:: MOD_4 semantic presentation

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 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;

definition

let K be non empty doubleLoopStr ;
func opp K -> strict doubleLoopStr equals :: MOD_4:def 1
doubleLoopStr(# the carrier of K,the add of K,(~ the mult of K),the unity of K,the Zero of K #);
correctness ;

end;

:: deftheorem defines opp MOD_4:def 1 :

registration

let K be non empty doubleLoopStr ;
cluster opp K -> non empty strict ;
coherence

proof end;

end;

Lm1: for K being non empty 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 )
by Th1;

Lm2: for K being non empty 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 unital doubleLoopStr ;
cluster opp K -> non empty unital strict ;
coherence

proof end;

end;

Lm3: for K being non empty unital doubleLoopStr holds 1. K = 1. (opp K)

proof end;

Lm4: now

let K be non empty add-associative right_zeroed right_complementable doubleLoopStr ; :: thesis:
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:

proof

let x, y, z be Scalar of (opp K); :: thesis:
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:
thus x + (0. (opp K)) = a + (0. K)
.= x by RLVECT_1:def 7 ; :: thesis:

end;

registration

let K be non empty add-associative right_zeroed right_complementable doubleLoopStr ;
cluster opp K -> non empty add-associative right_zeroed right_complementable strict ;
coherence

proof end;

end;

Lm5: for K being non empty add-associative right_zeroed right_complementable 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
( LoopStr(# the carrier of (opp K),the add of (opp K),the Zero of (opp K) #) = LoopStr(# the carrier of K,the add of K,the Zero 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;

theorem Th3: :: MOD_4:3

for K being non empty doubleLoopStr holds opp (opp K) = doubleLoopStr(# the carrier of K,the add of K,the mult of K,the unity of K,the Zero of K #) by FUNCT_4:56;

Lm6: for K being non empty add-associative right_zeroed right_complementable 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_4:4

( ( for K being non empty unital doubleLoopStr holds 1. K = 1. (opp K) ) & ( for K being non empty add-associative right_zeroed right_complementable 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 Lm5, Lm6, Lm3;

theorem Th5: :: MOD_4:5

for K being Ring holds opp K is strict Ring

proof end;

registration

let K be Ring;
cluster opp K -> non empty Abelian add-associative right_zeroed right_complementable unital strict distributive ;
coherence by Th5;

end;

theorem Th6: :: MOD_4:6

for K being Ring holds opp K is Ring

proof end;

registration

let K be Ring;
cluster opp K -> non empty Abelian add-associative right_zeroed right_complementable unital associative strict distributive ;
coherence by Th6;

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 empty Abelian add-associative right_zeroed right_complementable unital associative strict distributive Field-like non degenerated ;
coherence by Th7;

end;

Lm7: 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 -> non empty Abelian add-associative right_zeroed right_complementable unital associative strict distributive Field-like non degenerated ;
coherence ;

end;

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 add of V,the Zero of V,o #);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines opp MOD_4:def 2 :

registration

let K be non empty doubleLoopStr ;
let V be non empty VectSpStr of K;
cluster opp V -> non empty strict ;
coherence

proof end;

end;

theorem Th9: :: MOD_4:9

for K being non empty doubleLoopStr
for V being non empty VectSpStr of K holds
( LoopStr(# the carrier of (opp V),the add of (opp V),the Zero of (opp V) #) = LoopStr(# the carrier of V,the add of V,the Zero 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

proof end;

end;

:: deftheorem defines opp MOD_4:def 3 :

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 add of W,the Zero of W,o #);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines opp MOD_4:def 4 :

registration

let K be non empty doubleLoopStr ;
let W be non empty RightModStr of K;
cluster opp W -> non empty strict ;
coherence

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
( LoopStr(# the carrier of (opp W),the add of (opp W),the Zero of (opp W) #) = LoopStr(# the carrier of W,the add of W,the Zero 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

proof end;

end;

:: deftheorem defines opp MOD_4:def 5 :

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 Th15: :: MOD_4:15

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 (opp o) = o by FUNCT_4:56;

theorem Th16: :: 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 L = opp K & W = opp V & v1 = w1 & v2 = w2 holds
w1 + w2 = v1 + v2

proof end;

theorem Th19: :: MOD_4:19

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 (opp o) = o by FUNCT_4:56;

theorem Th20: :: 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 K = opp L & 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 K = opp L & 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 add of V,the Zero 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 add of W,the Zero 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 -> non empty Abelian add-associative right_zeroed right_complementable strict RightMod-like ;
coherence 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 -> non empty Abelian add-associative right_zeroed right_complementable strict VectSp-like ;
coherence by Th26;

end;

definition

let K, L be non empty doubleLoopStr ;
let IT be Function of K,L;
attr IT is antilinear means :Def6: :: MOD_4:def 6
( ( for x, y being Scalar of K holds IT . (x + y) = (IT . x) + (IT . y) ) & ( for x, y being Scalar of K holds IT . (x * y) = (IT . y) * (IT . x) ) & IT . (1. K) = 1. L );

end;

:: deftheorem Def6 defines antilinear MOD_4:def 6 :

definition

let K, L be non empty doubleLoopStr ;
let IT be Function of K,L;
attr IT is monomorphism means :Def7: :: MOD_4:def 7
( IT is linear & IT is one-to-one );
attr IT is antimonomorphism means :Def8: :: MOD_4:def 8
( IT is antilinear & IT is one-to-one );

end;

:: deftheorem Def7 defines monomorphism MOD_4:def 7 :

:: deftheorem Def8 defines antimonomorphism MOD_4:def 8 :

definition

let K, L be non empty doubleLoopStr ;
let IT be Function of K,L;
attr IT is epimorphism means :Def9: :: MOD_4:def 9
( IT is linear & rng IT = the carrier of L );
attr IT is antiepimorphism means :Def10: :: MOD_4:def 10
( IT is antilinear & rng IT = the carrier of L );

end;

:: deftheorem Def9 defines epimorphism MOD_4:def 9 :

:: deftheorem Def10 defines antiepimorphism MOD_4:def 10 :

definition

let K, L be non empty doubleLoopStr ;
let IT be Function of K,L;
attr IT is isomorphism means :Def11: :: MOD_4:def 11
( IT is monomorphism & rng IT = the carrier of L );
attr IT is antiisomorphism means :Def12: :: MOD_4:def 12
( IT is antimonomorphism & rng IT = the carrier of L );

end;

:: deftheorem Def11 defines isomorphism MOD_4:def 11 :

:: deftheorem Def12 defines antiisomorphism MOD_4:def 12 :

definition

let K be non empty doubleLoopStr ;
let IT be Function of K,K;
attr IT is endomorphism means :Def13: :: MOD_4:def 13
IT is linear;
attr IT is antiendomorphism means :Def14: :: MOD_4:def 14
IT is antilinear;
attr IT is automorphism means :Def15: :: MOD_4:def 15
IT is isomorphism;
attr IT is antiautomorphism means :Def16: :: MOD_4:def 16
IT is antiisomorphism;

end;

:: deftheorem Def13 defines endomorphism MOD_4:def 13 :

:: deftheorem Def14 defines antiendomorphism MOD_4:def 14 :

:: deftheorem Def15 defines automorphism MOD_4:def 15 :

:: deftheorem Def16 defines antiautomorphism MOD_4:def 16 :

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;

Lm8: 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;

Lm9: for K, L being Ring
for J being Function of K,L st J is linear holds
J . (0. K) = 0. L

proof end;

Lm10: 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;

Lm11: 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 Lm9, Lm10, Lm11;

Lm12: for K, L being Ring
for J being Function of K,L st J is antilinear holds
J . (0. K) = 0. L

proof end;

Lm13: 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;

Lm14: 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 Lm12, Lm13, Lm14;

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;

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 17
J;
coherence ;

end;

:: deftheorem defines opp MOD_4:def 17 :

theorem :: MOD_4:34

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

theorem Th35: :: MOD_4:35

for K being non empty add-associative right_zeroed right_complementable doubleLoopStr
for L being non empty add-associative right_zeroed right_complementable unital doubleLoopStr
for J being Function of K,L holds
( J is linear iff opp J is antilinear )

proof end;

theorem Th36: :: MOD_4:36

proof end;

theorem Th37: :: MOD_4:37

proof end;

theorem Th38: :: MOD_4:38

proof end;

theorem :: MOD_4:39

proof end;

theorem :: MOD_4:40

proof end;

theorem Th41: :: MOD_4:41

proof end;

theorem Th42: :: MOD_4:42

proof end;

theorem :: MOD_4:43

for K being non empty add-associative right_zeroed right_complementable unital 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 add-associative right_zeroed right_complementable unital 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 add-associative right_zeroed right_complementable unital 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 add-associative right_zeroed right_complementable unital doubleLoopStr
for J being Function of K,K holds
( J is antiautomorphism iff opp J is isomorphism )

proof end;

Lm15: 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 :Def18: :: MOD_4:def 18
for x, y being Element of G holds it . (x + y) = (it . x) + (it . y);
existence

proof end;

end;

:: deftheorem Def18 defines Homomorphism MOD_4:def 18 :

definition

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

proof end;

end;

definition

let G, H be AddGroup;
let IT be Homomorphism of G,H;
attr IT is monomorphism means :: MOD_4:def 19
IT is one-to-one;

end;

:: deftheorem defines monomorphism MOD_4:def 19 :

definition

let G, H be AddGroup;
let IT be Homomorphism of G,H;
attr IT is epimorphism means :: MOD_4:def 20
rng IT = the carrier of H;

end;

:: deftheorem defines epimorphism MOD_4:def 20 :

definition

let G, H be AddGroup;
let IT be Homomorphism of G,H;
attr IT is isomorphism means :Def21: :: MOD_4:def 21
( IT is one-to-one & rng IT = the carrier of H );

end;

:: deftheorem Def21 defines isomorphism MOD_4:def 21 :

definition

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

end;

Lm16: for G being AddGroup
for x being Element of G holds (id G) . x = x

proof end;

Lm17: 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 & rng (id G) = the carrier of G )

proof end;

registration

let G be AddGroup;
cluster isomorphism Homomorphism of G,G;
existence

proof end;

end;

definition

let G be AddGroup;
mode Automorphism of G is isomorphism Endomorphism of G;

end;

definition

let G be AddGroup;
:: original: id
redefine func id G -> Automorphism of G;
coherence

proof end;

end;

Lm18: for G, H being AddGroup
for f being Homomorphism of G,H holds f . (0. G) = 0. H

proof end;

Lm19: 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;

Lm20: 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 Lm18, Lm19, Lm20;

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;

Lm21: 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;

Lm22: 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;
mode Homomorphism of J,V,W -> Function of V,W means :Def23: :: MOD_4:def 23
( ( 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

proof end;

end;

:: deftheorem MOD_4:def 22 :

:: deftheorem Def23 defines Homomorphism MOD_4:def 23 :

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 24
f is one-to-one;
pred f is_epimorphism_wrp J means :: MOD_4:def 25
rng f = the carrier of W;
pred f is_isomorphism_wrp J means :: MOD_4:def 26
( f is one-to-one & rng f = the carrier of W );

end;

:: deftheorem defines is_monomorphism_wrp MOD_4:def 24 :

:: deftheorem defines is_epimorphism_wrp MOD_4:def 25 :

:: deftheorem defines is_isomorphism_wrp MOD_4:def 26 :

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 27
( f is one-to-one & rng f = the carrier of V );

end;

:: deftheorem defines is_automorphism_wrp MOD_4:def 27 :

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 28
IT is one-to-one;
attr IT is epimorphism means :: MOD_4:def 29
rng IT = the carrier of W;
attr IT is isomorphism means :: MOD_4:def 30
( IT is one-to-one & rng IT = the carrier of W );

end;

:: deftheorem defines monomorphism MOD_4:def 28 :

:: deftheorem defines epimorphism MOD_4:def 29 :

:: deftheorem defines isomorphism MOD_4:def 30 :

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 31
( IT is one-to-one & rng IT = the carrier of V );

end;

:: deftheorem defines automorphism MOD_4:def 31 :