:: REALSET2 semantic presentation
:: deftheorem Def1 defines zeroed REALSET2:def 1 :
:: deftheorem Def2 defines complementable REALSET2:def 2 :
definition
let L be non
empty LoopStr ;
redefine attr a1 is
add-associative means :
Def3:
:: REALSET2:def 3
for
a,
b,
c being
Element of
L holds the
add of
L . (the add of L . a,b),
c = the
add of
L . a,
(the add of L . b,c);
compatibility
( L is add-associative iff for a, b, c being Element of L holds the add of L . (the add of L . a,b),c = the add of L . a,(the add of L . b,c) )
redefine attr a1 is
Abelian means :
Def4:
:: REALSET2:def 4
for
a,
b being
Element of
L holds the
add of
L . a,
b = the
add of
L . b,
a;
compatibility
( L is Abelian iff for a, b being Element of L holds the add of L . a,b = the add of L . b,a )
end;
:: deftheorem Def3 defines add-associative REALSET2:def 3 :
:: deftheorem Def4 defines Abelian REALSET2:def 4 :
E29:
now
consider x being
set ;
set B =
{x};
E17:
x in {x}
by TARSKI:def 1;
consider og being
BinOp of
{x};
E18:
og . [x,x] = x
reconsider ng =
x as
Element of
{x} by TARSKI:def 1;
E20:
for
a,
b,
c being
Element of
{x} holds
og . (og . a,b),
c = og . a,
(og . b,c)
E24:
for
a being
Element of
{x} holds
(
og . a,
ng = a &
og . ng,
a = a )
E25:
for
a being
Element of
{x} ex
b being
Element of
{x} st
(
og . a,
b = ng &
og . b,
a = ng )
for
a,
b being
Element of
{x} holds
og . a,
b = og . b,
a
hence
ex
A being non
empty set ex
og being
BinOp of
A ex
ng being
Element of
A st
( ( for
a,
b,
c being
Element of
A holds
og . (og . a,b),
c = og . a,
(og . b,c) ) & ( for
a being
Element of
A holds
(
og . a,
ng = a &
og . ng,
a = a ) ) & ( for
a being
Element of
A ex
b being
Element of
A st
(
og . a,
b = ng &
og . b,
a = ng ) ) & ( for
a,
b being
Element of
A holds
og . a,
b = og . b,
a ) )
by , , Def1;
end;
:: deftheorem Def5 defines trivial REALSET2:def 5 :
Lemma35:
for S being 1-sorted holds
( S is trivial iff for x, y being Element of S holds x = y )
E41:
now
let A be non
trivial set ;
let od be
BinOp of
A,
om be
BinOp of
A;
let nd be
Element of
A;
let nm be
Element of
A \ {nd};
let nm0 be
Element of
A;
assume E17:
nm0 = nm
;
set F =
doubleLoopStr(#
A,
od,
om,
nm0,
nd #);
thus
not
doubleLoopStr(#
A,
od,
om,
nm0,
nd #) is
trivial
thus
doubleLoopStr(#
A,
od,
om,
nm0,
nd #) is
strict
;
end;
:: deftheorem Def6 defines field REALSET2:def 6 :
:: deftheorem Def7 defines trivial REALSET2:def 7 :
consider x being set , y being set such that
Lemma49:
x <> y
by VECTSP_1:78;
set Z = {x,y};
Lemma51:
x in {x,y}
by TARSKI:def 2;
Lemma52:
y in {x,y}
by TARSKI:def 2;
for s being Element of {x,y} holds {x,y} \ {s} is non empty set
then reconsider A = {x,y} as non trivial set by REALSET1:6;
reconsider nd = x as Element of A by TARSKI:def 2;
Lemma54:
for t being set holds
( t in [:A,A:] iff ( t = [x,x] or t = [x,y] or t = [y,x] or t = [y,y] ) )
Lemma58:
[:A,A:] = {[x,x],[x,y],[y,x],[y,y]}
by Def2, ENUMSET1:def 2;
set x1 = [[x,x],x];
set x2 = [[x,y],y];
set x3 = [[y,x],y];
set x4 = [[y,y],x];
set F = {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]};
Lemma63:
for p being set st p in {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} holds
ex q, r being set st [q,r] = p
for q, r1, r2 being set st [q,r1] in {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} & [q,r2] in {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} holds
r1 = r2
then reconsider od = {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} as Function by , FUNCT_1:2;
for t being set holds
( t in [:A,A:] iff ex r being set st [t,r] in {[[x,x],x],[[x,y],y],[[y,x],y],[[y,y],x]} )
then Lemma69:
[:A,A:] = dom od
by RELAT_1:def 4;
then Lemma70:
[x,x] in dom od
by Def2;
Lemma71:
[[x,x],x] in od
by ENUMSET1:def 2;
then Lemma72:
od . x,x = x
by , FUNCT_1:def 4;
Lemma73:
[x,y] in dom od
by Def2, Def3;
Lemma74:
[[x,y],y] in od
by ENUMSET1:def 2;
then Lemma75:
od . [x,y] = y
by , FUNCT_1:def 4;
Lemma76:
[y,x] in dom od
by Def2, Def3;
Lemma77:
[[y,x],y] in od
by ENUMSET1:def 2;
then Lemma78:
od . [y,x] = y
by , FUNCT_1:def 4;
Lemma79:
[y,y] in dom od
by Def2, Def3;
[[y,y],x] in od
by ENUMSET1:def 2;
then Lemma80:
od . y,y = x
by , FUNCT_1:def 4;
then Lemma81:
for t being set st t in [:A,A:] holds
od . t in A
by Def1, , Def2, , , ;
set s2 = [[x,y],x];
set s3 = [[y,x],x];
set s4 = [[y,y],y];
set D = {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]};
Lemma86:
for p being set st p in {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} holds
ex q, r being set st [q,r] = p
for q, r1, r2 being set st [q,r1] in {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} & [q,r2] in {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} holds
r1 = r2
then reconsider om = {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} as Function by , FUNCT_1:2;
for t being set holds
( t in [:A,A:] iff ex r being set st [t,r] in {[[x,x],x],[[x,y],x],[[y,x],x],[[y,y],y]} )
then Lemma87:
[:A,A:] = dom om
by RELAT_1:def 4;
then Lemma88:
[x,x] in dom om
by Def2;
[[x,x],x] in om
by ENUMSET1:def 2;
then Lemma89:
om . [x,x] = x
by , FUNCT_1:def 4;
Lemma90:
[x,y] in dom om
by Def2, Lemma29;
[[x,y],x] in om
by ENUMSET1:def 2;
then Lemma91:
om . [x,y] = x
by , FUNCT_1:def 4;
Lemma92:
[y,x] in dom om
by Def2, Lemma29;
[[y,x],x] in om
by ENUMSET1:def 2;
then Lemma93:
om . [y,x] = x
by , FUNCT_1:def 4;
Lemma94:
[y,y] in dom om
by Def2, Lemma29;
[[y,y],y] in om
by ENUMSET1:def 2;
then Lemma95:
om . [y,y] = y
by , FUNCT_1:def 4;
then Lemma96:
for t being set st t in [:A,A:] holds
om . t in A
by Def1, , Def2, , , Lemma35;
then Lemma97:
om is BinOp of A
by Lemma29, FUNCT_2:5;
Lemma98:
A \ {x} = {y}
by , ZFMISC_1:23;
then Lemma99:
[:(A \ {x}),(A \ {x}):] = {[y,y]}
by ZFMISC_1:35;
Lemma100:
for t being set st t in [:(A \ {x}),(A \ {x}):] holds
om . t in A \ {x}
reconsider nm = y as Element of A \ {nd} by , TARSKI:def 1;
reconsider od0 = od as BinOp of A by Def3, , FUNCT_2:5;
reconsider om0 = om as BinOp of A by Lemma29, , FUNCT_2:5;
Lemma103:
for a, b, d being Element of A holds od0 . (od0 . a,b),d = od0 . a,(od0 . b,d)
Lemma105:
for a being Element of A holds
( od0 . a,nd = a & od0 . nd,a = a )
Lemma106:
for a being Element of A ex b being Element of A st
( od0 . a,b = nd & od0 . b,a = nd )
for a, b being Element of A holds od0 . a,b = od0 . b,a
then Lemma107:
LoopStr(# A,od0,nd #) is Group
by , , , , , , ;
reconsider om1 = om as DnT of nd,A by , , REALSET1:def 8;
Lemma109:
for B being non empty set
for P being BinOp of B
for e being Element of B st B = A \ {nd} & e = nm & P = om1 ! A,nd holds
LoopStr(# B,P,e #) is Group
Lemma112:
for a, b, d being Element of A holds
( om0 . [a,(od0 . [b,d])] = od0 . [(om0 . [a,b]),(om0 . [a,d])] & om0 . [(od0 . [a,b]),d] = od0 . [(om0 . [a,d]),(om0 . [b,d])] )
definition
let IT be
doubleLoopStr ;
attr a1 is
Field-like means :
Def8:
:: REALSET2:def 8
ex
A being non
trivial set ex
od being
BinOp of
A ex
nd being
Element of
A ex
om being
DnT of
nd,
A ex
nm being
Element of
A \ {nd} st
(
x = field A,
od,
om,
nd,
nm &
LoopStr(#
A,
od,
nd #) is
Group & ( for
B being non
empty set for
P being
BinOp of
B for
e being
Element of
B st
B = A \ {nd} &
e = nm &
P = om ! A,
nd holds
LoopStr(#
B,
P,
e #) is
Group ) & ( for
x,
y,
z being
Element of
A holds
(
om . [x,(od . [y,z])] = od . [(om . [x,y]),(om . [x,z])] &
om . [(od . [x,y]),z] = od . [(om . [x,z]),(om . [y,z])] ) ) );
end;
:: deftheorem Def8 defines Field-like REALSET2:def 8 :
for
IT being
doubleLoopStr holds
(
IT is
Field-like iff ex
A being non
trivial set ex
od being
BinOp of
A ex
nd being
Element of
A ex
om being
DnT of
nd,
A ex
nm being
Element of
A \ {nd} st
(
IT = field A,
od,
om,
nd,
nm &
LoopStr(#
A,
od,
nd #) is
Group & ( for
B being non
empty set for
P being
BinOp of
B for
e being
Element of
B st
B = A \ {nd} &
e = nm &
P = om ! A,
nd holds
LoopStr(#
B,
P,
e #) is
Group ) & ( for
x,
y,
z being
Element of
A holds
(
om . [x,(od . [y,z])] = od . [(om . [x,y]),(om . [x,z])] &
om . [(od . [x,y]),z] = od . [(om . [x,z]),(om . [y,z])] ) ) ) );
definition
let F be
Field;
func suppf c1 -> non
trivial set means :
Def9:
:: REALSET2:def 9
ex
od being
BinOp of
it ex
nd being
Element of
it ex
om being
DnT of
nd,
it ex
nm being
Element of
it \ {nd} st
x = field it,
od,
om,
nd,
nm;
existence
ex b1 being non trivial set ex od being BinOp of b1 ex nd being Element of b1 ex om being DnT of nd,b1 ex nm being Element of b1 \ {nd} st F = field b1,od,om,nd,nm
uniqueness
for b1, b2 being non trivial set st ex od being BinOp of b1 ex nd being Element of b1 ex om being DnT of nd,b1 ex nm being Element of b1 \ {nd} st F = field b1,od,om,nd,nm & ex od being BinOp of b2 ex nd being Element of b2 ex om being DnT of nd,b2 ex nm being Element of b2 \ {nd} st F = field b2,od,om,nd,nm holds
b1 = b2
end;
:: deftheorem Def9 defines suppf REALSET2:def 9 :
definition
let F be
Field;
func odf c1 -> BinOp of
suppf a1 means :
Def10:
:: REALSET2:def 10
ex
nd being
Element of
suppf x ex
om being
DnT of
nd,
suppf x ex
nm being
Element of
(suppf x) \ {nd} st
x = field (suppf x),
it,
om,
nd,
nm;
existence
ex b1 being BinOp of suppf F ex nd being Element of suppf F ex om being DnT of nd, suppf F ex nm being Element of (suppf F) \ {nd} st F = field (suppf F),b1,om,nd,nm
by ;
uniqueness
for b1, b2 being BinOp of suppf F st ex nd being Element of suppf F ex om being DnT of nd, suppf F ex nm being Element of (suppf F) \ {nd} st F = field (suppf F),b1,om,nd,nm & ex nd being Element of suppf F ex om being DnT of nd, suppf F ex nm being Element of (suppf F) \ {nd} st F = field (suppf F),b2,om,nd,nm holds
b1 = b2
end;
:: deftheorem Def10 defines odf REALSET2:def 10 :
definition
let F be
Field;
func ndf c1 -> Element of
suppf a1 means :
Def11:
:: REALSET2:def 11
ex
om being
DnT of
it,
suppf x ex
nm being
Element of
(suppf x) \ {it} st
x = field (suppf x),
(odf x),
om,
it,
nm;
existence
ex b1 being Element of suppf F ex om being DnT of b1, suppf F ex nm being Element of (suppf F) \ {b1} st F = field (suppf F),(odf F),om,b1,nm
by Lemma51;
uniqueness
for b1, b2 being Element of suppf F st ex om being DnT of b1, suppf F ex nm being Element of (suppf F) \ {b1} st F = field (suppf F),(odf F),om,b1,nm & ex om being DnT of b2, suppf F ex nm being Element of (suppf F) \ {b2} st F = field (suppf F),(odf F),om,b2,nm holds
b1 = b2
end;
:: deftheorem Def11 defines ndf REALSET2:def 11 :
definition
let F be
Field;
func omf c1 -> DnT of
ndf a1,
suppf a1 means :
Def12:
:: REALSET2:def 12
ex
nm being
Element of
(suppf x) \ {(ndf x)} st
x = field (suppf x),
(odf x),
it,
(ndf x),
nm;
existence
ex b1 being DnT of ndf F, suppf F ex nm being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),b1,(ndf F),nm
by Lemma52;
uniqueness
for b1, b2 being DnT of ndf F, suppf F st ex nm being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),b1,(ndf F),nm & ex nm being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),b2,(ndf F),nm holds
b1 = b2
end;
:: deftheorem Def12 defines omf REALSET2:def 12 :
definition
let F be
Field;
func nmf c1 -> Element of
(suppf a1) \ {(ndf a1)} means :
Def13:
:: REALSET2:def 13
x = field (suppf x),
(odf x),
(omf x),
(ndf x),
it;
existence
ex b1 being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),(omf F),(ndf F),b1
by ;
uniqueness
for b1, b2 being Element of (suppf F) \ {(ndf F)} st F = field (suppf F),(odf F),(omf F),(ndf F),b1 & F = field (suppf F),(odf F),(omf F),(ndf F),b2 holds
b1 = b2
end;
:: deftheorem Def13 defines nmf REALSET2:def 13 :
theorem Th1: :: REALSET2:1
theorem Th2: :: REALSET2:2
theorem Th3: :: REALSET2:3
for
F being
Field for
x,
y,
z being
Element of
suppf F holds
(
(omf F) . [x,((odf F) . [y,z])] = (odf F) . [((omf F) . [x,y]),((omf F) . [x,z])] &
(omf F) . [((odf F) . [x,y]),z] = (odf F) . [((omf F) . [x,z]),((omf F) . [y,z])] )
theorem Th4: :: REALSET2:4
theorem Th5: :: REALSET2:5
theorem Th6: :: REALSET2:6
theorem Th7: :: REALSET2:7
theorem Th8: :: REALSET2:8
theorem Th9: :: REALSET2:9
theorem Th10: :: REALSET2:10
theorem Th11: :: REALSET2:11
:: deftheorem Def14 defines compf REALSET2:def 14 :
theorem Th12: :: REALSET2:12
theorem Th13: :: REALSET2:13
theorem Th14: :: REALSET2:14
theorem Th15: :: REALSET2:15
theorem Th16: :: REALSET2:16
theorem Th17: :: REALSET2:17
theorem Th18: :: REALSET2:18
theorem Th19: :: REALSET2:19
theorem Th20: :: REALSET2:20
theorem Th21: :: REALSET2:21
theorem Th22: :: REALSET2:22
theorem Th23: :: REALSET2:23
theorem Th24: :: REALSET2:24
theorem Th25: :: REALSET2:25
:: deftheorem Def15 defines revf REALSET2:def 15 :
theorem Th26: :: REALSET2:26
theorem Th27: :: REALSET2:27
theorem Th28: :: REALSET2:28
theorem Th29: :: REALSET2:29
theorem Th30: :: REALSET2:30