD_6:23

for K being Ring
for V being LeftMod of K
for W1, W2 being Subspace of V holds
( W1 /\ W2 c= W1 & W1 /\ W2 c= W2 ) by VECTSP_5:15;

theorem :: LMOD_6:24

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V st W1 c= W2 holds
W1 /\ W3 c= W2 /\ W3 by VECTSP_5:17;

theorem :: LMOD_6:25

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V st W1 c= W3 holds
W1 /\ W2 c= W3 by VECTSP_5:18;

theorem :: LMOD_6:26

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V st W1 c= W2 & W1 c= W3 holds
W1 c= W2 /\ W3 by VECTSP_5:19;

theorem :: LMOD_6:27

for K being Ring
for V being LeftMod of K
for W1, W2 being Subspace of V holds W1 /\ W2 c= W1 + W2 by VECTSP_5:23;

theorem :: LMOD_6:28

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V holds (W1 /\ W2) + (W2 /\ W3) c= W2 /\ (W1 + W3) by VECTSP_5:26;

theorem :: LMOD_6:29

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V st W1 c= W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3) by VECTSP_5:27;

theorem :: LMOD_6:30

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V holds W2 + (W1 /\ W3) c= (W1 + W2) /\ (W2 + W3) by VECTSP_5:28;

theorem :: LMOD_6:31

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V st W1 c= W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3) by VECTSP_5:29;

theorem :: LMOD_6:32

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V st W1 c= W2 holds
W1 c= W2 + W3 by VECTSP_5:33;

theorem :: LMOD_6:33

for K being Ring
for V being LeftMod of K
for W1, W2, W3 being Subspace of V st W1 c= W3 & W2 c= W3 holds
W1 + W2 c= W3 by VECTSP_5:34;

theorem :: LMOD_6:34

for K being Ring
for V being LeftMod of K
for A, B being Subset of V st A c= B holds
Lin A c= Lin B by MOD_3:10;

theorem :: LMOD_6:35

for K being Ring
for V being LeftMod of K
for A, B being Subset of V holds Lin (A /\ B) c= (Lin A) /\ (Lin B) by MOD_3:13;

theorem Th36: :: LMOD_6:36

for K being Ring
for M1, M2 being LeftMod of K st M1 c= M2 holds
[#] M1 c= [#] M2

proof end;

theorem Th37: :: LMOD_6:37

for K being Ring
for V being LeftMod of K
for W1, W2 being Subspace of V holds
( W1 c= W2 iff for a being Vector of V st a in W1 holds
a in W2 ) by VECTSP_4:8, VECTSP_4:28;

theorem Th38: :: LMOD_6:38

for K being Ring
for V being LeftMod of K
for W1, W2 being Subspace of V holds
( W1 c= W2 iff [#] W1 c= [#] W2 )

proof end;

theorem :: LMOD_6:39

for K being Ring
for V being LeftMod of K
for W1, W2 being Subspace of V holds
( W1 c= W2 iff @ ([#] W1) c= @ ([#] W2) ) by Th38;

theorem :: LMOD_6:40

for K being Ring
for V being LeftMod of K
for W, W1, W2 being Subspace of V holds
( (0). W c= V & (0). V c= W & (0). W1 c= W2 ) by VECTSP_4:38, VECTSP_4:39, VECTSP_4:40;