:: by Michal Muzalewski

::

:: Received October 18, 1991

:: Copyright (c) 1991-2021 Association of Mizar Users

Lm1: for R being Ring

for a being Scalar of R st - a = 0. R holds

a = 0. R

proof end;

theorem Th1: :: MOD_3:1

for R being non empty non degenerated right_complementable add-associative right_zeroed doubleLoopStr holds 0. R <> - (1. R)

proof end;

theorem :: MOD_3:2

for R being Ring

for V being LeftMod of R

for L being Linear_Combination of V

for C being finite Subset of V st Carrier L c= C holds

ex F being FinSequence of the carrier of V st

( F is one-to-one & rng F = C & Sum L = Sum (L (#) F) ) by VECTSP_7:21;

for V being LeftMod of R

for L being Linear_Combination of V

for C being finite Subset of V st Carrier L c= C holds

ex F being FinSequence of the carrier of V st

( F is one-to-one & rng F = C & Sum L = Sum (L (#) F) ) by VECTSP_7:21;

theorem :: MOD_3:3

for R being Ring

for V being LeftMod of R

for L being Linear_Combination of V

for a being Scalar of R holds Sum (a * L) = a * (Sum L) by VECTSP_7:22;

for V being LeftMod of R

for L being Linear_Combination of V

for a being Scalar of R holds Sum (a * L) = a * (Sum L) by VECTSP_7:22;

theorem Th4: :: MOD_3:4

for x being object

for R being Ring

for V being LeftMod of R

for A being Subset of V holds

( x in Lin A iff ex l being Linear_Combination of A st x = Sum l ) by VECTSP_7:7;

for R being Ring

for V being LeftMod of R

for A being Subset of V holds

( x in Lin A iff ex l being Linear_Combination of A st x = Sum l ) by VECTSP_7:7;

theorem :: MOD_3:7

theorem Th8: :: MOD_3:8

for R being non degenerated Ring

for V being LeftMod of R

for A being Subset of V

for W being strict Subspace of V st A = the carrier of W holds

Lin A = W

for V being LeftMod of R

for A being Subset of V

for W being strict Subspace of V st A = the carrier of W holds

Lin A = W

proof end;

theorem :: MOD_3:9

for R being non degenerated Ring

for V being strict LeftMod of R

for A being Subset of V st A = the carrier of V holds

Lin A = V

for V being strict LeftMod of R

for A being Subset of V st A = the carrier of V holds

Lin A = V

proof end;

theorem :: MOD_3:11

for R being Ring

for V being LeftMod of R

for A, B being Subset of V st Lin A = V & A c= B holds

Lin B = V

for V being LeftMod of R

for A, B being Subset of V st Lin A = V & A c= B holds

Lin B = V

proof end;

theorem :: MOD_3:12

theorem :: MOD_3:13

registration

let R be Ring;

ex b_{1} being LeftMod of R st

( b_{1} is strict & b_{1} is free )

end;
cluster non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed free for ModuleStr over R;

existence ex b

( b

proof end;

Lm2: for R being Skew-Field

for a being Scalar of R

for V being LeftMod of R

for v being Vector of V st a <> 0. R holds

( (a ") * (a * v) = (1. R) * v & ((a ") * a) * v = (1. R) * v )

proof end;

theorem :: MOD_3:15

for R being Skew-Field

for V being LeftMod of R

for v being Vector of V holds

( {v} is linearly-independent iff v <> 0. V )

for V being LeftMod of R

for v being Vector of V holds

( {v} is linearly-independent iff v <> 0. V )

proof end;

theorem Th16: :: MOD_3:16

for R being Skew-Field

for V being LeftMod of R

for v1, v2 being Vector of V holds

( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a being Scalar of R holds v1 <> a * v2 ) ) )

for V being LeftMod of R

for v1, v2 being Vector of V holds

( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a being Scalar of R holds v1 <> a * v2 ) ) )

proof end;

theorem :: MOD_3:17

for R being Skew-Field

for V being LeftMod of R

for v1, v2 being Vector of V holds

( ( v1 <> v2 & {v1,v2} is linearly-independent ) iff for a, b being Scalar of R st (a * v1) + (b * v2) = 0. V holds

( a = 0. R & b = 0. R ) )

for V being LeftMod of R

for v1, v2 being Vector of V holds

( ( v1 <> v2 & {v1,v2} is linearly-independent ) iff for a, b being Scalar of R st (a * v1) + (b * v2) = 0. V holds

( a = 0. R & b = 0. R ) )

proof end;

theorem Th18: :: MOD_3:18

for R being Skew-Field

for V being LeftMod of R

for A being Subset of V st A is linearly-independent holds

ex B being Subset of V st

( A c= B & B is base )

for V being LeftMod of R

for A being Subset of V st A is linearly-independent holds

ex B being Subset of V st

( A c= B & B is base )

proof end;

theorem Th19: :: MOD_3:19

for R being non degenerated almost_left_invertible Ring

for V being LeftMod of R

for A being Subset of V st Lin A = V holds

ex B being Subset of V st

( B c= A & B is base )

for V being LeftMod of R

for A being Subset of V st Lin A = V holds

ex B being Subset of V st

( B c= A & B is base )

proof end;

Lm3: for R being non degenerated almost_left_invertible Ring

for V being LeftMod of R ex B being Subset of V st B is base

proof end;

registration

let R be non degenerated almost_left_invertible Ring;

let V be LeftMod of R;

existence

ex b_{1} being Subset of V st b_{1} is base

end;
let V be LeftMod of R;

existence

ex b

proof end;

registration

let R be Skew-Field;

for b_{1} being LeftMod of R holds b_{1} is free
by Lm3;

end;
cluster non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed -> free for ModuleStr over R;

coherence for b

theorem :: MOD_3:21

for R being non degenerated almost_left_invertible Ring

for V being LeftMod of R

for A being Subset of V st A is linearly-independent holds

ex I being Basis of V st A c= I

for V being LeftMod of R

for A being Subset of V st A is linearly-independent holds

ex I being Basis of V st A c= I

proof end;

theorem :: MOD_3:22

for R being Skew-Field

for V being LeftMod of R

for A being Subset of V st Lin A = V holds

ex I being Basis of V st I c= A

for V being LeftMod of R

for A being Subset of V st Lin A = V holds

ex I being Basis of V st I c= A

proof end;