:: BINOP_2 semantic presentation
begin
scheme :: BINOP_2:sch 1
FuncDefUniq{ F1() -> non empty set , F2() -> non empty set , F3( Element of F1()) -> set } :
for f1, f2 being Function of F1(),F2() st ( for x being Element of F1() holds f1 . x = F3(x) ) & ( for x being Element of F1() holds f2 . x = F3(x) ) holds
f1 = f2
proof
let f1, f2 be Function of F1(),F2(); ::_thesis: ( ( for x being Element of F1() holds f1 . x = F3(x) ) & ( for x being Element of F1() holds f2 . x = F3(x) ) implies f1 = f2 )
assume that
A1: for x being Element of F1() holds f1 . x = F3(x) and
A2: for x being Element of F1() holds f2 . x = F3(x) ; ::_thesis: f1 = f2
now__::_thesis:_for_x_being_Element_of_F1()_holds_f1_._x_=_f2_._x
let x be Element of F1(); ::_thesis: f1 . x = f2 . x
thus f1 . x = F3(x) by A1
.= f2 . x by A2 ; ::_thesis: verum
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
end;
scheme :: BINOP_2:sch 2
BinOpDefuniq{ F1() -> non empty set , F2( Element of F1(), Element of F1()) -> set } :
for o1, o2 being BinOp of F1() st ( for a, b being Element of F1() holds o1 . (a,b) = F2(a,b) ) & ( for a, b being Element of F1() holds o2 . (a,b) = F2(a,b) ) holds
o1 = o2
proof
let o1, o2 be BinOp of F1(); ::_thesis: ( ( for a, b being Element of F1() holds o1 . (a,b) = F2(a,b) ) & ( for a, b being Element of F1() holds o2 . (a,b) = F2(a,b) ) implies o1 = o2 )
assume that
A1: for a, b being Element of F1() holds o1 . (a,b) = F2(a,b) and
A2: for a, b being Element of F1() holds o2 . (a,b) = F2(a,b) ; ::_thesis: o1 = o2
now__::_thesis:_for_a,_b_being_Element_of_F1()_holds_o1_._(a,b)_=_o2_._(a,b)
let a, b be Element of F1(); ::_thesis: o1 . (a,b) = o2 . (a,b)
thus o1 . (a,b) = F2(a,b) by A1
.= o2 . (a,b) by A2 ; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:2; ::_thesis: verum
end;
scheme :: BINOP_2:sch 3
CFuncDefUniq{ F1( complex number ) -> set } :
for f1, f2 being Function of COMPLEX,COMPLEX st ( for x being complex number holds f1 . x = F1(x) ) & ( for x being complex number holds f2 . x = F1(x) ) holds
f1 = f2
proof
let f1, f2 be Function of COMPLEX,COMPLEX; ::_thesis: ( ( for x being complex number holds f1 . x = F1(x) ) & ( for x being complex number holds f2 . x = F1(x) ) implies f1 = f2 )
assume that
A1: for x being complex number holds f1 . x = F1(x) and
A2: for x being complex number holds f2 . x = F1(x) ; ::_thesis: f1 = f2
now__::_thesis:_for_x_being_Element_of_COMPLEX_holds_f1_._x_=_f2_._x
let x be Element of COMPLEX ; ::_thesis: f1 . x = f2 . x
thus f1 . x = F1(x) by A1
.= f2 . x by A2 ; ::_thesis: verum
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
end;
scheme :: BINOP_2:sch 4
RFuncDefUniq{ F1( real number ) -> set } :
for f1, f2 being Function of REAL,REAL st ( for x being real number holds f1 . x = F1(x) ) & ( for x being real number holds f2 . x = F1(x) ) holds
f1 = f2
proof
let f1, f2 be Function of REAL,REAL; ::_thesis: ( ( for x being real number holds f1 . x = F1(x) ) & ( for x being real number holds f2 . x = F1(x) ) implies f1 = f2 )
assume that
A1: for x being real number holds f1 . x = F1(x) and
A2: for x being real number holds f2 . x = F1(x) ; ::_thesis: f1 = f2
now__::_thesis:_for_x_being_Element_of_REAL_holds_f1_._x_=_f2_._x
let x be Element of REAL ; ::_thesis: f1 . x = f2 . x
thus f1 . x = F1(x) by A1
.= f2 . x by A2 ; ::_thesis: verum
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
end;
registration
cluster -> rational for Element of RAT ;
coherence
for b1 being Element of RAT holds b1 is rational by RAT_1:def_2;
end;
scheme :: BINOP_2:sch 5
WFuncDefUniq{ F1( rational number ) -> set } :
for f1, f2 being Function of RAT,RAT st ( for x being rational number holds f1 . x = F1(x) ) & ( for x being rational number holds f2 . x = F1(x) ) holds
f1 = f2
proof
let f1, f2 be Function of RAT,RAT; ::_thesis: ( ( for x being rational number holds f1 . x = F1(x) ) & ( for x being rational number holds f2 . x = F1(x) ) implies f1 = f2 )
assume that
A1: for x being rational number holds f1 . x = F1(x) and
A2: for x being rational number holds f2 . x = F1(x) ; ::_thesis: f1 = f2
now__::_thesis:_for_x_being_Element_of_RAT_holds_f1_._x_=_f2_._x
let x be Element of RAT ; ::_thesis: f1 . x = f2 . x
thus f1 . x = F1(x) by A1
.= f2 . x by A2 ; ::_thesis: verum
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
end;
scheme :: BINOP_2:sch 6
IFuncDefUniq{ F1( integer number ) -> set } :
for f1, f2 being Function of INT,INT st ( for x being integer number holds f1 . x = F1(x) ) & ( for x being integer number holds f2 . x = F1(x) ) holds
f1 = f2
proof
let f1, f2 be Function of INT,INT; ::_thesis: ( ( for x being integer number holds f1 . x = F1(x) ) & ( for x being integer number holds f2 . x = F1(x) ) implies f1 = f2 )
assume that
A1: for x being integer number holds f1 . x = F1(x) and
A2: for x being integer number holds f2 . x = F1(x) ; ::_thesis: f1 = f2
now__::_thesis:_for_x_being_Element_of_INT_holds_f1_._x_=_f2_._x
let x be Element of INT ; ::_thesis: f1 . x = f2 . x
thus f1 . x = F1(x) by A1
.= f2 . x by A2 ; ::_thesis: verum
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
end;
scheme :: BINOP_2:sch 7
NFuncDefUniq{ F1( Nat) -> set } :
for f1, f2 being Function of NAT,NAT st ( for x being Nat holds f1 . x = F1(x) ) & ( for x being Nat holds f2 . x = F1(x) ) holds
f1 = f2
proof
let f1, f2 be Function of NAT,NAT; ::_thesis: ( ( for x being Nat holds f1 . x = F1(x) ) & ( for x being Nat holds f2 . x = F1(x) ) implies f1 = f2 )
assume that
A1: for x being Nat holds f1 . x = F1(x) and
A2: for x being Nat holds f2 . x = F1(x) ; ::_thesis: f1 = f2
now__::_thesis:_for_x_being_Element_of_NAT_holds_f1_._x_=_f2_._x
let x be Element of NAT ; ::_thesis: f1 . x = f2 . x
thus f1 . x = F1(x) by A1
.= f2 . x by A2 ; ::_thesis: verum
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
end;
scheme :: BINOP_2:sch 8
CBinOpDefuniq{ F1( complex number , complex number ) -> set } :
for o1, o2 being BinOp of COMPLEX st ( for a, b being complex number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being complex number holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proof
let o1, o2 be BinOp of COMPLEX; ::_thesis: ( ( for a, b being complex number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being complex number holds o2 . (a,b) = F1(a,b) ) implies o1 = o2 )
assume that
A1: for a, b being complex number holds o1 . (a,b) = F1(a,b) and
A2: for a, b being complex number holds o2 . (a,b) = F1(a,b) ; ::_thesis: o1 = o2
now__::_thesis:_for_a,_b_being_Element_of_COMPLEX_holds_o1_._(a,b)_=_o2_._(a,b)
let a, b be Element of COMPLEX ; ::_thesis: o1 . (a,b) = o2 . (a,b)
thus o1 . (a,b) = F1(a,b) by A1
.= o2 . (a,b) by A2 ; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:2; ::_thesis: verum
end;
scheme :: BINOP_2:sch 9
RBinOpDefuniq{ F1( real number , real number ) -> set } :
for o1, o2 being BinOp of REAL st ( for a, b being real number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being real number holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proof
let o1, o2 be BinOp of REAL; ::_thesis: ( ( for a, b being real number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being real number holds o2 . (a,b) = F1(a,b) ) implies o1 = o2 )
assume that
A1: for a, b being real number holds o1 . (a,b) = F1(a,b) and
A2: for a, b being real number holds o2 . (a,b) = F1(a,b) ; ::_thesis: o1 = o2
now__::_thesis:_for_a,_b_being_Element_of_REAL_holds_o1_._(a,b)_=_o2_._(a,b)
let a, b be Element of REAL ; ::_thesis: o1 . (a,b) = o2 . (a,b)
thus o1 . (a,b) = F1(a,b) by A1
.= o2 . (a,b) by A2 ; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:2; ::_thesis: verum
end;
scheme :: BINOP_2:sch 10
WBinOpDefuniq{ F1( rational number , rational number ) -> set } :
for o1, o2 being BinOp of RAT st ( for a, b being rational number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being rational number holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proof
let o1, o2 be BinOp of RAT; ::_thesis: ( ( for a, b being rational number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being rational number holds o2 . (a,b) = F1(a,b) ) implies o1 = o2 )
assume that
A1: for a, b being rational number holds o1 . (a,b) = F1(a,b) and
A2: for a, b being rational number holds o2 . (a,b) = F1(a,b) ; ::_thesis: o1 = o2
now__::_thesis:_for_a,_b_being_Element_of_RAT_holds_o1_._(a,b)_=_o2_._(a,b)
let a, b be Element of RAT ; ::_thesis: o1 . (a,b) = o2 . (a,b)
thus o1 . (a,b) = F1(a,b) by A1
.= o2 . (a,b) by A2 ; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:2; ::_thesis: verum
end;
scheme :: BINOP_2:sch 11
IBinOpDefuniq{ F1( integer number , integer number ) -> set } :
for o1, o2 being BinOp of INT st ( for a, b being integer number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being integer number holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proof
let o1, o2 be BinOp of INT; ::_thesis: ( ( for a, b being integer number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being integer number holds o2 . (a,b) = F1(a,b) ) implies o1 = o2 )
assume that
A1: for a, b being integer number holds o1 . (a,b) = F1(a,b) and
A2: for a, b being integer number holds o2 . (a,b) = F1(a,b) ; ::_thesis: o1 = o2
now__::_thesis:_for_a,_b_being_Element_of_INT_holds_o1_._(a,b)_=_o2_._(a,b)
let a, b be Element of INT ; ::_thesis: o1 . (a,b) = o2 . (a,b)
thus o1 . (a,b) = F1(a,b) by A1
.= o2 . (a,b) by A2 ; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:2; ::_thesis: verum
end;
scheme :: BINOP_2:sch 12
NBinOpDefuniq{ F1( Nat, Nat) -> set } :
for o1, o2 being BinOp of NAT st ( for a, b being Nat holds o1 . (a,b) = F1(a,b) ) & ( for a, b being Nat holds o2 . (a,b) = F1(a,b) ) holds
o1 = o2
proof
let o1, o2 be BinOp of NAT; ::_thesis: ( ( for a, b being Nat holds o1 . (a,b) = F1(a,b) ) & ( for a, b being Nat holds o2 . (a,b) = F1(a,b) ) implies o1 = o2 )
assume that
A1: for a, b being Nat holds o1 . (a,b) = F1(a,b) and
A2: for a, b being Nat holds o2 . (a,b) = F1(a,b) ; ::_thesis: o1 = o2
now__::_thesis:_for_a,_b_being_Element_of_NAT_holds_o1_._(a,b)_=_o2_._(a,b)
let a, b be Element of NAT ; ::_thesis: o1 . (a,b) = o2 . (a,b)
thus o1 . (a,b) = F1(a,b) by A1
.= o2 . (a,b) by A2 ; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:2; ::_thesis: verum
end;
scheme :: BINOP_2:sch 13
CLambda2D{ F1( complex number , complex number ) -> complex number } :
ex f being Function of [:COMPLEX,COMPLEX:],COMPLEX st
for x, y being complex number holds f . (x,y) = F1(x,y)
proof
defpred S1[ complex number , complex number , set ] means $3 = F1($1,$2);
A1: for x, y being Element of COMPLEX ex z being Element of COMPLEX st S1[x,y,z]
proof
let x, y be Element of COMPLEX ; ::_thesis: ex z being Element of COMPLEX st S1[x,y,z]
reconsider z = F1(x,y) as Element of COMPLEX by XCMPLX_0:def_2;
take z ; ::_thesis: S1[x,y,z]
thus S1[x,y,z] ; ::_thesis: verum
end;
consider f being Function of [:COMPLEX,COMPLEX:],COMPLEX such that
A2: for x, y being Element of COMPLEX holds S1[x,y,f . (x,y)] from BINOP_1:sch_3(A1);
take f ; ::_thesis: for x, y being complex number holds f . (x,y) = F1(x,y)
let x, y be complex number ; ::_thesis: f . (x,y) = F1(x,y)
reconsider x = x, y = y as Element of COMPLEX by XCMPLX_0:def_2;
S1[x,y,f . (x,y)] by A2;
then f . (x,y) = F1(x,y) ;
hence f . (x,y) = F1(x,y) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 14
RLambda2D{ F1( real number , real number ) -> real number } :
ex f being Function of [:REAL,REAL:],REAL st
for x, y being real number holds f . (x,y) = F1(x,y)
proof
defpred S1[ real number , real number , set ] means $3 = F1($1,$2);
A1: for x, y being Element of REAL ex z being Element of REAL st S1[x,y,z]
proof
let x, y be Element of REAL ; ::_thesis: ex z being Element of REAL st S1[x,y,z]
reconsider z = F1(x,y) as Element of REAL by XREAL_0:def_1;
take z ; ::_thesis: S1[x,y,z]
thus S1[x,y,z] ; ::_thesis: verum
end;
consider f being Function of [:REAL,REAL:],REAL such that
A2: for x, y being Element of REAL holds S1[x,y,f . (x,y)] from BINOP_1:sch_3(A1);
take f ; ::_thesis: for x, y being real number holds f . (x,y) = F1(x,y)
let x, y be real number ; ::_thesis: f . (x,y) = F1(x,y)
reconsider x = x, y = y as Element of REAL by XREAL_0:def_1;
S1[x,y,f . (x,y)] by A2;
then f . (x,y) = F1(x,y) ;
hence f . (x,y) = F1(x,y) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 15
WLambda2D{ F1( rational number , rational number ) -> rational number } :
ex f being Function of [:RAT,RAT:],RAT st
for x, y being rational number holds f . (x,y) = F1(x,y)
proof
defpred S1[ rational number , rational number , set ] means $3 = F1($1,$2);
A1: for x, y being Element of RAT ex z being Element of RAT st S1[x,y,z]
proof
let x, y be Element of RAT ; ::_thesis: ex z being Element of RAT st S1[x,y,z]
reconsider z = F1(x,y) as Element of RAT by RAT_1:def_2;
take z ; ::_thesis: S1[x,y,z]
thus S1[x,y,z] ; ::_thesis: verum
end;
consider f being Function of [:RAT,RAT:],RAT such that
A2: for x, y being Element of RAT holds S1[x,y,f . (x,y)] from BINOP_1:sch_3(A1);
take f ; ::_thesis: for x, y being rational number holds f . (x,y) = F1(x,y)
let x, y be rational number ; ::_thesis: f . (x,y) = F1(x,y)
reconsider x = x, y = y as Element of RAT by RAT_1:def_2;
S1[x,y,f . (x,y)] by A2;
then f . (x,y) = F1(x,y) ;
hence f . (x,y) = F1(x,y) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 16
ILambda2D{ F1( integer number , integer number ) -> integer number } :
ex f being Function of [:INT,INT:],INT st
for x, y being integer number holds f . (x,y) = F1(x,y)
proof
defpred S1[ integer number , integer number , set ] means $3 = F1($1,$2);
A1: for x, y being Element of INT ex z being Element of INT st S1[x,y,z]
proof
let x, y be Element of INT ; ::_thesis: ex z being Element of INT st S1[x,y,z]
reconsider z = F1(x,y) as Element of INT by INT_1:def_2;
take z ; ::_thesis: S1[x,y,z]
thus S1[x,y,z] ; ::_thesis: verum
end;
consider f being Function of [:INT,INT:],INT such that
A2: for x, y being Element of INT holds S1[x,y,f . (x,y)] from BINOP_1:sch_3(A1);
take f ; ::_thesis: for x, y being integer number holds f . (x,y) = F1(x,y)
let x, y be integer number ; ::_thesis: f . (x,y) = F1(x,y)
reconsider x = x, y = y as Element of INT by INT_1:def_2;
S1[x,y,f . (x,y)] by A2;
then f . (x,y) = F1(x,y) ;
hence f . (x,y) = F1(x,y) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 17
NLambda2D{ F1( Nat, Nat) -> Nat } :
ex f being Function of [:NAT,NAT:],NAT st
for x, y being Nat holds f . (x,y) = F1(x,y)
proof
defpred S1[ Nat, Nat, set ] means $3 = F1($1,$2);
A1: for x, y being Element of NAT ex z being Element of NAT st S1[x,y,z]
proof
let x, y be Element of NAT ; ::_thesis: ex z being Element of NAT st S1[x,y,z]
reconsider z = F1(x,y) as Element of NAT by ORDINAL1:def_12;
take z ; ::_thesis: S1[x,y,z]
thus S1[x,y,z] ; ::_thesis: verum
end;
consider f being Function of [:NAT,NAT:],NAT such that
A2: for x, y being Element of NAT holds S1[x,y,f . (x,y)] from BINOP_1:sch_3(A1);
take f ; ::_thesis: for x, y being Nat holds f . (x,y) = F1(x,y)
let x, y be Nat; ::_thesis: f . (x,y) = F1(x,y)
reconsider x = x, y = y as Element of NAT by ORDINAL1:def_12;
S1[x,y,f . (x,y)] by A2;
then f . (x,y) = F1(x,y) ;
hence f . (x,y) = F1(x,y) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 18
CLambdaD{ F1( complex number ) -> complex number } :
ex f being Function of COMPLEX,COMPLEX st
for x being complex number holds f . x = F1(x)
proof
defpred S1[ Element of COMPLEX , set ] means $2 = F1($1);
A1: for x being Element of COMPLEX ex y being Element of COMPLEX st S1[x,y]
proof
let x be Element of COMPLEX ; ::_thesis: ex y being Element of COMPLEX st S1[x,y]
reconsider y = F1(x) as Element of COMPLEX by XCMPLX_0:def_2;
take y ; ::_thesis: S1[x,y]
thus S1[x,y] ; ::_thesis: verum
end;
consider f being Function of COMPLEX,COMPLEX such that
A2: for x being Element of COMPLEX holds S1[x,f . x] from FUNCT_2:sch_3(A1);
take f ; ::_thesis: for x being complex number holds f . x = F1(x)
let c be complex number ; ::_thesis: f . c = F1(c)
reconsider c = c as Element of COMPLEX by XCMPLX_0:def_2;
S1[c,f . c] by A2;
hence f . c = F1(c) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 19
RLambdaD{ F1( real number ) -> real number } :
ex f being Function of REAL,REAL st
for x being real number holds f . x = F1(x)
proof
defpred S1[ Element of REAL , set ] means $2 = F1($1);
A1: for x being Element of REAL ex y being Element of REAL st S1[x,y]
proof
let x be Element of REAL ; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider y = F1(x) as Element of REAL by XREAL_0:def_1;
take y ; ::_thesis: S1[x,y]
thus S1[x,y] ; ::_thesis: verum
end;
consider f being Function of REAL,REAL such that
A2: for x being Element of REAL holds S1[x,f . x] from FUNCT_2:sch_3(A1);
take f ; ::_thesis: for x being real number holds f . x = F1(x)
let c be real number ; ::_thesis: f . c = F1(c)
reconsider c = c as Element of REAL by XREAL_0:def_1;
S1[c,f . c] by A2;
hence f . c = F1(c) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 20
WLambdaD{ F1( rational number ) -> rational number } :
ex f being Function of RAT,RAT st
for x being rational number holds f . x = F1(x)
proof
defpred S1[ Element of RAT , set ] means $2 = F1($1);
A1: for x being Element of RAT ex y being Element of RAT st S1[x,y]
proof
let x be Element of RAT ; ::_thesis: ex y being Element of RAT st S1[x,y]
reconsider y = F1(x) as Element of RAT by RAT_1:def_2;
take y ; ::_thesis: S1[x,y]
thus S1[x,y] ; ::_thesis: verum
end;
consider f being Function of RAT,RAT such that
A2: for x being Element of RAT holds S1[x,f . x] from FUNCT_2:sch_3(A1);
take f ; ::_thesis: for x being rational number holds f . x = F1(x)
let c be rational number ; ::_thesis: f . c = F1(c)
reconsider c = c as Element of RAT by RAT_1:def_2;
S1[c,f . c] by A2;
hence f . c = F1(c) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 21
ILambdaD{ F1( integer number ) -> integer number } :
ex f being Function of INT,INT st
for x being integer number holds f . x = F1(x)
proof
defpred S1[ Element of INT , set ] means $2 = F1($1);
A1: for x being Element of INT ex y being Element of INT st S1[x,y]
proof
let x be Element of INT ; ::_thesis: ex y being Element of INT st S1[x,y]
reconsider y = F1(x) as Element of INT by INT_1:def_2;
take y ; ::_thesis: S1[x,y]
thus S1[x,y] ; ::_thesis: verum
end;
consider f being Function of INT,INT such that
A2: for x being Element of INT holds S1[x,f . x] from FUNCT_2:sch_3(A1);
take f ; ::_thesis: for x being integer number holds f . x = F1(x)
let c be integer number ; ::_thesis: f . c = F1(c)
reconsider c = c as Element of INT by INT_1:def_2;
S1[c,f . c] by A2;
hence f . c = F1(c) ; ::_thesis: verum
end;
scheme :: BINOP_2:sch 22
NLambdaD{ F1( Nat) -> Nat } :
ex f being Function of NAT,NAT st
for x being Nat holds f . x = F1(x)
proof
defpred S1[ Element of NAT , set ] means $2 = F1($1);
A1: for x being Element of NAT ex y being Element of NAT st S1[x,y]
proof
let x be Element of NAT ; ::_thesis: ex y being Element of NAT st S1[x,y]
reconsider y = F1(x) as Element of NAT by ORDINAL1:def_12;
take y ; ::_thesis: S1[x,y]
thus S1[x,y] ; ::_thesis: verum
end;
consider f being Function of NAT,NAT such that
A2: for x being Element of NAT holds S1[x,f . x] from FUNCT_2:sch_3(A1);
take f ; ::_thesis: for x being Nat holds f . x = F1(x)
let c be Nat; ::_thesis: f . c = F1(c)
reconsider c = c as Element of NAT by ORDINAL1:def_12;
S1[c,f . c] by A2;
hence f . c = F1(c) ; ::_thesis: verum
end;
definition
let c1 be complex number ;
:: original: -
redefine func - c1 -> Element of COMPLEX ;
coherence
- c1 is Element of COMPLEX by XCMPLX_0:def_2;
:: original: "
redefine funcc1 " -> Element of COMPLEX ;
coherence
c1 " is Element of COMPLEX by XCMPLX_0:def_2;
let c2 be complex number ;
:: original: +
redefine funcc1 + c2 -> Element of COMPLEX ;
coherence
c1 + c2 is Element of COMPLEX by XCMPLX_0:def_2;
:: original: -
redefine funcc1 - c2 -> Element of COMPLEX ;
coherence
c1 - c2 is Element of COMPLEX by XCMPLX_0:def_2;
:: original: *
redefine funcc1 * c2 -> Element of COMPLEX ;
coherence
c1 * c2 is Element of COMPLEX by XCMPLX_0:def_2;
:: original: /
redefine funcc1 / c2 -> Element of COMPLEX ;
coherence
c1 / c2 is Element of COMPLEX by XCMPLX_0:def_2;
end;
definition
let r1 be real number ;
:: original: -
redefine func - r1 -> Element of REAL ;
coherence
- r1 is Element of REAL by XREAL_0:def_1;
:: original: "
redefine funcr1 " -> Element of REAL ;
coherence
r1 " is Element of REAL by XREAL_0:def_1;
let r2 be real number ;
:: original: +
redefine funcr1 + r2 -> Element of REAL ;
coherence
r1 + r2 is Element of REAL by XREAL_0:def_1;
:: original: -
redefine funcr1 - r2 -> Element of REAL ;
coherence
r1 - r2 is Element of REAL by XREAL_0:def_1;
:: original: *
redefine funcr1 * r2 -> Element of REAL ;
coherence
r1 * r2 is Element of REAL by XREAL_0:def_1;
:: original: /
redefine funcr1 / r2 -> Element of REAL ;
coherence
r1 / r2 is Element of REAL by XREAL_0:def_1;
end;
definition
let w1 be rational number ;
:: original: -
redefine func - w1 -> Element of RAT ;
coherence
- w1 is Element of RAT by RAT_1:def_2;
:: original: "
redefine funcw1 " -> Element of RAT ;
coherence
w1 " is Element of RAT by RAT_1:def_2;
let w2 be rational number ;
:: original: +
redefine funcw1 + w2 -> Element of RAT ;
coherence
w1 + w2 is Element of RAT by RAT_1:def_2;
:: original: -
redefine funcw1 - w2 -> Element of RAT ;
coherence
w1 - w2 is Element of RAT by RAT_1:def_2;
:: original: *
redefine funcw1 * w2 -> Element of RAT ;
coherence
w1 * w2 is Element of RAT by RAT_1:def_2;
:: original: /
redefine funcw1 / w2 -> Element of RAT ;
coherence
w1 / w2 is Element of RAT by RAT_1:def_2;
end;
definition
let i1 be integer number ;
:: original: -
redefine func - i1 -> Element of INT ;
coherence
- i1 is Element of INT by INT_1:def_2;
let i2 be integer number ;
:: original: +
redefine funci1 + i2 -> Element of INT ;
coherence
i1 + i2 is Element of INT by INT_1:def_2;
:: original: -
redefine funci1 - i2 -> Element of INT ;
coherence
i1 - i2 is Element of INT by INT_1:def_2;
:: original: *
redefine funci1 * i2 -> Element of INT ;
coherence
i1 * i2 is Element of INT by INT_1:def_2;
end;
definition
let n1, n2 be Nat;
:: original: +
redefine funcn1 + n2 -> Element of NAT ;
coherence
n1 + n2 is Element of NAT by ORDINAL1:def_12;
:: original: *
redefine funcn1 * n2 -> Element of NAT ;
coherence
n1 * n2 is Element of NAT by ORDINAL1:def_12;
end;
definition
func compcomplex -> UnOp of COMPLEX means :: BINOP_2:def 1
for c being complex number holds it . c = - c;
existence
ex b1 being UnOp of COMPLEX st
for c being complex number holds b1 . c = - c from BINOP_2:sch_18();
uniqueness
for b1, b2 being UnOp of COMPLEX st ( for c being complex number holds b1 . c = - c ) & ( for c being complex number holds b2 . c = - c ) holds
b1 = b2 from BINOP_2:sch_3();
func invcomplex -> UnOp of COMPLEX means :: BINOP_2:def 2
for c being complex number holds it . c = c " ;
existence
ex b1 being UnOp of COMPLEX st
for c being complex number holds b1 . c = c " from BINOP_2:sch_18();
uniqueness
for b1, b2 being UnOp of COMPLEX st ( for c being complex number holds b1 . c = c " ) & ( for c being complex number holds b2 . c = c " ) holds
b1 = b2 from BINOP_2:sch_3();
func addcomplex -> BinOp of COMPLEX means :Def3: :: BINOP_2:def 3
for c1, c2 being complex number holds it . (c1,c2) = c1 + c2;
existence
ex b1 being BinOp of COMPLEX st
for c1, c2 being complex number holds b1 . (c1,c2) = c1 + c2 from BINOP_2:sch_13();
uniqueness
for b1, b2 being BinOp of COMPLEX st ( for c1, c2 being complex number holds b1 . (c1,c2) = c1 + c2 ) & ( for c1, c2 being complex number holds b2 . (c1,c2) = c1 + c2 ) holds
b1 = b2 from BINOP_2:sch_8();
func diffcomplex -> BinOp of COMPLEX means :: BINOP_2:def 4
for c1, c2 being complex number holds it . (c1,c2) = c1 - c2;
existence
ex b1 being BinOp of COMPLEX st
for c1, c2 being complex number holds b1 . (c1,c2) = c1 - c2 from BINOP_2:sch_13();
uniqueness
for b1, b2 being BinOp of COMPLEX st ( for c1, c2 being complex number holds b1 . (c1,c2) = c1 - c2 ) & ( for c1, c2 being complex number holds b2 . (c1,c2) = c1 - c2 ) holds
b1 = b2 from BINOP_2:sch_8();
func multcomplex -> BinOp of COMPLEX means :Def5: :: BINOP_2:def 5
for c1, c2 being complex number holds it . (c1,c2) = c1 * c2;
existence
ex b1 being BinOp of COMPLEX st
for c1, c2 being complex number holds b1 . (c1,c2) = c1 * c2 from BINOP_2:sch_13();
uniqueness
for b1, b2 being BinOp of COMPLEX st ( for c1, c2 being complex number holds b1 . (c1,c2) = c1 * c2 ) & ( for c1, c2 being complex number holds b2 . (c1,c2) = c1 * c2 ) holds
b1 = b2 from BINOP_2:sch_8();
func divcomplex -> BinOp of COMPLEX means :: BINOP_2:def 6
for c1, c2 being complex number holds it . (c1,c2) = c1 / c2;
existence
ex b1 being BinOp of COMPLEX st
for c1, c2 being complex number holds b1 . (c1,c2) = c1 / c2 from BINOP_2:sch_13();
uniqueness
for b1, b2 being BinOp of COMPLEX st ( for c1, c2 being complex number holds b1 . (c1,c2) = c1 / c2 ) & ( for c1, c2 being complex number holds b2 . (c1,c2) = c1 / c2 ) holds
b1 = b2 from BINOP_2:sch_8();
end;
:: deftheorem defines compcomplex BINOP_2:def_1_:_
for b1 being UnOp of COMPLEX holds
( b1 = compcomplex iff for c being complex number holds b1 . c = - c );
:: deftheorem defines invcomplex BINOP_2:def_2_:_
for b1 being UnOp of COMPLEX holds
( b1 = invcomplex iff for c being complex number holds b1 . c = c " );
:: deftheorem Def3 defines addcomplex BINOP_2:def_3_:_
for b1 being BinOp of COMPLEX holds
( b1 = addcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 + c2 );
:: deftheorem defines diffcomplex BINOP_2:def_4_:_
for b1 being BinOp of COMPLEX holds
( b1 = diffcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 - c2 );
:: deftheorem Def5 defines multcomplex BINOP_2:def_5_:_
for b1 being BinOp of COMPLEX holds
( b1 = multcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 * c2 );
:: deftheorem defines divcomplex BINOP_2:def_6_:_
for b1 being BinOp of COMPLEX holds
( b1 = divcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 / c2 );
definition
func compreal -> UnOp of REAL means :: BINOP_2:def 7
for r being real number holds it . r = - r;
existence
ex b1 being UnOp of REAL st
for r being real number holds b1 . r = - r from BINOP_2:sch_19();
uniqueness
for b1, b2 being UnOp of REAL st ( for r being real number holds b1 . r = - r ) & ( for r being real number holds b2 . r = - r ) holds
b1 = b2 from BINOP_2:sch_4();
func invreal -> UnOp of REAL means :: BINOP_2:def 8
for r being real number holds it . r = r " ;
existence
ex b1 being UnOp of REAL st
for r being real number holds b1 . r = r " from BINOP_2:sch_19();
uniqueness
for b1, b2 being UnOp of REAL st ( for r being real number holds b1 . r = r " ) & ( for r being real number holds b2 . r = r " ) holds
b1 = b2 from BINOP_2:sch_4();
func addreal -> BinOp of REAL means :Def9: :: BINOP_2:def 9
for r1, r2 being real number holds it . (r1,r2) = r1 + r2;
existence
ex b1 being BinOp of REAL st
for r1, r2 being real number holds b1 . (r1,r2) = r1 + r2 from BINOP_2:sch_14();
uniqueness
for b1, b2 being BinOp of REAL st ( for r1, r2 being real number holds b1 . (r1,r2) = r1 + r2 ) & ( for r1, r2 being real number holds b2 . (r1,r2) = r1 + r2 ) holds
b1 = b2 from BINOP_2:sch_9();
func diffreal -> BinOp of REAL means :: BINOP_2:def 10
for r1, r2 being real number holds it . (r1,r2) = r1 - r2;
existence
ex b1 being BinOp of REAL st
for r1, r2 being real number holds b1 . (r1,r2) = r1 - r2 from BINOP_2:sch_14();
uniqueness
for b1, b2 being BinOp of REAL st ( for r1, r2 being real number holds b1 . (r1,r2) = r1 - r2 ) & ( for r1, r2 being real number holds b2 . (r1,r2) = r1 - r2 ) holds
b1 = b2 from BINOP_2:sch_9();
func multreal -> BinOp of REAL means :Def11: :: BINOP_2:def 11
for r1, r2 being real number holds it . (r1,r2) = r1 * r2;
existence
ex b1 being BinOp of REAL st
for r1, r2 being real number holds b1 . (r1,r2) = r1 * r2 from BINOP_2:sch_14();
uniqueness
for b1, b2 being BinOp of REAL st ( for r1, r2 being real number holds b1 . (r1,r2) = r1 * r2 ) & ( for r1, r2 being real number holds b2 . (r1,r2) = r1 * r2 ) holds
b1 = b2 from BINOP_2:sch_9();
func divreal -> BinOp of REAL means :: BINOP_2:def 12
for r1, r2 being real number holds it . (r1,r2) = r1 / r2;
existence
ex b1 being BinOp of REAL st
for r1, r2 being real number holds b1 . (r1,r2) = r1 / r2 from BINOP_2:sch_14();
uniqueness
for b1, b2 being BinOp of REAL st ( for r1, r2 being real number holds b1 . (r1,r2) = r1 / r2 ) & ( for r1, r2 being real number holds b2 . (r1,r2) = r1 / r2 ) holds
b1 = b2 from BINOP_2:sch_9();
end;
:: deftheorem defines compreal BINOP_2:def_7_:_
for b1 being UnOp of REAL holds
( b1 = compreal iff for r being real number holds b1 . r = - r );
:: deftheorem defines invreal BINOP_2:def_8_:_
for b1 being UnOp of REAL holds
( b1 = invreal iff for r being real number holds b1 . r = r " );
:: deftheorem Def9 defines addreal BINOP_2:def_9_:_
for b1 being BinOp of REAL holds
( b1 = addreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 + r2 );
:: deftheorem defines diffreal BINOP_2:def_10_:_
for b1 being BinOp of REAL holds
( b1 = diffreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 - r2 );
:: deftheorem Def11 defines multreal BINOP_2:def_11_:_
for b1 being BinOp of REAL holds
( b1 = multreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 * r2 );
:: deftheorem defines divreal BINOP_2:def_12_:_
for b1 being BinOp of REAL holds
( b1 = divreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 / r2 );
definition
func comprat -> UnOp of RAT means :: BINOP_2:def 13
for w being rational number holds it . w = - w;
existence
ex b1 being UnOp of RAT st
for w being rational number holds b1 . w = - w from BINOP_2:sch_20();
uniqueness
for b1, b2 being UnOp of RAT st ( for w being rational number holds b1 . w = - w ) & ( for w being rational number holds b2 . w = - w ) holds
b1 = b2 from BINOP_2:sch_5();
func invrat -> UnOp of RAT means :: BINOP_2:def 14
for w being rational number holds it . w = w " ;
existence
ex b1 being UnOp of RAT st
for w being rational number holds b1 . w = w " from BINOP_2:sch_20();
uniqueness
for b1, b2 being UnOp of RAT st ( for w being rational number holds b1 . w = w " ) & ( for w being rational number holds b2 . w = w " ) holds
b1 = b2 from BINOP_2:sch_5();
func addrat -> BinOp of RAT means :Def15: :: BINOP_2:def 15
for w1, w2 being rational number holds it . (w1,w2) = w1 + w2;
existence
ex b1 being BinOp of RAT st
for w1, w2 being rational number holds b1 . (w1,w2) = w1 + w2 from BINOP_2:sch_15();
uniqueness
for b1, b2 being BinOp of RAT st ( for w1, w2 being rational number holds b1 . (w1,w2) = w1 + w2 ) & ( for w1, w2 being rational number holds b2 . (w1,w2) = w1 + w2 ) holds
b1 = b2 from BINOP_2:sch_10();
func diffrat -> BinOp of RAT means :: BINOP_2:def 16
for w1, w2 being rational number holds it . (w1,w2) = w1 - w2;
existence
ex b1 being BinOp of RAT st
for w1, w2 being rational number holds b1 . (w1,w2) = w1 - w2 from BINOP_2:sch_15();
uniqueness
for b1, b2 being BinOp of RAT st ( for w1, w2 being rational number holds b1 . (w1,w2) = w1 - w2 ) & ( for w1, w2 being rational number holds b2 . (w1,w2) = w1 - w2 ) holds
b1 = b2 from BINOP_2:sch_10();
func multrat -> BinOp of RAT means :Def17: :: BINOP_2:def 17
for w1, w2 being rational number holds it . (w1,w2) = w1 * w2;
existence
ex b1 being BinOp of RAT st
for w1, w2 being rational number holds b1 . (w1,w2) = w1 * w2 from BINOP_2:sch_15();
uniqueness
for b1, b2 being BinOp of RAT st ( for w1, w2 being rational number holds b1 . (w1,w2) = w1 * w2 ) & ( for w1, w2 being rational number holds b2 . (w1,w2) = w1 * w2 ) holds
b1 = b2 from BINOP_2:sch_10();
func divrat -> BinOp of RAT means :: BINOP_2:def 18
for w1, w2 being rational number holds it . (w1,w2) = w1 / w2;
existence
ex b1 being BinOp of RAT st
for w1, w2 being rational number holds b1 . (w1,w2) = w1 / w2 from BINOP_2:sch_15();
uniqueness
for b1, b2 being BinOp of RAT st ( for w1, w2 being rational number holds b1 . (w1,w2) = w1 / w2 ) & ( for w1, w2 being rational number holds b2 . (w1,w2) = w1 / w2 ) holds
b1 = b2 from BINOP_2:sch_10();
end;
:: deftheorem defines comprat BINOP_2:def_13_:_
for b1 being UnOp of RAT holds
( b1 = comprat iff for w being rational number holds b1 . w = - w );
:: deftheorem defines invrat BINOP_2:def_14_:_
for b1 being UnOp of RAT holds
( b1 = invrat iff for w being rational number holds b1 . w = w " );
:: deftheorem Def15 defines addrat BINOP_2:def_15_:_
for b1 being BinOp of RAT holds
( b1 = addrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 + w2 );
:: deftheorem defines diffrat BINOP_2:def_16_:_
for b1 being BinOp of RAT holds
( b1 = diffrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 - w2 );
:: deftheorem Def17 defines multrat BINOP_2:def_17_:_
for b1 being BinOp of RAT holds
( b1 = multrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 * w2 );
:: deftheorem defines divrat BINOP_2:def_18_:_
for b1 being BinOp of RAT holds
( b1 = divrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 / w2 );
definition
func compint -> UnOp of INT means :: BINOP_2:def 19
for i being integer number holds it . i = - i;
existence
ex b1 being UnOp of INT st
for i being integer number holds b1 . i = - i from BINOP_2:sch_21();
uniqueness
for b1, b2 being UnOp of INT st ( for i being integer number holds b1 . i = - i ) & ( for i being integer number holds b2 . i = - i ) holds
b1 = b2 from BINOP_2:sch_6();
func addint -> BinOp of INT means :Def20: :: BINOP_2:def 20
for i1, i2 being integer number holds it . (i1,i2) = i1 + i2;
existence
ex b1 being BinOp of INT st
for i1, i2 being integer number holds b1 . (i1,i2) = i1 + i2 from BINOP_2:sch_16();
uniqueness
for b1, b2 being BinOp of INT st ( for i1, i2 being integer number holds b1 . (i1,i2) = i1 + i2 ) & ( for i1, i2 being integer number holds b2 . (i1,i2) = i1 + i2 ) holds
b1 = b2 from BINOP_2:sch_11();
func diffint -> BinOp of INT means :: BINOP_2:def 21
for i1, i2 being integer number holds it . (i1,i2) = i1 - i2;
existence
ex b1 being BinOp of INT st
for i1, i2 being integer number holds b1 . (i1,i2) = i1 - i2 from BINOP_2:sch_16();
uniqueness
for b1, b2 being BinOp of INT st ( for i1, i2 being integer number holds b1 . (i1,i2) = i1 - i2 ) & ( for i1, i2 being integer number holds b2 . (i1,i2) = i1 - i2 ) holds
b1 = b2 from BINOP_2:sch_11();
func multint -> BinOp of INT means :Def22: :: BINOP_2:def 22
for i1, i2 being integer number holds it . (i1,i2) = i1 * i2;
existence
ex b1 being BinOp of INT st
for i1, i2 being integer number holds b1 . (i1,i2) = i1 * i2 from BINOP_2:sch_16();
uniqueness
for b1, b2 being BinOp of INT st ( for i1, i2 being integer number holds b1 . (i1,i2) = i1 * i2 ) & ( for i1, i2 being integer number holds b2 . (i1,i2) = i1 * i2 ) holds
b1 = b2 from BINOP_2:sch_11();
end;
:: deftheorem defines compint BINOP_2:def_19_:_
for b1 being UnOp of INT holds
( b1 = compint iff for i being integer number holds b1 . i = - i );
:: deftheorem Def20 defines addint BINOP_2:def_20_:_
for b1 being BinOp of INT holds
( b1 = addint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 + i2 );
:: deftheorem defines diffint BINOP_2:def_21_:_
for b1 being BinOp of INT holds
( b1 = diffint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 - i2 );
:: deftheorem Def22 defines multint BINOP_2:def_22_:_
for b1 being BinOp of INT holds
( b1 = multint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 * i2 );
definition
func addnat -> BinOp of NAT means :Def23: :: BINOP_2:def 23
for n1, n2 being Nat holds it . (n1,n2) = n1 + n2;
existence
ex b1 being BinOp of NAT st
for n1, n2 being Nat holds b1 . (n1,n2) = n1 + n2 from BINOP_2:sch_17();
uniqueness
for b1, b2 being BinOp of NAT st ( for n1, n2 being Nat holds b1 . (n1,n2) = n1 + n2 ) & ( for n1, n2 being Nat holds b2 . (n1,n2) = n1 + n2 ) holds
b1 = b2 from BINOP_2:sch_12();
func multnat -> BinOp of NAT means :Def24: :: BINOP_2:def 24
for n1, n2 being Nat holds it . (n1,n2) = n1 * n2;
existence
ex b1 being BinOp of NAT st
for n1, n2 being Nat holds b1 . (n1,n2) = n1 * n2 from BINOP_2:sch_17();
uniqueness
for b1, b2 being BinOp of NAT st ( for n1, n2 being Nat holds b1 . (n1,n2) = n1 * n2 ) & ( for n1, n2 being Nat holds b2 . (n1,n2) = n1 * n2 ) holds
b1 = b2 from BINOP_2:sch_12();
end;
:: deftheorem Def23 defines addnat BINOP_2:def_23_:_
for b1 being BinOp of NAT holds
( b1 = addnat iff for n1, n2 being Nat holds b1 . (n1,n2) = n1 + n2 );
:: deftheorem Def24 defines multnat BINOP_2:def_24_:_
for b1 being BinOp of NAT holds
( b1 = multnat iff for n1, n2 being Nat holds b1 . (n1,n2) = n1 * n2 );
registration
cluster addcomplex -> commutative associative ;
coherence
( addcomplex is commutative & addcomplex is associative )
proof
thus addcomplex is commutative ::_thesis: addcomplex is associative
proof
let c1, c2 be Element of COMPLEX ; :: according to BINOP_1:def_2 ::_thesis: addcomplex . (c1,c2) = addcomplex . (c2,c1)
thus addcomplex . (c1,c2) = c1 + c2 by Def3
.= addcomplex . (c2,c1) by Def3 ; ::_thesis: verum
end;
let c1, c2, c3 be Element of COMPLEX ; :: according to BINOP_1:def_3 ::_thesis: addcomplex . (c1,(addcomplex . (c2,c3))) = addcomplex . ((addcomplex . (c1,c2)),c3)
thus addcomplex . (c1,(addcomplex . (c2,c3))) = c1 + (addcomplex . (c2,c3)) by Def3
.= c1 + (c2 + c3) by Def3
.= (c1 + c2) + c3
.= (addcomplex . (c1,c2)) + c3 by Def3
.= addcomplex . ((addcomplex . (c1,c2)),c3) by Def3 ; ::_thesis: verum
end;
cluster multcomplex -> commutative associative ;
coherence
( multcomplex is commutative & multcomplex is associative )
proof
thus multcomplex is commutative ::_thesis: multcomplex is associative
proof
let c1, c2 be Element of COMPLEX ; :: according to BINOP_1:def_2 ::_thesis: multcomplex . (c1,c2) = multcomplex . (c2,c1)
thus multcomplex . (c1,c2) = c1 * c2 by Def5
.= multcomplex . (c2,c1) by Def5 ; ::_thesis: verum
end;
let c1, c2, c3 be Element of COMPLEX ; :: according to BINOP_1:def_3 ::_thesis: multcomplex . (c1,(multcomplex . (c2,c3))) = multcomplex . ((multcomplex . (c1,c2)),c3)
thus multcomplex . (c1,(multcomplex . (c2,c3))) = c1 * (multcomplex . (c2,c3)) by Def5
.= c1 * (c2 * c3) by Def5
.= (c1 * c2) * c3
.= (multcomplex . (c1,c2)) * c3 by Def5
.= multcomplex . ((multcomplex . (c1,c2)),c3) by Def5 ; ::_thesis: verum
end;
cluster addreal -> commutative associative ;
coherence
( addreal is commutative & addreal is associative )
proof
thus addreal is commutative ::_thesis: addreal is associative
proof
let r1, r2 be Element of REAL ; :: according to BINOP_1:def_2 ::_thesis: addreal . (r1,r2) = addreal . (r2,r1)
thus addreal . (r1,r2) = r1 + r2 by Def9
.= addreal . (r2,r1) by Def9 ; ::_thesis: verum
end;
let r1, r2, r3 be Element of REAL ; :: according to BINOP_1:def_3 ::_thesis: addreal . (r1,(addreal . (r2,r3))) = addreal . ((addreal . (r1,r2)),r3)
thus addreal . (r1,(addreal . (r2,r3))) = r1 + (addreal . (r2,r3)) by Def9
.= r1 + (r2 + r3) by Def9
.= (r1 + r2) + r3
.= (addreal . (r1,r2)) + r3 by Def9
.= addreal . ((addreal . (r1,r2)),r3) by Def9 ; ::_thesis: verum
end;
cluster multreal -> commutative associative ;
coherence
( multreal is commutative & multreal is associative )
proof
thus multreal is commutative ::_thesis: multreal is associative
proof
let r1, r2 be Element of REAL ; :: according to BINOP_1:def_2 ::_thesis: multreal . (r1,r2) = multreal . (r2,r1)
thus multreal . (r1,r2) = r1 * r2 by Def11
.= multreal . (r2,r1) by Def11 ; ::_thesis: verum
end;
let r1, r2, r3 be Element of REAL ; :: according to BINOP_1:def_3 ::_thesis: multreal . (r1,(multreal . (r2,r3))) = multreal . ((multreal . (r1,r2)),r3)
thus multreal . (r1,(multreal . (r2,r3))) = r1 * (multreal . (r2,r3)) by Def11
.= r1 * (r2 * r3) by Def11
.= (r1 * r2) * r3
.= (multreal . (r1,r2)) * r3 by Def11
.= multreal . ((multreal . (r1,r2)),r3) by Def11 ; ::_thesis: verum
end;
cluster addrat -> commutative associative ;
coherence
( addrat is commutative & addrat is associative )
proof
thus addrat is commutative ::_thesis: addrat is associative
proof
let w1, w2 be Element of RAT ; :: according to BINOP_1:def_2 ::_thesis: addrat . (w1,w2) = addrat . (w2,w1)
thus addrat . (w1,w2) = w1 + w2 by Def15
.= addrat . (w2,w1) by Def15 ; ::_thesis: verum
end;
let w1, w2, w3 be Element of RAT ; :: according to BINOP_1:def_3 ::_thesis: addrat . (w1,(addrat . (w2,w3))) = addrat . ((addrat . (w1,w2)),w3)
thus addrat . (w1,(addrat . (w2,w3))) = w1 + (addrat . (w2,w3)) by Def15
.= w1 + (w2 + w3) by Def15
.= (w1 + w2) + w3
.= (addrat . (w1,w2)) + w3 by Def15
.= addrat . ((addrat . (w1,w2)),w3) by Def15 ; ::_thesis: verum
end;
cluster multrat -> commutative associative ;
coherence
( multrat is commutative & multrat is associative )
proof
thus multrat is commutative ::_thesis: multrat is associative
proof
let w1, w2 be Element of RAT ; :: according to BINOP_1:def_2 ::_thesis: multrat . (w1,w2) = multrat . (w2,w1)
thus multrat . (w1,w2) = w1 * w2 by Def17
.= multrat . (w2,w1) by Def17 ; ::_thesis: verum
end;
let w1, w2, w3 be Element of RAT ; :: according to BINOP_1:def_3 ::_thesis: multrat . (w1,(multrat . (w2,w3))) = multrat . ((multrat . (w1,w2)),w3)
thus multrat . (w1,(multrat . (w2,w3))) = w1 * (multrat . (w2,w3)) by Def17
.= w1 * (w2 * w3) by Def17
.= (w1 * w2) * w3
.= (multrat . (w1,w2)) * w3 by Def17
.= multrat . ((multrat . (w1,w2)),w3) by Def17 ; ::_thesis: verum
end;
cluster addint -> commutative associative ;
coherence
( addint is commutative & addint is associative )
proof
thus addint is commutative ::_thesis: addint is associative
proof
let i1, i2 be Element of INT ; :: according to BINOP_1:def_2 ::_thesis: addint . (i1,i2) = addint . (i2,i1)
thus addint . (i1,i2) = i1 + i2 by Def20
.= addint . (i2,i1) by Def20 ; ::_thesis: verum
end;
let i1, i2, i3 be Element of INT ; :: according to BINOP_1:def_3 ::_thesis: addint . (i1,(addint . (i2,i3))) = addint . ((addint . (i1,i2)),i3)
thus addint . (i1,(addint . (i2,i3))) = i1 + (addint . (i2,i3)) by Def20
.= i1 + (i2 + i3) by Def20
.= (i1 + i2) + i3
.= (addint . (i1,i2)) + i3 by Def20
.= addint . ((addint . (i1,i2)),i3) by Def20 ; ::_thesis: verum
end;
cluster multint -> commutative associative ;
coherence
( multint is commutative & multint is associative )
proof
thus multint is commutative ::_thesis: multint is associative
proof
let i1, i2 be Element of INT ; :: according to BINOP_1:def_2 ::_thesis: multint . (i1,i2) = multint . (i2,i1)
thus multint . (i1,i2) = i1 * i2 by Def22
.= multint . (i2,i1) by Def22 ; ::_thesis: verum
end;
let i1, i2, i3 be Element of INT ; :: according to BINOP_1:def_3 ::_thesis: multint . (i1,(multint . (i2,i3))) = multint . ((multint . (i1,i2)),i3)
thus multint . (i1,(multint . (i2,i3))) = i1 * (multint . (i2,i3)) by Def22
.= i1 * (i2 * i3) by Def22
.= (i1 * i2) * i3
.= (multint . (i1,i2)) * i3 by Def22
.= multint . ((multint . (i1,i2)),i3) by Def22 ; ::_thesis: verum
end;
cluster addnat -> commutative associative ;
coherence
( addnat is commutative & addnat is associative )
proof
thus addnat is commutative ::_thesis: addnat is associative
proof
let n1, n2 be Element of NAT ; :: according to BINOP_1:def_2 ::_thesis: addnat . (n1,n2) = addnat . (n2,n1)
thus addnat . (n1,n2) = n1 + n2 by Def23
.= addnat . (n2,n1) by Def23 ; ::_thesis: verum
end;
let n1, n2, n3 be Element of NAT ; :: according to BINOP_1:def_3 ::_thesis: addnat . (n1,(addnat . (n2,n3))) = addnat . ((addnat . (n1,n2)),n3)
thus addnat . (n1,(addnat . (n2,n3))) = n1 + (addnat . (n2,n3)) by Def23
.= n1 + (n2 + n3) by Def23
.= (n1 + n2) + n3
.= (addnat . (n1,n2)) + n3 by Def23
.= addnat . ((addnat . (n1,n2)),n3) by Def23 ; ::_thesis: verum
end;
cluster multnat -> commutative associative ;
coherence
( multnat is commutative & multnat is associative )
proof
thus multnat is commutative ::_thesis: multnat is associative
proof
let n1, n2 be Element of NAT ; :: according to BINOP_1:def_2 ::_thesis: multnat . (n1,n2) = multnat . (n2,n1)
thus multnat . (n1,n2) = n1 * n2 by Def24
.= multnat . (n2,n1) by Def24 ; ::_thesis: verum
end;
let n1, n2, n3 be Element of NAT ; :: according to BINOP_1:def_3 ::_thesis: multnat . (n1,(multnat . (n2,n3))) = multnat . ((multnat . (n1,n2)),n3)
thus multnat . (n1,(multnat . (n2,n3))) = n1 * (multnat . (n2,n3)) by Def24
.= n1 * (n2 * n3) by Def24
.= (n1 * n2) * n3
.= (multnat . (n1,n2)) * n3 by Def24
.= multnat . ((multnat . (n1,n2)),n3) by Def24 ; ::_thesis: verum
end;
end;
Lm1: 0 in NAT
;
then reconsider z = 0 as Element of COMPLEX by NUMBERS:20;
Lm2: z is_a_unity_wrt addcomplex
proof
thus for c being Element of COMPLEX holds addcomplex . (z,c) = c :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: z is_a_right_unity_wrt addcomplex
proof
let c be Element of COMPLEX ; ::_thesis: addcomplex . (z,c) = c
thus addcomplex . (z,c) = 0 + c by Def3
.= c ; ::_thesis: verum
end;
let c be Element of COMPLEX ; :: according to BINOP_1:def_17 ::_thesis: addcomplex . (c,z) = c
thus addcomplex . (c,z) = c + 0 by Def3
.= c ; ::_thesis: verum
end;
Lm3: 0 is_a_unity_wrt addreal
proof
thus for r being Element of REAL holds addreal . (0,r) = r :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: 0 is_a_right_unity_wrt addreal
proof
let r be Element of REAL ; ::_thesis: addreal . (0,r) = r
thus addreal . (0,r) = 0 + r by Def9
.= r ; ::_thesis: verum
end;
let r be Element of REAL ; :: according to BINOP_1:def_17 ::_thesis: addreal . (r,0) = r
thus addreal . (r,0) = r + 0 by Def9
.= r ; ::_thesis: verum
end;
reconsider z = 0 as Element of RAT by Lm1, NUMBERS:18;
Lm4: z is_a_unity_wrt addrat
proof
thus for w being Element of RAT holds addrat . (z,w) = w :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: z is_a_right_unity_wrt addrat
proof
let w be Element of RAT ; ::_thesis: addrat . (z,w) = w
thus addrat . (z,w) = 0 + w by Def15
.= w ; ::_thesis: verum
end;
let w be Element of RAT ; :: according to BINOP_1:def_17 ::_thesis: addrat . (w,z) = w
thus addrat . (w,z) = w + 0 by Def15
.= w ; ::_thesis: verum
end;
reconsider z = 0 as Element of INT by Lm1, NUMBERS:17;
Lm5: z is_a_unity_wrt addint
proof
thus for i being Element of INT holds addint . (z,i) = i :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: z is_a_right_unity_wrt addint
proof
let i be Element of INT ; ::_thesis: addint . (z,i) = i
thus addint . (z,i) = 0 + i by Def20
.= i ; ::_thesis: verum
end;
let i be Element of INT ; :: according to BINOP_1:def_17 ::_thesis: addint . (i,z) = i
thus addint . (i,z) = i + 0 by Def20
.= i ; ::_thesis: verum
end;
Lm6: 0 is_a_unity_wrt addnat
proof
thus for n being Element of NAT holds addnat . (0,n) = n :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: 0 is_a_right_unity_wrt addnat
proof
let n be Element of NAT ; ::_thesis: addnat . (0,n) = n
thus addnat . (0,n) = 0 + n by Def23
.= n ; ::_thesis: verum
end;
let n be Element of NAT ; :: according to BINOP_1:def_17 ::_thesis: addnat . (n,0) = n
thus addnat . (n,0) = n + 0 by Def23
.= n ; ::_thesis: verum
end;
Lm7: 1 in NAT
;
then reconsider z = 1 as Element of COMPLEX by NUMBERS:20;
Lm8: z is_a_unity_wrt multcomplex
proof
thus for c being Element of COMPLEX holds multcomplex . (z,c) = c :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: z is_a_right_unity_wrt multcomplex
proof
let c be Element of COMPLEX ; ::_thesis: multcomplex . (z,c) = c
thus multcomplex . (z,c) = 1 * c by Def5
.= c ; ::_thesis: verum
end;
let c be Element of COMPLEX ; :: according to BINOP_1:def_17 ::_thesis: multcomplex . (c,z) = c
thus multcomplex . (c,z) = c * 1 by Def5
.= c ; ::_thesis: verum
end;
Lm9: 1 is_a_unity_wrt multreal
proof
thus for r being Element of REAL holds multreal . (1,r) = r :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: 1 is_a_right_unity_wrt multreal
proof
let r be Element of REAL ; ::_thesis: multreal . (1,r) = r
thus multreal . (1,r) = 1 * r by Def11
.= r ; ::_thesis: verum
end;
let r be Element of REAL ; :: according to BINOP_1:def_17 ::_thesis: multreal . (r,1) = r
thus multreal . (r,1) = r * 1 by Def11
.= r ; ::_thesis: verum
end;
reconsider z = 1 as Element of RAT by Lm7, NUMBERS:18;
Lm10: z is_a_unity_wrt multrat
proof
thus for w being Element of RAT holds multrat . (z,w) = w :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: z is_a_right_unity_wrt multrat
proof
let w be Element of RAT ; ::_thesis: multrat . (z,w) = w
thus multrat . (z,w) = 1 * w by Def17
.= w ; ::_thesis: verum
end;
let w be Element of RAT ; :: according to BINOP_1:def_17 ::_thesis: multrat . (w,z) = w
thus multrat . (w,z) = w * 1 by Def17
.= w ; ::_thesis: verum
end;
reconsider z = 1 as Element of INT by Lm7, NUMBERS:17;
Lm11: z is_a_unity_wrt multint
proof
thus for i being Element of INT holds multint . (z,i) = i :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: z is_a_right_unity_wrt multint
proof
let i be Element of INT ; ::_thesis: multint . (z,i) = i
thus multint . (z,i) = 1 * i by Def22
.= i ; ::_thesis: verum
end;
let i be Element of INT ; :: according to BINOP_1:def_17 ::_thesis: multint . (i,z) = i
thus multint . (i,z) = i * 1 by Def22
.= i ; ::_thesis: verum
end;
Lm12: 1 is_a_unity_wrt multnat
proof
thus for n being Element of NAT holds multnat . (1,n) = n :: according to BINOP_1:def_7,BINOP_1:def_16 ::_thesis: 1 is_a_right_unity_wrt multnat
proof
let n be Element of NAT ; ::_thesis: multnat . (1,n) = n
thus multnat . (1,n) = 1 * n by Def24
.= n ; ::_thesis: verum
end;
let n be Element of NAT ; :: according to BINOP_1:def_17 ::_thesis: multnat . (n,1) = n
thus multnat . (n,1) = n * 1 by Def24
.= n ; ::_thesis: verum
end;
registration
cluster addcomplex -> having_a_unity ;
coherence
addcomplex is having_a_unity by Lm2, SETWISEO:def_2;
cluster addreal -> having_a_unity ;
coherence
addreal is having_a_unity by Lm3, SETWISEO:def_2;
cluster addrat -> having_a_unity ;
coherence
addrat is having_a_unity by Lm4, SETWISEO:def_2;
cluster addint -> having_a_unity ;
coherence
addint is having_a_unity by Lm5, SETWISEO:def_2;
cluster addnat -> having_a_unity ;
coherence
addnat is having_a_unity by Lm6, SETWISEO:def_2;
cluster multcomplex -> having_a_unity ;
coherence
multcomplex is having_a_unity by Lm8, SETWISEO:def_2;
cluster multreal -> having_a_unity ;
coherence
multreal is having_a_unity by Lm9, SETWISEO:def_2;
cluster multrat -> having_a_unity ;
coherence
multrat is having_a_unity by Lm10, SETWISEO:def_2;
cluster multint -> having_a_unity ;
coherence
multint is having_a_unity by Lm11, SETWISEO:def_2;
cluster multnat -> having_a_unity ;
coherence
multnat is having_a_unity by Lm12, SETWISEO:def_2;
end;
theorem :: BINOP_2:1
the_unity_wrt addcomplex = 0 by Lm2, BINOP_1:def_8;
theorem :: BINOP_2:2
the_unity_wrt addreal = 0 by Lm3, BINOP_1:def_8;
theorem :: BINOP_2:3
the_unity_wrt addrat = 0 by Lm4, BINOP_1:def_8;
theorem :: BINOP_2:4
the_unity_wrt addint = 0 by Lm5, BINOP_1:def_8;
theorem :: BINOP_2:5
the_unity_wrt addnat = 0 by Lm6, BINOP_1:def_8;
theorem :: BINOP_2:6
the_unity_wrt multcomplex = 1 by Lm8, BINOP_1:def_8;
theorem :: BINOP_2:7
the_unity_wrt multreal = 1 by Lm9, BINOP_1:def_8;
theorem :: BINOP_2:8
the_unity_wrt multrat = 1 by Lm10, BINOP_1:def_8;
theorem :: BINOP_2:9
the_unity_wrt multint = 1 by Lm11, BINOP_1:def_8;
theorem :: BINOP_2:10
the_unity_wrt multnat = 1 by Lm12, BINOP_1:def_8;