:: Opposite Rings, Modules and their Morphisms
:: by Micha{\l} Muzalewski
::
:: Received June 22, 1992
:: Copyright (c) 1992 Association of Mizar Users


begin

definition
let A, B, C be non empty set ;
let f be Function of [:A,B:],C;
:: original: ~
redefine func ~ f -> Function of [:B,A:],C;
coherence
~ f is Function of [:B,A:],C by FUNCT_4:51;
end;

theorem Th1: :: MOD_4:1
for C, A, B being non empty set
for f being Function of [:A,B:],C
for x being Element of A
for y being Element of B holds f . x,y = (~ f) . y,x
proof end;

begin

definition
let K be non empty doubleLoopStr ;
func opp K -> strict doubleLoopStr equals :: MOD_4:def 1
 doubleLoopStr(# the carrier of K,the addF of K,(~ the multF of K),(1. K),(0. K) #);
correctness
coherence
doubleLoopStr(# the carrier of K,the addF of K,(~ the multF of K),(1. K),(0. K) #) is strict doubleLoopStr ;
;
end;

:: deftheorem defines opp MOD_4:def 1 :
for K being non empty doubleLoopStr holds opp K = doubleLoopStr(# the carrier of K,the addF of K,(~ the multF of K),(1. K),(0. K) #);

registration
let K be non empty doubleLoopStr ;
cluster opp K -> non empty strict ;
coherence
not opp K is empty ;
end;

Lm1: for K being non empty well-unital doubleLoopStr
for h, e being Element of (opp K) st e = 1. K holds
( h * e = h & e * h = h )
proof end;

registration
let K be non empty well-unital doubleLoopStr ;
cluster opp K -> strict well-unital ;
coherence
opp K is well-unital
proof end;
end;

Lm2: now
let K be non empty right_complementable add-associative right_zeroed doubleLoopStr ; :: thesis: for x, y, z being Scalar of (opp K) holds
( (x + y) + z = x + (y + z) & x + (0. (opp K)) = x )
set L = opp K;
thus for x, y, z being Scalar of (opp K) holds
( (x + y) + z = x + (y + z) & x + (0. (opp K)) = x ) :: thesis: verum
proof
let x, y, z be Scalar of (opp K); :: thesis: ( (x + y) + z = x + (y + z) & x + (0. (opp K)) = x )
reconsider a = x, b = y, c = z as Scalar of K ;
thus (x + y) + z = (a + b) + c
.= a + (b + c) by RLVECT_1:def 6
.= x + (y + z) ; :: thesis: x + (0. (opp K)) = x
thus x + (0. (opp K)) = a + (0. K)
.= x by RLVECT_1:def 7 ; :: thesis: verum
end;
end;

registration
let K be non empty right_complementable add-associative right_zeroed doubleLoopStr ;
cluster opp K -> right_complementable strict add-associative right_zeroed ;
coherence
( opp K is add-associative & opp K is right_zeroed & opp K is right_complementable )
proof end;
end;

Lm3: for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for x, y being Scalar of K
for a, b being Scalar of (opp K) st x = a & y = b holds
( x + y = a + b & x * y = b * a & - x = - a )
proof end;

theorem :: MOD_4:2
for K being non empty doubleLoopStr holds
( addLoopStr(# the carrier of (opp K),the addF of (opp K),the ZeroF of (opp K) #) = addLoopStr(# the carrier of K,the addF of K,the ZeroF of K #) & ( K is add-associative & K is right_zeroed & K is right_complementable implies comp (opp K) = comp K ) & ( for x being set holds
( x is Scalar of (opp K) iff x is Scalar of K ) ) )
proof end;

Lm4: for K being non empty doubleLoopStr
for x, y, z, u being Scalar of K
for a, b, c, d being Scalar of (opp K) st x = a & y = b & z = c & u = d holds
( (x + y) + z = (a + b) + c & x + (y + z) = a + (b + c) & (x * y) * z = c * (b * a) & x * (y * z) = (c * b) * a & x * (y + z) = (b + c) * a & (y + z) * x = a * (b + c) & (x * y) + (z * u) = (b * a) + (d * c) )
proof end;

theorem :: MOD_4:3
canceled;

theorem :: MOD_4:4
( ( for K being non empty unital doubleLoopStr holds 1. K = 1. (opp K) ) & ( for K being non empty right_complementable add-associative right_zeroed doubleLoopStr holds
( 0. K = 0. (opp K) & ( for x, y, z, u being Scalar of K
for a, b, c, d being Scalar of (opp K) st x = a & y = b & z = c & u = d holds
( x + y = a + b & x * y = b * a & - x = - a & (x + y) + z = (a + b) + c & x + (y + z) = a + (b + c) & (x * y) * z = c * (b * a) & x * (y * z) = (c * b) * a & x * (y + z) = (b + c) * a & (y + z) * x = a * (b + c) & (x * y) + (z * u) = (b * a) + (d * c) ) ) ) ) ) by Lm3, Lm4;

registration
let K be non empty Abelian doubleLoopStr ;
cluster opp K -> strict Abelian ;
coherence
opp K is Abelian
proof end;
end;

registration
let K be non empty add-associative doubleLoopStr ;
cluster opp K -> strict add-associative ;
coherence
opp K is add-associative
proof end;
end;

registration
let K be non empty right_zeroed doubleLoopStr ;
cluster opp K -> strict right_zeroed ;
coherence
opp K is right_zeroed
proof end;
end;

registration
let K be non empty right_complementable doubleLoopStr ;
cluster opp K -> right_complementable strict ;
coherence
opp K is right_complementable
proof end;
end;

registration
let K be non empty associative doubleLoopStr ;
cluster opp K -> strict associative ;
coherence
opp K is associative
proof end;
end;

registration
let K be non empty well-unital doubleLoopStr ;
cluster opp K -> strict well-unital ;
coherence
opp K is well-unital ;
end;

registration
let K be non empty distributive doubleLoopStr ;
cluster opp K -> strict distributive ;
coherence
opp K is distributive
proof end;
end;

theorem :: MOD_4:5
for K being Ring holds opp K is strict Ring ;

registration
let K be Ring;
cluster opp K -> right_complementable strict unital distributive Abelian add-associative right_zeroed ;
coherence
( opp K is Abelian & opp K is add-associative & opp K is right_zeroed & opp K is right_complementable & opp K is unital & opp K is distributive ) ;
end;

theorem :: MOD_4:6
for K being Ring holds opp K is Ring ;

registration
let K be Ring;
cluster opp K -> strict associative ;
coherence
opp K is associative ;
end;

theorem Th7: :: MOD_4:7
for K being Skew-Field holds opp K is Skew-Field
proof end;

registration
let K be Skew-Field;
cluster opp K -> non degenerated right_complementable almost_left_invertible strict unital associative distributive Abelian add-associative right_zeroed ;
coherence
( not opp K is degenerated & opp K is almost_left_invertible & opp K is associative & opp K is Abelian & opp K is add-associative & opp K is right_zeroed & opp K is right_complementable & opp K is unital & opp K is distributive ) by Th7;
end;

Lm5: for F being non empty commutative doubleLoopStr
for x, y being Element of F holds x * y = y * x
;

theorem :: MOD_4:8
for K being Field holds opp K is strict Field
proof end;

registration
let K be Field;
cluster opp K -> almost_left_invertible strict ;
coherence
( opp K is strict & opp K is almost_left_invertible ) ;
end;

begin

definition
let K be non empty doubleLoopStr ;
let V be non empty VectSpStr of K;
func opp V -> strict RightModStr of opp K means :Def2: :: MOD_4:def 2
for o being Function of [:the carrier of V,the carrier of (opp K):],the carrier of V st o = ~ the lmult of V holds
it = RightModStr(# the carrier of V,the addF of V,(0. V),o #);
existence
ex b1 being strict RightModStr of opp K st
for o being Function of [:the carrier of V,the carrier of (opp K):],the carrier of V st o = ~ the lmult of V holds
b1 = RightModStr(# the carrier of V,the addF of V,(0. V),o #)
proof end;
uniqueness
for b1, b2 being strict RightModStr of opp K st ( for o being Function of [:the carrier of V,the carrier of (opp K):],the carrier of V st o = ~ the lmult of V holds
b1 = RightModStr(# the carrier of V,the addF of V,(0. V),o #) ) & ( for o being Function of [:the carrier of V,the carrier of (opp K):],the carrier of V st o = ~ the lmult of V holds
b2 = RightModStr(# the carrier of V,the addF of V,(0. V),o #) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines opp MOD_4:def 2 :
for K being non empty doubleLoopStr
for V being non empty VectSpStr of K
for b3 being strict RightModStr of opp K holds
( b3 = opp V iff for o being Function of [:the carrier of V,the carrier of (opp K):],the carrier of V st o = ~ the lmult of V holds
b3 = RightModStr(# the carrier of V,the addF of V,(0. V),o #) );

registration
let K be non empty doubleLoopStr ;
let V be non empty VectSpStr of K;
cluster opp V -> non empty strict ;
coherence
not opp V is empty
proof end;
end;

theorem Th9: :: MOD_4:9
for K being non empty doubleLoopStr
for V being non empty VectSpStr of K holds
( addLoopStr(# the carrier of (opp V),the addF of (opp V),the ZeroF of (opp V) #) = addLoopStr(# the carrier of V,the addF of V,the ZeroF of V #) & ( for x being set holds
( x is Vector of V iff x is Vector of (opp V) ) ) )
proof end;

definition
let K be non empty doubleLoopStr ;
let V be non empty VectSpStr of K;
let o be Function of [:the carrier of K,the carrier of V:],the carrier of V;
func opp o -> Function of [:the carrier of (opp V),the carrier of (opp K):],the carrier of (opp V) equals :: MOD_4:def 3
 ~ o;
coherence
~ o is Function of [:the carrier of (opp V),the carrier of (opp K):],the carrier of (opp V)
proof end;
end;

:: deftheorem defines opp MOD_4:def 3 :
for K being non empty doubleLoopStr
for V being non empty VectSpStr of K
for o being Function of [:the carrier of K,the carrier of V:],the carrier of V holds opp o = ~ o;

theorem Th10: :: MOD_4:10
for K being non empty doubleLoopStr
for V being non empty VectSpStr of K holds the rmult of (opp V) = opp the lmult of V
proof end;

definition
let K be non empty doubleLoopStr ;
let W be non empty RightModStr of K;
func opp W -> strict VectSpStr of opp K means :Def4: :: MOD_4:def 4
for o being Function of [:the carrier of (opp K),the carrier of W:],the carrier of W st o = ~ the rmult of W holds
it = VectSpStr(# the carrier of W,the addF of W,(0. W),o #);
existence
ex b1 being strict VectSpStr of opp K st
for o being Function of [:the carrier of (opp K),the carrier of W:],the carrier of W st o = ~ the rmult of W holds
b1 = VectSpStr(# the carrier of W,the addF of W,(0. W),o #)
proof end;
uniqueness
for b1, b2 being strict VectSpStr of opp K st ( for o being Function of [:the carrier of (opp K),the carrier of W:],the carrier of W st o = ~ the rmult of W holds
b1 = VectSpStr(# the carrier of W,the addF of W,(0. W),o #) ) & ( for o being Function of [:the carrier of (opp K),the carrier of W:],the carrier of W st o = ~ the rmult of W holds
b2 = VectSpStr(# the carrier of W,the addF of W,(0. W),o #) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines opp MOD_4:def 4 :
for K being non empty doubleLoopStr
for W being non empty RightModStr of K
for b3 being strict VectSpStr of opp K holds
( b3 = opp W iff for o being Function of [:the carrier of (opp K),the carrier of W:],the carrier of W st o = ~ the rmult of W holds
b3 = VectSpStr(# the carrier of W,the addF of W,(0. W),o #) );

registration
let K be non empty doubleLoopStr ;
let W be non empty RightModStr of K;
cluster opp W -> non empty strict ;
coherence
not opp W is empty
proof end;
end;

theorem :: MOD_4:11
canceled;

theorem Th12: :: MOD_4:12
for K being non empty doubleLoopStr
for W being non empty RightModStr of K holds
( addLoopStr(# the carrier of (opp W),the addF of (opp W),the ZeroF of (opp W) #) = addLoopStr(# the carrier of W,the addF of W,the ZeroF of W #) & ( for x being set holds
( x is Vector of W iff x is Vector of (opp W) ) ) )
proof end;

definition
let K be non empty doubleLoopStr ;
let W be non empty RightModStr of K;
let o be Function of [:the carrier of W,the carrier of K:],the carrier of W;
func opp o -> Function of [:the carrier of (opp K),the carrier of (opp W):],the carrier of (opp W) equals :: MOD_4:def 5
 ~ o;
coherence
~ o is Function of [:the carrier of (opp K),the carrier of (opp W):],the carrier of (opp W)
proof end;
end;

:: deftheorem defines opp MOD_4:def 5 :
for K being non empty doubleLoopStr
for W being non empty RightModStr of K
for o being Function of [:the carrier of W,the carrier of K:],the carrier of W holds opp o = ~ o;

theorem Th13: :: MOD_4:13
for K being non empty doubleLoopStr
for W being non empty RightModStr of K holds the lmult of (opp W) = opp the rmult of W
proof end;

theorem :: MOD_4:14
canceled;

theorem :: MOD_4:15
canceled;

theorem :: MOD_4:16
for K being non empty doubleLoopStr
for V being non empty VectSpStr of K
for o being Function of [:the carrier of K,the carrier of V:],the carrier of V
for x being Scalar of K
for y being Scalar of (opp K)
for v being Vector of V
for w being Vector of (opp V) st x = y & v = w holds
(opp o) . w,y = o . x,v by Th1;

theorem Th17: :: MOD_4:17
for K, L being Ring
for V being non empty VectSpStr of K
for W being non empty RightModStr of L
for x being Scalar of K
for y being Scalar of L
for v being Vector of V
for w being Vector of W st L = opp K & W = opp V & x = y & v = w holds
w * y = x * v
proof end;

theorem Th18: :: MOD_4:18
for K, L being Ring
for V being non empty VectSpStr of K
for W being non empty RightModStr of L
for v1, v2 being Vector of V
for w1, w2 being Vector of W st W = opp V & v1 = w1 & v2 = w2 holds
w1 + w2 = v1 + v2
proof end;

theorem :: MOD_4:19
canceled;

theorem :: MOD_4:20
for K being non empty doubleLoopStr
for W being non empty RightModStr of K
for o being Function of [:the carrier of W,the carrier of K:],the carrier of W
for x being Scalar of K
for y being Scalar of (opp K)
for v being Vector of W
for w being Vector of (opp W) st x = y & v = w holds
(opp o) . y,w = o . v,x by Th1;

theorem Th21: :: MOD_4:21
for K, L being Ring
for V being non empty VectSpStr of K
for W being non empty RightModStr of L
for x being Scalar of K
for y being Scalar of L
for v being Vector of V
for w being Vector of W st V = opp W & x = y & v = w holds
w * y = x * v
proof end;

theorem Th22: :: MOD_4:22
for K, L being Ring
for V being non empty VectSpStr of K
for W being non empty RightModStr of L
for v1, v2 being Vector of V
for w1, w2 being Vector of W st V = opp W & v1 = w1 & v2 = w2 holds
w1 + w2 = v1 + v2
proof end;

theorem :: MOD_4:23
for K being non empty strict doubleLoopStr
for V being non empty VectSpStr of K holds opp (opp V) = VectSpStr(# the carrier of V,the addF of V,the ZeroF of V,the lmult of V #)
proof end;

theorem :: MOD_4:24
for K being non empty strict doubleLoopStr
for W being non empty RightModStr of K holds opp (opp W) = RightModStr(# the carrier of W,the addF of W,the ZeroF of W,the rmult of W #)
proof end;

theorem Th25: :: MOD_4:25
for K being Ring
for V being LeftMod of K holds opp V is strict RightMod of opp K
proof end;

registration
let K be Ring;
let V be LeftMod of K;
cluster opp V -> right_complementable Abelian add-associative right_zeroed strict RightMod-like ;
coherence
( opp V is Abelian & opp V is add-associative & opp V is right_zeroed & opp V is right_complementable & opp V is RightMod-like ) by Th25;
end;

theorem Th26: :: MOD_4:26
for K being Ring
for W being RightMod of K holds opp W is strict LeftMod of opp K
proof end;

registration
let K be Ring;
let W be RightMod of K;
cluster opp W -> right_complementable strict VectSp-like Abelian add-associative right_zeroed ;
coherence
( opp W is Abelian & opp W is add-associative & opp W is right_zeroed & opp W is right_complementable & opp W is VectSp-like ) by Th26;
end;

begin

definition
let K, L be non empty multMagma ;
let IT be Function of K,L;
attr IT is antimultiplicative means :Def6: :: MOD_4:def 6
for x, y being Scalar of K holds IT . (x * y) = (IT . y) * (IT . x);
end;

:: deftheorem Def6 defines antimultiplicative MOD_4:def 6 :
for K, L being non empty multMagma
for IT being Function of K,L holds
( IT is antimultiplicative iff for x, y being Scalar of K holds IT . (x * y) = (IT . y) * (IT . x) );

definition
let K, L be non empty doubleLoopStr ;
let IT be Function of K,L;
attr IT is antilinear means :Def7: :: MOD_4:def 7
( IT is additive & IT is antimultiplicative & IT is unity-preserving );
end;

:: deftheorem Def7 defines antilinear MOD_4:def 7 :
for K, L being non empty doubleLoopStr
for IT being Function of K,L holds
( IT is antilinear iff ( IT is additive & IT is antimultiplicative & IT is unity-preserving ) );

registration
let K, L be non empty doubleLoopStr ;
cluster Function-like quasi_total antilinear -> unity-preserving additive antimultiplicative Element of bool [:the carrier of K,the carrier of L:];
coherence
for b1 being Function of K,L st b1 is antilinear holds
( b1 is additive & b1 is antimultiplicative & b1 is unity-preserving ) by Def7;
cluster Function-like quasi_total unity-preserving additive antimultiplicative -> antilinear Element of bool [:the carrier of K,the carrier of L:];
coherence
for b1 being Function of K,L st b1 is additive & b1 is antimultiplicative & b1 is unity-preserving holds
b1 is antilinear by Def7;
end;

definition
let K, L be non empty doubleLoopStr ;
let IT be Function of K,L;
attr IT is monomorphism means :Def8: :: MOD_4:def 8
( IT is linear & IT is one-to-one );
attr IT is antimonomorphism means :Def9: :: MOD_4:def 9
( IT is antilinear & IT is one-to-one );
end;

:: deftheorem Def8 defines monomorphism MOD_4:def 8 :
for K, L being non empty doubleLoopStr
for IT being Function of K,L holds
( IT is monomorphism iff ( IT is linear & IT is one-to-one ) );

:: deftheorem Def9 defines antimonomorphism MOD_4:def 9 :
for K, L being non empty doubleLoopStr
for IT being Function of K,L holds
( IT is antimonomorphism iff ( IT is antilinear & IT is one-to-one ) );

definition
let K, L be non empty doubleLoopStr ;
let IT be Function of K,L;
attr IT is epimorphism means :Def10: :: MOD_4:def 10
( IT is linear & rng IT = the carrier of L );
attr IT is antiepimorphism means :Def11: :: MOD_4:def 11
( IT is antilinear & rng IT = the carrier of L );
end;

:: deftheorem Def10 defines epimorphism MOD_4:def 10 :
for K, L being non empty doubleLoopStr
for IT being Function of K,L holds
( IT is epimorphism iff ( IT is linear & rng IT = the carrier of L ) );

:: deftheorem Def11 defines antiepimorphism MOD_4:def 11 :
for K, L being non empty doubleLoopStr
for IT being Function of K,L holds
( IT is antiepimorphism iff ( IT is antilinear & rng IT = the carrier of L ) );

definition
let K, L be non empty doubleLoopStr ;
let IT be Function of K,L;
attr IT is isomorphism means :Def12: :: MOD_4:def 12
( IT is monomorphism & rng IT = the carrier of L );
attr IT is antiisomorphism means :Def13: :: MOD_4:def 13
( IT is antimonomorphism & rng IT = the carrier of L );
end;

:: deftheorem Def12 defines isomorphism MOD_4:def 12 :
for K, L being non empty doubleLoopStr
for IT being Function of K,L holds
( IT is isomorphism iff ( IT is monomorphism & rng IT = the carrier of L ) );

:: deftheorem Def13 defines antiisomorphism MOD_4:def 13 :
for K, L being non empty doubleLoopStr
for IT being Function of K,L holds
( IT is antiisomorphism iff ( IT is antimonomorphism & rng IT = the carrier of L ) );

definition
let K be non empty doubleLoopStr ;
let IT be Function of K,K;
attr IT is endomorphism means :Def14: :: MOD_4:def 14
IT is linear ;
attr IT is antiendomorphism means :Def15: :: MOD_4:def 15
IT is antilinear ;
attr IT is automorphism means :Def16: :: MOD_4:def 16
IT is isomorphism ;
attr IT is antiautomorphism means :Def17: :: MOD_4:def 17
IT is antiisomorphism ;
end;

:: deftheorem Def14 defines endomorphism MOD_4:def 14 :
for K being non empty doubleLoopStr
for IT being Function of K,K holds
( IT is endomorphism iff IT is linear );

:: deftheorem Def15 defines antiendomorphism MOD_4:def 15 :
for K being non empty doubleLoopStr
for IT being Function of K,K holds
( IT is antiendomorphism iff IT is antilinear );

:: deftheorem Def16 defines automorphism MOD_4:def 16 :
for K being non empty doubleLoopStr
for IT being Function of K,K holds
( IT is automorphism iff IT is isomorphism );

:: deftheorem Def17 defines antiautomorphism MOD_4:def 17 :
for K being non empty doubleLoopStr
for IT being Function of K,K holds
( IT is antiautomorphism iff IT is antiisomorphism );

theorem :: MOD_4:27
for K being non empty doubleLoopStr
for J being Function of K,K holds
( J is automorphism iff ( ( for x, y being Scalar of K holds J . (x + y) = (J . x) + (J . y) ) & ( for x, y being Scalar of K holds J . (x * y) = (J . x) * (J . y) ) & J . (1_ K) = 1_ K & J is one-to-one & rng J = the carrier of K ) )
proof end;

theorem Th28: :: MOD_4:28
for K being non empty doubleLoopStr
for J being Function of K,K holds
( J is antiautomorphism iff ( ( for x, y being Scalar of K holds J . (x + y) = (J . x) + (J . y) ) & ( for x, y being Scalar of K holds J . (x * y) = (J . y) * (J . x) ) & J . (1_ K) = 1_ K & J is one-to-one & rng J = the carrier of K ) )
proof end;

Lm6: for K being non empty doubleLoopStr holds
( ( for x, y being Scalar of K holds (id K) . (x + y) = ((id K) . x) + ((id K) . y) ) & ( for x, y being Scalar of K holds (id K) . (x * y) = ((id K) . x) * ((id K) . y) ) & (id K) . (1_ K) = 1_ K & id K is one-to-one & rng (id K) = the carrier of K )
proof end;

theorem :: MOD_4:29
for K being non empty doubleLoopStr holds id K is automorphism
proof end;

Lm7: for K, L being Ring
for J being Function of K,L st J is linear holds
J . (0. K) = 0. L
proof end;

Lm8: for L, K being Ring
for J being Function of K,L
for x being Scalar of K st J is linear holds
J . (- x) = - (J . x)
proof end;

Lm9: for L, K being Ring
for J being Function of K,L
for x, y being Scalar of K st J is linear holds
J . (x - y) = (J . x) - (J . y)
proof end;

theorem :: MOD_4:30
for K, L being Ring
for J being Function of K,L
for x, y being Scalar of K st J is linear holds
( J . (0. K) = 0. L & J . (- x) = - (J . x) & J . (x - y) = (J . x) - (J . y) ) by Lm7, Lm8, Lm9;

Lm10: for K, L being Ring
for J being Function of K,L st J is antilinear holds
J . (0. K) = 0. L
proof end;

Lm11: for L, K being Ring
for J being Function of K,L
for x being Scalar of K st J is antilinear holds
J . (- x) = - (J . x)
proof end;

Lm12: for L, K being Ring
for J being Function of K,L
for x, y being Scalar of K st J is antilinear holds
J . (x - y) = (J . x) - (J . y)
proof end;

theorem :: MOD_4:31
for K, L being Ring
for J being Function of K,L
for x, y being Scalar of K st J is antilinear holds
( J . (0. K) = 0. L & J . (- x) = - (J . x) & J . (x - y) = (J . x) - (J . y) ) by Lm10, Lm11, Lm12;

theorem :: MOD_4:32
for K being Ring holds
( id K is antiautomorphism iff K is comRing )
proof end;

theorem :: MOD_4:33
for K being Skew-Field holds
( id K is antiautomorphism iff K is Field )
proof end;

begin

definition
let K, L be non empty doubleLoopStr ;
let J be Function of K,L;
func opp J -> Function of K,(opp L) equals :: MOD_4:def 18
J;
coherence
J is Function of K,(opp L) ;
end;

:: deftheorem defines opp MOD_4:def 18 :
for K, L being non empty doubleLoopStr
for J being Function of K,L holds opp J = J;

theorem :: MOD_4:34
for K, L being non empty right_complementable add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds opp (opp J) = J ;

theorem Th35: :: MOD_4:35
for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is linear iff opp J is antilinear )
proof end;

theorem Th36: :: MOD_4:36
for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is antilinear iff opp J is linear )
proof end;

theorem Th37: :: MOD_4:37
for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is monomorphism iff opp J is antimonomorphism )
proof end;

theorem Th38: :: MOD_4:38
for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is antimonomorphism iff opp J is monomorphism )
proof end;

theorem :: MOD_4:39
for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is epimorphism iff opp J is antiepimorphism )
proof end;

theorem :: MOD_4:40
for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is antiepimorphism iff opp J is epimorphism )
proof end;

theorem Th41: :: MOD_4:41
for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is isomorphism iff opp J is antiisomorphism )
proof end;

theorem Th42: :: MOD_4:42
for K being non empty right_complementable add-associative right_zeroed doubleLoopStr
for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is antiisomorphism iff opp J is isomorphism )
proof end;

theorem :: MOD_4:43
for K being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,K holds
( J is endomorphism iff opp J is antilinear )
proof end;

theorem :: MOD_4:44
for K being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,K holds
( J is antiendomorphism iff opp J is linear )
proof end;

theorem :: MOD_4:45
for K being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,K holds
( J is automorphism iff opp J is antiisomorphism )
proof end;

theorem :: MOD_4:46
for K being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,K holds
( J is antiautomorphism iff opp J is isomorphism )
proof end;

begin

Lm13: for G, H being AddGroup
for x, y being Element of G holds (ZeroMap G,H) . (x + y) = ((ZeroMap G,H) . x) + ((ZeroMap G,H) . y)
proof end;

definition
let G, H be AddGroup;
mode Homomorphism of G,H -> Function of G,H means :Def19: :: MOD_4:def 19
for x, y being Element of G holds it . (x + y) = (it . x) + (it . y);
existence
ex b1 being Function of G,H st
for x, y being Element of G holds b1 . (x + y) = (b1 . x) + (b1 . y)
proof end;
end;

:: deftheorem Def19 defines Homomorphism MOD_4:def 19 :
for G, H being AddGroup
for b3 being Function of G,H holds
( b3 is Homomorphism of G,H iff for x, y being Element of G holds b3 . (x + y) = (b3 . x) + (b3 . y) );

definition
let G, H be AddGroup;
:: original: ZeroMap
redefine func ZeroMap G,H -> Homomorphism of G,H;
coherence
ZeroMap G,H is Homomorphism of G,H
proof end;
end;

definition
let G be AddGroup;
mode Endomorphism of G is Homomorphism of G,G;
end;

Lm14: for G being AddGroup holds
( ( for x, y being Element of G holds (id G) . (x + y) = ((id G) . x) + ((id G) . y) ) & id G is one-to-one & id G is onto )
proof end;

registration
let G be AddGroup;
cluster Relation-like the carrier of G -defined the carrier of G -valued Function-like quasi_total bijective Homomorphism of G,G;
existence
ex b1 being Endomorphism of G st b1 is bijective
proof end;
end;

definition
let G be AddGroup;
mode Automorphism of G is bijective Endomorphism of G;
end;

definition
let G be AddGroup;
:: original: id
redefine func id G -> Automorphism of G;
coherence
id G is Automorphism of G
proof end;
end;

Lm15: for G, H being AddGroup
for f being Homomorphism of G,H holds f . (0. G) = 0. H
proof end;

Lm16: for H, G being AddGroup
for f being Homomorphism of G,H
for x being Element of G holds f . (- x) = - (f . x)
proof end;

Lm17: for H, G being AddGroup
for f being Homomorphism of G,H
for x, y being Element of G holds f . (x - y) = (f . x) - (f . y)
proof end;

theorem :: MOD_4:47
canceled;

theorem :: MOD_4:48
for G, H being AddGroup
for f being Homomorphism of G,H
for x, y being Element of G holds
( f . (0. G) = 0. H & f . (- x) = - (f . x) & f . (x - y) = (f . x) - (f . y) ) by Lm15, Lm16, Lm17;

theorem :: MOD_4:49
for H, G being AbGroup
for f being Homomorphism of G,H
for x, y being Element of G holds f . (x - y) = (f . x) - (f . y)
proof end;

begin

Lm18: for K, L being Ring
for V being LeftMod of K
for W being LeftMod of L
for x, y being Vector of V holds (ZeroMap V,W) . (x + y) = ((ZeroMap V,W) . x) + ((ZeroMap V,W) . y)
proof end;

Lm19: for L, K being Ring
for J being Function of K,L
for V being LeftMod of K
for W being LeftMod of L
for a being Scalar of K
for x being Vector of V holds (ZeroMap V,W) . (a * x) = (J . a) * ((ZeroMap V,W) . x)
proof end;

definition
let K, L be Ring;
let J be Function of K,L;
let V be LeftMod of K;
let W be LeftMod of L;
canceled;
canceled;
canceled;
canceled;
mode Homomorphism of J,V,W -> Function of V,W means :Def24: :: MOD_4:def 24
( ( for x, y being Vector of V holds it . (x + y) = (it . x) + (it . y) ) & ( for a being Scalar of K
for x being Vector of V holds it . (a * x) = (J . a) * (it . x) ) );
existence
ex b1 being Function of V,W st
( ( for x, y being Vector of V holds b1 . (x + y) = (b1 . x) + (b1 . y) ) & ( for a being Scalar of K
for x being Vector of V holds b1 . (a * x) = (J . a) * (b1 . x) ) )
proof end;
end;

:: deftheorem MOD_4:def 20 :
canceled;

:: deftheorem MOD_4:def 21 :
canceled;

:: deftheorem MOD_4:def 22 :
canceled;

:: deftheorem MOD_4:def 23 :
canceled;

:: deftheorem Def24 defines Homomorphism MOD_4:def 24 :
for K, L being Ring
for J being Function of K,L
for V being LeftMod of K
for W being LeftMod of L
for b6 being Function of V,W holds
( b6 is Homomorphism of J,V,W iff ( ( for x, y being Vector of V holds b6 . (x + y) = (b6 . x) + (b6 . y) ) & ( for a being Scalar of K
for x being Vector of V holds b6 . (a * x) = (J . a) * (b6 . x) ) ) );

theorem :: MOD_4:50
for K, L being Ring
for J being Function of K,L
for V being LeftMod of K
for W being LeftMod of L holds ZeroMap V,W is Homomorphism of J,V,W
proof end;

definition
let K, L be Ring;
let J be Function of K,L;
let V be LeftMod of K;
let W be LeftMod of L;
let f be Homomorphism of J,V,W;
pred f is_monomorphism_wrp J means :: MOD_4:def 25
f is one-to-one ;
pred f is_epimorphism_wrp J means :: MOD_4:def 26
 rng f = the carrier of W;
pred f is_isomorphism_wrp J means :: MOD_4:def 27
( f is one-to-one & rng f = the carrier of W );
end;

:: deftheorem defines is_monomorphism_wrp MOD_4:def 25 :
for K, L being Ring
for J being Function of K,L
for V being LeftMod of K
for W being LeftMod of L
for f being Homomorphism of J,V,W holds
( f is_monomorphism_wrp J iff f is one-to-one );

:: deftheorem defines is_epimorphism_wrp MOD_4:def 26 :
for K, L being Ring
for J being Function of K,L
for V being LeftMod of K
for W being LeftMod of L
for f being Homomorphism of J,V,W holds
( f is_epimorphism_wrp J iff rng f = the carrier of W );

:: deftheorem defines is_isomorphism_wrp MOD_4:def 27 :
for K, L being Ring
for J being Function of K,L
for V being LeftMod of K
for W being LeftMod of L
for f being Homomorphism of J,V,W holds
( f is_isomorphism_wrp J iff ( f is one-to-one & rng f = the carrier of W ) );

definition
let K be Ring;
let J be Function of K,K;
let V be LeftMod of K;
mode Endomorphism of J,V is Homomorphism of J,V,V;
end;

definition
let K be Ring;
let J be Function of K,K;
let V be LeftMod of K;
let f be Homomorphism of J,V,V;
pred f is_automorphism_wrp J means :: MOD_4:def 28
( f is one-to-one & rng f = the carrier of V );
end;

:: deftheorem defines is_automorphism_wrp MOD_4:def 28 :
for K being Ring
for J being Function of K,K
for V being LeftMod of K
for f being Homomorphism of J,V,V holds
( f is_automorphism_wrp J iff ( f is one-to-one & rng f = the carrier of V ) );

definition
let K be Ring;
let V, W be LeftMod of K;
mode Homomorphism of V,W is Homomorphism of id K,V,W;
end;

theorem :: MOD_4:51
for K being Ring
for V, W being LeftMod of K
for f being Function of V,W holds
( f is Homomorphism of V,W iff ( ( for x, y being Vector of V holds f . (x + y) = (f . x) + (f . y) ) & ( for a being Scalar of K
for x being Vector of V holds f . (a * x) = a * (f . x) ) ) )
proof end;

definition
let K be Ring;
let V, W be LeftMod of K;
let IT be Homomorphism of V,W;
attr IT is monomorphism means :: MOD_4:def 29
IT is one-to-one ;
attr IT is epimorphism means :: MOD_4:def 30
 rng IT = the carrier of W;
attr IT is isomorphism means :: MOD_4:def 31
( IT is one-to-one & rng IT = the carrier of W );
end;

:: deftheorem defines monomorphism MOD_4:def 29 :
for K being Ring
for V, W being LeftMod of K
for IT being Homomorphism of V,W holds
( IT is monomorphism iff IT is one-to-one );

:: deftheorem defines epimorphism MOD_4:def 30 :
for K being Ring
for V, W being LeftMod of K
for IT being Homomorphism of V,W holds
( IT is epimorphism iff rng IT = the carrier of W );

:: deftheorem defines isomorphism MOD_4:def 31 :
for K being Ring
for V, W being LeftMod of K
for IT being Homomorphism of V,W holds
( IT is isomorphism iff ( IT is one-to-one & rng IT = the carrier of W ) );

definition
let K be Ring;
let V be LeftMod of K;
mode Endomorphism of V is Homomorphism of V,V;
end;

definition
let K be Ring;
let V be LeftMod of K;
let IT be Endomorphism of V;
attr IT is automorphism means :: MOD_4:def 32
( IT is one-to-one & rng IT = the carrier of V );
end;

:: deftheorem defines automorphism MOD_4:def 32 :
for K being Ring
for V being LeftMod of K
for IT being Endomorphism of V holds
( IT is automorphism iff ( IT is one-to-one & rng IT = the carrier of V ) );