:: Groups, Rings, Left- and Right-Modules
:: by Micha{\l} Muzalewski and Wojciech Skaba
::
:: Received October 22, 1990
:: Copyright (c) 1990 Association of Mizar Users

:: MOD_1 semantic presentation

theorem :: MOD_1:1
canceled;

theorem :: MOD_1:2
canceled;

theorem :: MOD_1:3
canceled;

theorem :: MOD_1:4
canceled;

theorem :: MOD_1:5
canceled;

theorem :: MOD_1:6
canceled;

theorem :: MOD_1:7
canceled;

theorem :: MOD_1:8
canceled;

theorem :: MOD_1:9
canceled;

theorem :: MOD_1:10
canceled;

theorem :: MOD_1:11
canceled;

theorem :: MOD_1:12
canceled;

theorem :: MOD_1:13
for K being non empty right_complementable right-distributive right_unital add-associative right_zeroed doubleLoopStr
for a being Element of K holds a * (- (1. K)) = - a
proof end;

theorem :: MOD_1:14
for K being non empty right_complementable left-distributive left_unital add-associative right_zeroed doubleLoopStr
for a being Element of K holds (- (1. K)) * a = - a
proof end;

theorem :: MOD_1:15
canceled;

theorem :: MOD_1:16
canceled;

theorem :: MOD_1:17
canceled;

theorem :: MOD_1:18
canceled;

theorem :: MOD_1:19
canceled;

theorem :: MOD_1:20
canceled;

theorem :: MOD_1:21
canceled;

theorem :: MOD_1:22
canceled;

theorem :: MOD_1:23
canceled;

theorem :: MOD_1:24
canceled;

theorem :: MOD_1:25
for F being non degenerated almost_left_invertible Ring
for x being Scalar of F
for V being non empty right_complementable VectSp-like add-associative right_zeroed VectSpStr of F
for v being Vector of V holds
( x * v = 0. V iff ( x = 0. F or v = 0. V ) )
proof end;

theorem :: MOD_1:26
for F being non degenerated almost_left_invertible Ring
for x being Scalar of F
for V being non empty right_complementable VectSp-like add-associative right_zeroed VectSpStr of F
for v being Vector of V st x <> 0. F holds
(x " ) * (x * v) = v
proof end;

theorem :: MOD_1:27
canceled;

theorem :: MOD_1:28
canceled;

theorem :: MOD_1:29
canceled;

theorem :: MOD_1:30
canceled;

theorem :: MOD_1:31
canceled;

theorem :: MOD_1:32
canceled;

theorem :: MOD_1:33
canceled;

theorem :: MOD_1:34
canceled;

theorem :: MOD_1:35
canceled;

theorem :: MOD_1:36
canceled;

theorem Th37: :: MOD_1:37
for R being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of R
for x being Scalar of R
for v being Vector of V holds
( v * (0. R) = 0. V & v * (- (1_ R)) = - v & (0. V) * x = 0. V )
proof end;

theorem Th38: :: MOD_1:38
for R being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of R
for x being Scalar of R
for v, w being Vector of V holds
( - (v * x) = v * (- x) & w - (v * x) = w + (v * (- x)) )
proof end;

theorem Th39: :: MOD_1:39
for R being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of R
for x being Scalar of R
for v being Vector of V holds (- v) * x = - (v * x)
proof end;

theorem :: MOD_1:40
for R being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of R
for x being Scalar of R
for v, w being Vector of V holds (v - w) * x = (v * x) - (w * x)
proof end;

theorem :: MOD_1:41
canceled;

theorem :: MOD_1:42
for F being non degenerated almost_left_invertible Ring
for x being Scalar of F
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of F
for v being Vector of V holds
( v * x = 0. V iff ( x = 0. F or v = 0. V ) )
proof end;

theorem :: MOD_1:43
for F being non degenerated almost_left_invertible Ring
for x being Scalar of F
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of F
for v being Vector of V st x <> 0. F holds
(v * x) * (x " ) = v
proof end;