:: MONOID_0 semantic presentation

deffunc H₁( 1-sorted ) -> set = the carrier of $1;

deffunc H₂( HGrStr ) -> Relation of [:the carrier of $1,the carrier of $1:],the carrier of $1 = the mult of $1;

definition

let G be 1-sorted ;
mode BinOp of G is BinOp of the carrier of G;

end;

definition

let IT be 1-sorted ;
attr IT is constituted-Functions means :Def1: :: MONOID_0:def 1
for a being Element of IT holds a is Function;
attr IT is constituted-FinSeqs means :Def2: :: MONOID_0:def 2
for a being Element of IT holds a is FinSequence;

end;

:: deftheorem Def1 defines constituted-Functions MONOID_0:def 1 :

:: deftheorem Def2 defines constituted-FinSeqs MONOID_0:def 2 :

registration

cluster constituted-Functions 1-sorted ;
existence

proof end;

cluster constituted-FinSeqs 1-sorted ;
existence

proof end;

end;

registration

let X be constituted-Functions 1-sorted ;
cluster -> Relation-like Function-like Element of the carrier of X;
coherence by Def1;

end;

registration

cluster constituted-FinSeqs -> constituted-Functions 1-sorted ;
coherence

proof end;

end;

registration

cluster constituted-FinSeqs -> constituted-Functions HGrStr ;
coherence

proof end;

end;

registration

let X be constituted-FinSeqs 1-sorted ;
cluster -> Relation-like Function-like FinSequence-like Element of the carrier of X;
coherence by Def2;

end;

definition

let D be set ;
let p, q be FinSequence of D;
:: original: ^
redefine func p ^ q -> Element of D * ;
coherence

proof end;

end;

Lm1: ( ( for X being set
for f, g being Function of X,X holds f * g is Function of X,X ) & ( for X being set
for f, g being Permutation of X holds f * g is Permutation of X ) & ( for X, Y, Z being set
for f being PartFunc of X,Y
for g being PartFunc of Y,Z holds g * f is PartFunc of X,Z ) & ( for X being set
for A, B being non empty set
for f being Function of X,A
for g being Function of A,B holds g * f is Function of X,B ) )
;

notation

let g, f be Function;
synonym f (*) g for f * g;

end;

definition

let X be set ;
let g, f be Function of X,X;
:: original: (*)
redefine func f * g -> Function of X,X;
coherence by Lm1;

end;

definition

let X be set ;
let g, f be Permutation of X;
:: original: (*)
redefine func f * g -> Permutation of X;
coherence by Lm1;

end;

definition

let D be non empty set ;
let IT be BinOp of D;
attr IT is left-invertible means :: MONOID_0:def 3
for a, b being Element of D ex l being Element of D st IT . l,a = b;
attr IT is right-invertible means :: MONOID_0:def 4
for a, b being Element of D ex r being Element of D st IT . a,r = b;
attr IT is invertible means :Def5: :: MONOID_0:def 5
for a, b being Element of D ex r, l being Element of D st
( IT . a,r = b & IT . l,a = b );
attr IT is left-cancelable means :: MONOID_0:def 6
for a, b, c being Element of D st IT . a,b = IT . a,c holds
b = c;
attr IT is right-cancelable means :: MONOID_0:def 7
for a, b, c being Element of D st IT . b,a = IT . c,a holds
b = c;
attr IT is cancelable means :: MONOID_0:def 8
for a, b, c being Element of D st ( IT . a,b = IT . a,c or IT . b,a = IT . c,a ) holds
b = c;
attr IT is uniquely-decomposable means :: MONOID_0:def 9
( IT has_a_unity & ( for a, b being Element of D st IT . a,b = the_unity_wrt IT holds
( a = b & b = the_unity_wrt IT ) ) );

end;

:: deftheorem defines left-invertible MONOID_0:def 3 :

:: deftheorem defines right-invertible MONOID_0:def 4 :

:: deftheorem Def5 defines invertible MONOID_0:def 5 :

:: deftheorem defines left-cancelable MONOID_0:def 6 :

:: deftheorem defines right-cancelable MONOID_0:def 7 :

:: deftheorem defines cancelable MONOID_0:def 8 :

:: deftheorem defines uniquely-decomposable MONOID_0:def 9 :

theorem Th1: :: MONOID_0:1

for D being non empty set
for f being BinOp of D holds
( f is invertible iff ( f is left-invertible & f is right-invertible ) )

proof end;

theorem Th2: :: MONOID_0:2

for D being non empty set
for f being BinOp of D holds
( f is cancelable iff ( f is left-cancelable & f is right-cancelable ) )

proof end;

theorem Th3: :: MONOID_0:3

for x being set
for f being BinOp of {x} holds
( f = {[x,x]} --> x & f has_a_unity & f is commutative & f is associative & f is idempotent & f is invertible & f is cancelable & f is uniquely-decomposable )

proof end;

definition

let IT be non empty HGrStr ;
redefine attr IT is unital means :Def10: :: MONOID_0:def 10
the mult of IT has_a_unity ;
compatibility

proof end;

end;

:: deftheorem Def10 defines unital MONOID_0:def 10 :

definition

let G be non empty HGrStr ;
redefine attr G is commutative means :: MONOID_0:def 11
the mult of G is commutative;
compatibility

proof end;

redefine attr G is associative means :Def12: :: MONOID_0:def 12
the mult of G is associative;
compatibility

proof end;

end;

:: deftheorem defines commutative MONOID_0:def 11 :

:: deftheorem Def12 defines associative MONOID_0:def 12 :

definition

end;

:: deftheorem defines idempotent MONOID_0:def 13 :

:: deftheorem defines left-invertible MONOID_0:def 14 :

:: deftheorem defines right-invertible MONOID_0:def 15 :

:: deftheorem Def16 defines invertible MONOID_0:def 16 :

:: deftheorem defines left-cancelable MONOID_0:def 17 :

:: deftheorem defines right-cancelable MONOID_0:def 18 :

:: deftheorem defines cancelable MONOID_0:def 19 :

:: deftheorem Def20 defines uniquely-decomposable MONOID_0:def 20 :

registration

cluster non empty strict unital associative commutative constituted-Functions constituted-FinSeqs idempotent invertible cancelable uniquely-decomposable HGrStr ;
existence

proof end;

end;

theorem Th4: :: MONOID_0:4

for G being non empty HGrStr st G is unital holds
the_unity_wrt the mult of G is_a_unity_wrt the mult of G

proof end;

theorem :: MONOID_0:5

for G being non empty HGrStr holds
( G is unital iff for a being Element of G holds
( (the_unity_wrt the mult of G) * a = a & a * (the_unity_wrt the mult of G) = a ) )

proof end;

theorem Th6: :: MONOID_0:6

for G being non empty HGrStr holds
( G is unital iff ex a being Element of G st
for b being Element of G holds
( a * b = b & b * a = b ) )

proof end;

Lm2: for G being non empty HGrStr holds
( G is commutative iff for a, b being Element of G holds a * b = b * a )

proof end;

Lm3: for G being non empty HGrStr holds
( G is associative iff for a, b, c being Element of G holds (a * b) * c = a * (b * c) )

proof end;

theorem :: MONOID_0:7

canceled;

theorem :: MONOID_0:8

canceled;

theorem Th9: :: MONOID_0:9

for G being non empty HGrStr holds
( G is idempotent iff for a being Element of G holds a * a = a )

proof end;

theorem :: MONOID_0:10

for G being non empty HGrStr holds
( G is left-invertible iff for a, b being Element of G ex l being Element of G st l * a = b )

proof end;

theorem :: MONOID_0:11

for G being non empty HGrStr holds
( G is right-invertible iff for a, b being Element of G ex r being Element of G st a * r = b )

proof end;

theorem Th12: :: MONOID_0:12

for G being non empty HGrStr holds
( G is invertible iff for a, b being Element of G ex r, l being Element of G st
( a * r = b & l * a = b ) )

proof end;

theorem :: MONOID_0:13

for G being non empty HGrStr holds
( G is left-cancelable iff for a, b, c being Element of G st a * b = a * c holds
b = c )

proof end;

theorem :: MONOID_0:14

for G being non empty HGrStr holds
( G is right-cancelable iff for a, b, c being Element of G st b * a = c * a holds
b = c )

proof end;

theorem Th15: :: MONOID_0:15

for G being non empty HGrStr holds
( G is cancelable iff for a, b, c being Element of G st ( a * b = a * c or b * a = c * a ) holds
b = c )

proof end;

theorem Th16: :: MONOID_0:16

for G being non empty HGrStr holds
( G is uniquely-decomposable iff ( the mult of G has_a_unity & ( for a, b being Element of G st a * b = the_unity_wrt the mult of G holds
( a = b & b = the_unity_wrt the mult of G ) ) ) )

proof end;

theorem Th17: :: MONOID_0:17

for G being non empty HGrStr st G is associative holds
( G is invertible iff ( G is unital & the mult of G has_an_inverseOp ) )

proof end;

Lm4: for G being non empty HGrStr holds
( G is invertible iff ( G is left-invertible & G is right-invertible ) )

proof end;

Lm5: for G being non empty HGrStr holds
( G is cancelable iff ( G is left-cancelable & G is right-cancelable ) )

proof end;

Lm6: for G being non empty HGrStr st G is associative & G is invertible holds
G is Group-like

proof end;

registration

cluster non empty Group-like associative -> non empty invertible HGrStr ;
coherence

proof end;

cluster non empty associative invertible -> non empty Group-like HGrStr ;
coherence by Lm6;

end;

registration

cluster non empty invertible -> non empty left-invertible right-invertible HGrStr ;
coherence by Lm4;
cluster non empty left-invertible right-invertible -> non empty invertible HGrStr ;
coherence by Lm4;
cluster non empty cancelable -> non empty left-cancelable right-cancelable HGrStr ;
coherence by Lm5;
cluster non empty left-cancelable right-cancelable -> non empty cancelable HGrStr ;
coherence by Lm5;
cluster non empty associative invertible -> non empty unital cancelable HGrStr ;
coherence

proof end;

end;

deffunc H₃( multLoopStr ) -> Element of the carrier of $1 = the unity of $1;

definition

let IT be non empty multLoopStr ;
redefine attr IT is well-unital means :Def21: :: MONOID_0:def 21
the unity of IT is_a_unity_wrt the mult of IT;
compatibility

proof end;

end;

:: deftheorem Def21 defines well-unital MONOID_0:def 21 :

theorem Th18: :: MONOID_0:18

for M being non empty multLoopStr holds
( M is well-unital iff for a being Element of M holds
( the unity of M * a = a & a * the unity of M = a ) )

proof end;

Lm7: for M being non empty multLoopStr st M is well-unital holds
M is unital

proof end;

registration

cluster non empty well-unital -> non empty unital multLoopStr ;
coherence by Lm7;

end;

theorem Th19: :: MONOID_0:19

for M being non empty multLoopStr st M is well-unital holds
the unity of M = the_unity_wrt the mult of M

proof end;

registration

let A be non empty set ;
let m be BinOp of A;
let u be Element of A;
cluster multLoopStr(# A,m,u #) -> non empty ;
coherence by STRUCT_0:def 1;

end;

registration

cluster non empty unital Group-like associative commutative strict well-unital constituted-Functions constituted-FinSeqs idempotent left-invertible right-invertible invertible left-cancelable right-cancelable cancelable uniquely-decomposable multLoopStr ;
existence

proof end;

end;

definition

mode Monoid is non empty associative well-unital multLoopStr ;

end;

definition

let G be HGrStr ;
mode MonoidalExtension of G -> multLoopStr means :Def22: :: MONOID_0:def 22
HGrStr(# the carrier of it,the mult of it #) = HGrStr(# the carrier of G,the mult of G #);
existence

proof end;

end;

:: deftheorem Def22 defines MonoidalExtension MONOID_0:def 22 :

registration

let G be non empty HGrStr ;
cluster -> non empty MonoidalExtension of G;
coherence

proof end;

end;

theorem Th20: :: MONOID_0:20

for G being non empty HGrStr
for M being MonoidalExtension of G holds
( the carrier of M = the carrier of G & the mult of M = the mult of G & ( for a, b being Element of M
for a', b' being Element of G st a = a' & b = b' holds
a * b = a' * b' ) )

proof end;

registration

let G be HGrStr ;
cluster strict MonoidalExtension of G;
existence

proof end;

end;

theorem Th21: :: MONOID_0:21

for G being non empty HGrStr
for M being MonoidalExtension of G holds
( ( G is unital implies M is unital ) & ( G is commutative implies M is commutative ) & ( G is associative implies M is associative ) & ( G is invertible implies M is invertible ) & ( G is uniquely-decomposable implies M is uniquely-decomposable ) & ( G is cancelable implies M is cancelable ) )

proof end;

registration

let G be constituted-Functions HGrStr ;
cluster -> constituted-Functions MonoidalExtension of G;
coherence

proof end;

end;

registration

let G be constituted-FinSeqs HGrStr ;
cluster -> constituted-Functions constituted-FinSeqs MonoidalExtension of G;
coherence

proof end;

end;

registration

let G be non empty unital HGrStr ;
cluster -> non empty unital MonoidalExtension of G;
coherence by Th21;

end;

registration

let G be non empty associative HGrStr ;
cluster -> non empty associative MonoidalExtension of G;
coherence by Th21;

end;

registration

let G be non empty commutative HGrStr ;
cluster -> non empty commutative MonoidalExtension of G;
coherence by Th21;

end;

registration

let G be non empty invertible HGrStr ;
cluster -> non empty left-invertible right-invertible invertible MonoidalExtension of G;
coherence by Th21;

end;

registration

let G be non empty cancelable HGrStr ;
cluster -> non empty left-cancelable right-cancelable cancelable MonoidalExtension of G;
coherence by Th21;

end;

registration

let G be non empty uniquely-decomposable HGrStr ;
cluster -> non empty uniquely-decomposable MonoidalExtension of G;
coherence by Th21;

end;

registration

let G be non empty unital HGrStr ;
cluster non empty unital strict well-unital MonoidalExtension of G;
existence

proof end;

end;

theorem Th22: :: MONOID_0:22

for G being non empty unital HGrStr
for M1, M2 being strict well-unital MonoidalExtension of G holds M1 = M2

proof end;

definition

let G be HGrStr ;
mode SubStr of G -> HGrStr means :Def23: :: MONOID_0:def 23
the mult of it <= the mult of G;
existence ;

end;

:: deftheorem Def23 defines SubStr MONOID_0:def 23 :

registration

let G be HGrStr ;
cluster strict SubStr of G;
existence

proof end;

end;

registration

let G be non empty HGrStr ;
cluster non empty strict SubStr of G;
existence

proof end;

end;

registration

let G be non empty unital HGrStr ;
cluster non empty strict unital Group-like associative commutative idempotent left-invertible right-invertible invertible left-cancelable right-cancelable cancelable uniquely-decomposable SubStr of G;
existence

proof end;

end;

definition

let G be HGrStr ;
mode MonoidalSubStr of G -> multLoopStr means :Def24: :: MONOID_0:def 24
( the mult of it <= the mult of G & ( for M being multLoopStr st G = M holds
the unity of it = the unity of M ) );
existence

proof end;

end;

:: deftheorem Def24 defines MonoidalSubStr MONOID_0:def 24 :

registration

let G be HGrStr ;
cluster strict MonoidalSubStr of G;
existence

proof end;

end;

registration

let G be non empty HGrStr ;
cluster non empty strict MonoidalSubStr of G;
existence

proof end;

end;

definition

let M be multLoopStr ;
redefine mode MonoidalSubStr of M means :Def25: :: MONOID_0:def 25
( the mult of it <= the mult of M & the unity of it = the unity of M );
compatibility

proof end;

end;

:: deftheorem Def25 defines MonoidalSubStr MONOID_0:def 25 :

registration

let G be non empty well-unital multLoopStr ;
cluster non empty unital Group-like associative commutative strict well-unital idempotent left-invertible right-invertible invertible left-cancelable right-cancelable cancelable uniquely-decomposable MonoidalSubStr of G;
existence

proof end;

end;

theorem Th23: :: MONOID_0:23

for G being HGrStr
for M being MonoidalSubStr of G holds M is SubStr of G

proof end;

definition

let G be HGrStr ;
let M be MonoidalExtension of G;
:: original: SubStr
redefine mode SubStr of M -> SubStr of G;
coherence

proof end;

end;

definition

let G1 be HGrStr ;
let G2 be SubStr of G1;
:: original: SubStr
redefine mode SubStr of G2 -> SubStr of G1;
coherence

proof end;

end;

definition

let G1 be HGrStr ;
let G2 be MonoidalSubStr of G1;
:: original: SubStr
redefine mode SubStr of G2 -> SubStr of G1;
coherence

proof end;

end;

definition

let G be HGrStr ;
let M be MonoidalSubStr of G;
:: original: MonoidalSubStr
redefine mode MonoidalSubStr of M -> MonoidalSubStr of G;
coherence

proof end;

end;

theorem :: MONOID_0:24

for G being non empty HGrStr
for M being non empty multLoopStr holds
( G is SubStr of G & M is MonoidalSubStr of M )

proof end;

theorem Th25: :: MONOID_0:25

for G being non empty HGrStr
for H being non empty SubStr of G
for N being non empty MonoidalSubStr of G holds
( the carrier of H c= the carrier of G & the carrier of N c= the carrier of G )

proof end;

theorem Th26: :: MONOID_0:26

for G being non empty HGrStr
for H being non empty SubStr of G holds the mult of H = the mult of G || the carrier of H

proof end;

theorem Th27: :: MONOID_0:27

for G being non empty HGrStr
for H being non empty SubStr of G
for a, b being Element of H
for a', b' being Element of G st a = a' & b = b' holds
a * b = a' * b'

proof end;

theorem Th28: :: MONOID_0:28

for G being non empty HGrStr
for H1, H2 being non empty SubStr of G st the carrier of H1 = the carrier of H2 holds
HGrStr(# the carrier of H1,the mult of H1 #) = HGrStr(# the carrier of H2,the mult of H2 #)

proof end;

theorem :: MONOID_0:29

for M being non empty multLoopStr
for H1, H2 being non empty MonoidalSubStr of M st the carrier of H1 = the carrier of H2 holds
multLoopStr(# the carrier of H1,the mult of H1,the unity of H1 #) = multLoopStr(# the carrier of H2,the mult of H2,the unity of H2 #)

proof end;

theorem Th30: :: MONOID_0:30

for G being non empty HGrStr
for H1, H2 being non empty SubStr of G st the carrier of H1 c= the carrier of H2 holds
H1 is SubStr of H2

proof end;

theorem :: MONOID_0:31

for M being non empty multLoopStr
for H1, H2 being non empty MonoidalSubStr of M st the carrier of H1 c= the carrier of H2 holds
H1 is MonoidalSubStr of H2

proof end;

theorem Th32: :: MONOID_0:32

for G being non empty HGrStr
for H being non empty SubStr of G st G is unital & the_unity_wrt the mult of G in the carrier of H holds
( H is unital & the_unity_wrt the mult of G = the_unity_wrt the mult of H )

proof end;

theorem Th33: :: MONOID_0:33

for M being non empty well-unital multLoopStr
for N being non empty MonoidalSubStr of M holds N is well-unital

proof end;

theorem Th34: :: MONOID_0:34

for G being non empty HGrStr
for H being non empty SubStr of G st G is commutative holds
H is commutative

proof end;

theorem Th35: :: MONOID_0:35

for G being non empty HGrStr
for H being non empty SubStr of G st G is associative holds
H is associative

proof end;

theorem Th36: :: MONOID_0:36

for G being non empty HGrStr
for H being non empty SubStr of G st G is idempotent holds
H is idempotent

proof end;

theorem Th37: :: MONOID_0:37

for G being non empty HGrStr
for H being non empty SubStr of G st G is cancelable holds
H is cancelable

proof end;

theorem Th38: :: MONOID_0:38

for G being non empty HGrStr
for H being non empty SubStr of G st the_unity_wrt the mult of G in the carrier of H & G is uniquely-decomposable holds
H is uniquely-decomposable

proof end;

theorem Th39: :: MONOID_0:39

for M being non empty well-unital uniquely-decomposable multLoopStr
for N being non empty MonoidalSubStr of M holds N is uniquely-decomposable

proof end;

registration

let G be non empty constituted-Functions HGrStr ;
cluster non empty -> non empty constituted-Functions SubStr of G;
coherence

proof end;

cluster non empty -> non empty constituted-Functions MonoidalSubStr of G;
coherence

proof end;

end;

registration

let G be non empty constituted-FinSeqs HGrStr ;
cluster non empty -> non empty constituted-Functions constituted-FinSeqs SubStr of G;
coherence

proof end;

cluster non empty -> non empty constituted-Functions constituted-FinSeqs MonoidalSubStr of G;
coherence

proof end;

end;

registration

let M be non empty well-unital multLoopStr ;
cluster non empty -> non empty unital well-unital MonoidalSubStr of M;
coherence by Th33;

end;

registration

let G be non empty commutative HGrStr ;
cluster non empty -> non empty commutative SubStr of G;
coherence by Th34;
cluster non empty -> non empty commutative MonoidalSubStr of G;
coherence

proof end;

end;

registration

let G be non empty associative HGrStr ;
cluster non empty -> non empty associative SubStr of G;
coherence by Th35;
cluster non empty -> non empty associative MonoidalSubStr of G;
coherence

proof end;

end;

registration

let G be non empty idempotent HGrStr ;
cluster non empty -> non empty idempotent SubStr of G;
coherence by Th36;
cluster non empty -> non empty idempotent MonoidalSubStr of G;
coherence

proof end;

end;

registration

let G be non empty cancelable HGrStr ;
cluster non empty -> non empty left-cancelable right-cancelable cancelable SubStr of G;
coherence by Th37;
cluster non empty -> non empty left-cancelable right-cancelable cancelable MonoidalSubStr of G;
coherence

proof end;

end;

registration

let M be non empty well-unital uniquely-decomposable multLoopStr ;
cluster non empty -> non empty unital well-unital uniquely-decomposable MonoidalSubStr of M;
coherence by Th39;

end;

scheme :: MONOID_0:sch 1
SubStrEx1{ F₁() -> non empty HGrStr , F₂() -> non empty Subset of F₁() } :

ex H being non empty strict SubStr of F₁() st the carrier of H = F₂()

provided

A1: for x, y being Element of F₂() holds x * y in F₂()

proof end;

scheme :: MONOID_0:sch 2
SubStrEx2{ F₁() -> non empty HGrStr , P₁[ set ] } :

ex H being non empty strict SubStr of F₁() st
for x being Element of F₁() holds
( x in the carrier of H iff P₁[x] )

provided

A1: for x, y being Element of F₁() st P₁[x] & P₁[y] holds
P₁[x * y] and
A2: ex x being Element of F₁() st P₁[x]

proof end;

scheme :: MONOID_0:sch 3
MonoidalSubStrEx1{ F₁() -> non empty multLoopStr , F₂() -> non empty Subset of F₁() } :

ex H being non empty strict MonoidalSubStr of F₁() st the carrier of H = F₂()

provided

A1: for x, y being Element of F₂() holds x * y in F₂() and
A2: the unity of F₁() in F₂()

proof end;

scheme :: MONOID_0:sch 4
MonoidalSubStrEx2{ F₁() -> non empty multLoopStr , P₁[ set ] } :

ex M being non empty strict MonoidalSubStr of F₁() st
for x being Element of F₁() holds
( x in the carrier of M iff P₁[x] )

provided

A1: for x, y being Element of F₁() st P₁[x] & P₁[y] holds
P₁[x * y] and
A2: P₁[the unity of F₁()]

proof end;

notation

let G be non empty HGrStr ;
let a, b be Element of G;
synonym a [*] b for a * b;

end;

definition

let G be non empty HGrStr ;
let a, b be Element of G;
:: original: [*]
redefine func a [*] b -> Element of G;
coherence ;

end;

definition

func <REAL,+> -> non empty strict unital associative commutative invertible cancelable HGrStr equals :: MONOID_0:def 26
HGrStr(# REAL ,addreal #);
coherence

proof end;

end;

:: deftheorem defines <REAL,+> MONOID_0:def 26 :

theorem Th40: :: MONOID_0:40

( the carrier of <REAL,+> = REAL & the mult of <REAL,+> = addreal & ( for a, b being Element of <REAL,+>
for x, y being Real st a = x & b = y holds
a * b = x + y ) ) by BINOP_2:def 9;

theorem :: MONOID_0:41

for x being set holds
( x is Element of <REAL,+> iff x is Real ) ;

theorem :: MONOID_0:42

the_unity_wrt the mult of <REAL,+> = 0 by BINOP_2:2;

theorem Th43: :: MONOID_0:43

for N being non empty SubStr of <REAL,+>
for a, b being Element of N
for x, y being Real st a = x & b = y holds
a * b = x + y

proof end;

theorem Th44: :: MONOID_0:44

for N being non empty unital SubStr of <REAL,+> holds the_unity_wrt the mult of N = 0

proof end;

registration

let G be non empty unital HGrStr ;
cluster non empty associative invertible -> non empty unital Group-like left-cancelable right-cancelable cancelable SubStr of G;
coherence

proof end;

end;

definition

:: original: INT.Group
redefine func INT.Group -> non empty strict unital invertible SubStr of <REAL,+> ;
coherence

proof end;

end;

theorem :: MONOID_0:45

canceled;

theorem :: MONOID_0:46

for G being non empty strict SubStr of <REAL,+> holds
( G = INT.Group iff the carrier of G = INT ) by Th28, GR_CY_1:def 4;

theorem :: MONOID_0:47

for x being set holds
( x is Element of INT.Group iff x is Integer ) by GR_CY_1:def 4, INT_1:def 2;

definition

func <NAT,+> -> non empty strict unital uniquely-decomposable SubStr of INT.Group means :Def27: :: MONOID_0:def 27
the carrier of it = NAT ;
existence

proof end;

uniqueness by Th28;

end;

:: deftheorem Def27 defines <NAT,+> MONOID_0:def 27 :

definition

func <NAT,+,0> -> non empty strict well-unital MonoidalExtension of <NAT,+> means :: MONOID_0:def 28
verum;
existence ;
uniqueness by Th22;

end;

:: deftheorem defines <NAT,+,0> MONOID_0:def 28 :

definition

redefine func addnat equals :Def29: :: MONOID_0:def 29
the mult of <NAT,+> ;
compatibility

proof end;

end;

:: deftheorem Def29 defines addnat MONOID_0:def 29 :

theorem :: MONOID_0:48

canceled;

theorem Th49: :: MONOID_0:49

<NAT,+> = HGrStr(# NAT ,addnat #) by Def27, Def29;

theorem :: MONOID_0:50

for x being set holds
( x is Element of <NAT,+,0> iff x is Nat )

proof end;

theorem :: MONOID_0:51

for n1, n2 being Nat
for m1, m2 being Element of <NAT,+,0> st n1 = m1 & n2 = m2 holds
m1 * m2 = n1 + n2

proof end;

theorem :: MONOID_0:52

<NAT,+,0> = multLoopStr(# NAT ,addnat ,0 #)

proof end;

theorem :: MONOID_0:53

( addnat = addreal || NAT & addnat = addint || NAT )

proof end;

theorem :: MONOID_0:54

( 0 is_a_unity_wrt addnat & addnat is uniquely-decomposable )

proof end;

definition

func <REAL,*> -> non empty strict unital associative commutative HGrStr equals :: MONOID_0:def 30
HGrStr(# REAL ,multreal #);
coherence

proof end;

end;

:: deftheorem defines <REAL,*> MONOID_0:def 30 :

theorem Th55: :: MONOID_0:55

( the carrier of <REAL,*> = REAL & the mult of <REAL,*> = multreal & ( for a, b being Element of <REAL,*>
for x, y being Real st a = x & b = y holds
a * b = x * y ) ) by BINOP_2:def 11;

theorem :: MONOID_0:56

for x being set holds
( x is Element of <REAL,*> iff x is Real ) ;

theorem :: MONOID_0:57

the_unity_wrt the mult of <REAL,*> = 1 by BINOP_2:7;

theorem Th58: :: MONOID_0:58

for N being non empty SubStr of <REAL,*>
for a, b being Element of N
for x, y being Real st a = x & b = y holds
a * b = x * y

proof end;

theorem :: MONOID_0:59

canceled;

theorem :: MONOID_0:60

for N being non empty unital SubStr of <REAL,*> holds
( the_unity_wrt the mult of N = 0 or the_unity_wrt the mult of N = 1 )

proof end;

definition

func <NAT,*> -> non empty strict unital uniquely-decomposable SubStr of <REAL,*> means :Def31: :: MONOID_0:def 31
the carrier of it = NAT ;
existence

proof end;

uniqueness by Th28;

end;

:: deftheorem Def31 defines <NAT,*> MONOID_0:def 31 :

definition

func <NAT,*,1> -> non empty strict well-unital MonoidalExtension of <NAT,*> means :: MONOID_0:def 32
verum;
uniqueness by Th22;
existence ;

end;

:: deftheorem defines <NAT,*,1> MONOID_0:def 32 :

definition

redefine func multnat equals :Def33: :: MONOID_0:def 33
the mult of <NAT,*> ;
compatibility

proof end;

end;

:: deftheorem Def33 defines multnat MONOID_0:def 33 :

theorem Th61: :: MONOID_0:61

<NAT,*> = HGrStr(# NAT ,multnat #) by Def31, Def33;

theorem :: MONOID_0:62

for n1, n2 being Nat
for m1, m2 being Element of <NAT,*> st n1 = m1 & n2 = m2 holds
m1 * m2 = n1 * n2 by Th58;

theorem Th63: :: MONOID_0:63

the_unity_wrt the mult of <NAT,*> = 1

proof end;

theorem :: MONOID_0:64

for n1, n2 being Nat
for m1, m2 being Element of <NAT,*,1> st n1 = m1 & n2 = m2 holds
m1 * m2 = n1 * n2

proof end;

theorem :: MONOID_0:65

<NAT,*,1> = multLoopStr(# NAT ,multnat ,1 #)

proof end;

theorem :: MONOID_0:66

multnat = multreal || NAT by Th26, Th61;

theorem :: MONOID_0:67

( 1 is_a_unity_wrt multnat & multnat is uniquely-decomposable )

proof end;

definition

let D be non empty set ;
func D *+^ -> non empty strict unital associative constituted-FinSeqs cancelable uniquely-decomposable HGrStr means :Def34: :: MONOID_0:def 34
( the carrier of it = D * & ( for p, q being Element of it holds p [*] q = p ^ q ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def34 defines *+^ MONOID_0:def 34 :

definition

let D be non empty set ;
func D *+^+<0> -> non empty strict well-unital MonoidalExtension of D *+^ means :: MONOID_0:def 35
verum;
correctness by Th22;
func D -concatenation -> BinOp of D * equals :: MONOID_0:def 36
the mult of (D *+^ );
correctness

proof end;

end;

:: deftheorem defines *+^+<0> MONOID_0:def 35 :

:: deftheorem defines -concatenation MONOID_0:def 36 :

theorem Th68: :: MONOID_0:68

for D being non empty set holds D *+^ = HGrStr(# (D * ),(D -concatenation ) #) by Def34;

theorem Th69: :: MONOID_0:69

for D being non empty set holds the_unity_wrt the mult of (D *+^ ) = {}

proof end;

theorem :: MONOID_0:70

for D being non empty set holds
( the carrier of (D *+^+<0> ) = D * & the mult of (D *+^+<0> ) = D -concatenation & the unity of (D *+^+<0> ) = {} )

proof end;

theorem :: MONOID_0:71

for D being non empty set
for a, b being Element of (D *+^+<0> ) holds a [*] b = a ^ b

proof end;

theorem Th72: :: MONOID_0:72

for D being non empty set
for F being non empty SubStr of D *+^
for p, q being Element of F holds p [*] q = p ^ q

proof end;

theorem :: MONOID_0:73

for D being non empty set
for F being non empty unital SubStr of D *+^ holds the_unity_wrt the mult of F = {}

proof end;

theorem :: MONOID_0:74

for D being non empty set
for F being non empty SubStr of D *+^ st {} is Element of F holds
( F is unital & the_unity_wrt the mult of F = {} )

proof end;

theorem :: MONOID_0:75

for A, B being non empty set st A c= B holds
A *+^ is SubStr of B *+^

proof end;

theorem :: MONOID_0:76

for D being non empty set holds
( D -concatenation has_a_unity & the_unity_wrt (D -concatenation ) = {} & D -concatenation is associative )

proof end;

definition

let X be set ;
func GPFuncs X -> non empty strict unital associative constituted-Functions HGrStr means :Def37: :: MONOID_0:def 37
( the carrier of it = PFuncs X,X & ( for f, g being Element of it holds f [*] g = f (*) g ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def37 defines GPFuncs MONOID_0:def 37 :

definition

let X be set ;
func MPFuncs X -> non empty strict well-unital MonoidalExtension of GPFuncs X means :: MONOID_0:def 38
verum;
existence ;
uniqueness by Th22;
func X -composition -> BinOp of PFuncs X,X equals :: MONOID_0:def 39
the mult of (GPFuncs X);
correctness

proof end;

end;

:: deftheorem defines MPFuncs MONOID_0:def 38 :

:: deftheorem defines -composition MONOID_0:def 39 :

theorem :: MONOID_0:77

for x, X being set holds
( x is Element of (GPFuncs X) iff x is PartFunc of X,X )

proof end;

theorem Th78: :: MONOID_0:78

for X being set holds the_unity_wrt the mult of (GPFuncs X) = id X

proof end;

theorem Th79: :: MONOID_0:79

for X being set
for F being non empty SubStr of GPFuncs X
for f, g being Element of F holds f [*] g = f (*) g

proof end;

theorem Th80: :: MONOID_0:80

for X being set
for F being non empty SubStr of GPFuncs X st id X is Element of F holds
( F is unital & the_unity_wrt the mult of F = id X )

proof end;

theorem :: MONOID_0:81

for Y, X being set st Y c= X holds
GPFuncs Y is SubStr of GPFuncs X

proof end;

Lm8: for x, X being set st x in Funcs X,X holds
x is Function of X,X

proof end;

definition

let X be set ;
func GFuncs X -> non empty strict unital SubStr of GPFuncs X means :Def40: :: MONOID_0:def 40
the carrier of it = Funcs X,X;
existence

proof end;

uniqueness by Th28;

end;

:: deftheorem Def40 defines GFuncs MONOID_0:def 40 :

definition

let X be set ;
func MFuncs X -> strict well-unital MonoidalExtension of GFuncs X means :: MONOID_0:def 41
verum;
correctness by Th22;

end;

:: deftheorem defines MFuncs MONOID_0:def 41 :

theorem :: MONOID_0:82

for x, X being set holds
( x is Element of (GFuncs X) iff x is Function of X,X )

proof end;

theorem Th83: :: MONOID_0:83

for X being set holds the mult of (GFuncs X) = (X -composition ) || (Funcs X,X)

proof end;

theorem Th84: :: MONOID_0:84

for X being set holds the_unity_wrt the mult of (GFuncs X) = id X

proof end;

theorem :: MONOID_0:85

for X being set holds
( the carrier of (MFuncs X) = Funcs X,X & the mult of (MFuncs X) = (X -composition ) || (Funcs X,X) & the unity of (MFuncs X) = id X )

proof end;

definition

let X be set ;
func GPerms X -> non empty strict unital invertible SubStr of GFuncs X means :Def42: :: MONOID_0:def 42
for f being Element of (GFuncs X) holds
( f in the carrier of it iff f is Permutation of X );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def42 defines GPerms MONOID_0:def 42 :

theorem Th86: :: MONOID_0:86

for x, X being set holds
( x is Element of (GPerms X) iff x is Permutation of X )

proof end;

theorem Th87: :: MONOID_0:87

for X being set holds
( the_unity_wrt the mult of (GPerms X) = id X & 1. (GPerms X) = id X )

proof end;

theorem :: MONOID_0:88

for X being set
for f being Element of (GPerms X) holds f " = f "

proof end;

theorem :: MONOID_0:89

for S being 1-sorted st the carrier of S is functional holds
S is constituted-Functions

proof end;