:: REALSET2 semantic presentation
:: deftheorem Def1 defines zeroed REALSET2:def 1 :
:: deftheorem Def2 defines complementable REALSET2:def 2 :
definition
let c
1 be non
empty LoopStr ;
redefine attr a
1 is
add-associative means :
Def3:
:: REALSET2:def 3
for b
1, b
2, b
3 being
Element of a
1 holds the
add of a
1 . (the add of a1 . b1,b2),b
3 = the
add of a
1 . b
1,
(the add of a1 . b2,b3);
compatibility
( c1 is add-associative iff for b1, b2, b3 being Element of c1 holds the add of c1 . (the add of c1 . b1,b2),b3 = the add of c1 . b1,(the add of c1 . b2,b3) )
redefine attr a
1 is
Abelian means :
Def4:
:: REALSET2:def 4
for b
1, b
2 being
Element of a
1 holds the
add of a
1 . b
1,b
2 = the
add of a
1 . b
2,b
1;
compatibility
( c1 is Abelian iff for b1, b2 being Element of c1 holds the add of c1 . b1,b2 = the add of c1 . b2,b1 )
end;
:: deftheorem Def3 defines add-associative REALSET2:def 3 :
:: deftheorem Def4 defines Abelian REALSET2:def 4 :
E5:
now
consider c
1 being
set ;
set c
2 =
{c1};
E6:
c
1 in {c1}
by TARSKI:def 1;
consider c
3 being
BinOp of
{c1};
E7:
c
3 . [c1,c1] = c
1
reconsider c
4 = c
1 as
Element of
{c1} by TARSKI:def 1;
E8:
for b
1, b
2, b
3 being
Element of
{c1} holds c
3 . (c3 . b1,b2),b
3 = c
3 . b
1,
(c3 . b2,b3)
E9:
for b
1 being
Element of
{c1} holds
( c
3 . b
1,c
4 = b
1 & c
3 . c
4,b
1 = b
1 )
E10:
for b
1 being
Element of
{c1} holds
ex b
2 being
Element of
{c1} st
( c
3 . b
1,b
2 = c
4 & c
3 . b
2,b
1 = c
4 )
for b
1, b
2 being
Element of
{c1} holds c
3 . b
1,b
2 = c
3 . b
2,b
1
hence
ex b
1 being non
empty set ex b
2 being
BinOp of b
1ex b
3 being
Element of b
1 st
( ( for b
4, b
5, b
6 being
Element of b
1 holds b
2 . (b2 . b4,b5),b
6 = b
2 . b
4,
(b2 . b5,b6) ) & ( for b
4 being
Element of b
1 holds
( b
2 . b
4,b
3 = b
4 & b
2 . b
3,b
4 = b
4 ) ) & ( for b
4 being
Element of b
1 holds
ex b
5 being
Element of b
1 st
( b
2 . b
4,b
5 = b
3 & b
2 . b
5,b
4 = b
3 ) ) & ( for b
4, b
5 being
Element of b
1 holds b
2 . b
4,b
5 = b
2 . b
5,b
4 ) )
by E8, E9, E10;
end;
:: deftheorem Def5 defines trivial REALSET2:def 5 :
Lemma6:
for b1 being 1-sorted holds
( b1 is trivial iff for b2, b3 being Element of b1 holds b2 = b3 )
E7:
now
let c
1 be non
trivial set ;
let c
2, c
3 be
BinOp of c
1;
let c
4 be
Element of c
1;
let c
5 be
Element of c
1 \ {c4};
let c
6 be
Element of c
1;
assume E8:
c
6 = c
5
;
set c
7 =
doubleLoopStr(# c
1,c
2,c
3,c
6,c
4 #);
thus
not
doubleLoopStr(# c
1,c
2,c
3,c
6,c
4 #) is
trivial
thus
doubleLoopStr(# c
1,c
2,c
3,c
6,c
4 #) is
strict
;
end;
:: deftheorem Def6 defines field REALSET2:def 6 :
:: deftheorem Def7 defines trivial REALSET2:def 7 :
consider c1, c2 being set such that
Lemma9:
c1 <> c2
by VECTSP_1:78;
set c3 = {c1,c2};
Lemma10:
c1 in {c1,c2}
by TARSKI:def 2;
Lemma11:
c2 in {c1,c2}
by TARSKI:def 2;
for b1 being Element of {c1,c2} holds
{c1,c2} \ {b1} is non empty set
then reconsider c4 = {c1,c2} as non trivial set by REALSET1:6;
reconsider c5 = c1 as Element of c4 by TARSKI:def 2;
Lemma12:
for b1 being set holds
( b1 in [:c4,c4:] iff not ( not b1 = [c1,c1] & not b1 = [c1,c2] & not b1 = [c2,c1] & not b1 = [c2,c2] ) )
Lemma13:
[:c4,c4:] = {[c1,c1],[c1,c2],[c2,c1],[c2,c2]}
by Lemma12, ENUMSET1:def 2;
set c6 = [[c1,c1],c1];
set c7 = [[c1,c2],c2];
set c8 = [[c2,c1],c2];
set c9 = [[c2,c2],c1];
set c10 = {[[c1,c1],c1],[[c1,c2],c2],[[c2,c1],c2],[[c2,c2],c1]};
Lemma14:
for b1 being set holds
not ( b1 in {[[c1,c1],c1],[[c1,c2],c2],[[c2,c1],c2],[[c2,c2],c1]} & ( for b2, b3 being set holds
not [b2,b3] = b1 ) )
for b1, b2, b3 being set holds
( [b1,b2] in {[[c1,c1],c1],[[c1,c2],c2],[[c2,c1],c2],[[c2,c2],c1]} & [b1,b3] in {[[c1,c1],c1],[[c1,c2],c2],[[c2,c1],c2],[[c2,c2],c1]} implies b2 = b3 )
then reconsider c11 = {[[c1,c1],c1],[[c1,c2],c2],[[c2,c1],c2],[[c2,c2],c1]} as Function by Lemma14, FUNCT_1:2;
for b1 being set holds
( b1 in [:c4,c4:] iff ex b2 being set st [b1,b2] in {[[c1,c1],c1],[[c1,c2],c2],[[c2,c1],c2],[[c2,c2],c1]} )
then Lemma15:
[:c4,c4:] = dom c11
by RELAT_1:def 4;
then Lemma16:
[c1,c1] in dom c11
by Lemma12;
Lemma17:
[[c1,c1],c1] in c11
by ENUMSET1:def 2;
then Lemma18:
c11 . c1,c1 = c1
by Lemma16, FUNCT_1:def 4;
Lemma19:
[c1,c2] in dom c11
by Lemma12, Lemma15;
Lemma20:
[[c1,c2],c2] in c11
by ENUMSET1:def 2;
then Lemma21:
c11 . [c1,c2] = c2
by Lemma19, FUNCT_1:def 4;
Lemma22:
[c2,c1] in dom c11
by Lemma12, Lemma15;
Lemma23:
[[c2,c1],c2] in c11
by ENUMSET1:def 2;
then Lemma24:
c11 . [c2,c1] = c2
by Lemma22, FUNCT_1:def 4;
Lemma25:
[c2,c2] in dom c11
by Lemma12, Lemma15;
[[c2,c2],c1] in c11
by ENUMSET1:def 2;
then Lemma26:
c11 . c2,c2 = c1
by Lemma25, FUNCT_1:def 4;
then Lemma27:
for b1 being set holds
( b1 in [:c4,c4:] implies c11 . b1 in c4 )
by Lemma10, Lemma11, Lemma12, Lemma18, Lemma21, Lemma24;
set c12 = [[c1,c2],c1];
set c13 = [[c2,c1],c1];
set c14 = [[c2,c2],c2];
set c15 = {[[c1,c1],c1],[[c1,c2],c1],[[c2,c1],c1],[[c2,c2],c2]};
Lemma28:
for b1 being set holds
not ( b1 in {[[c1,c1],c1],[[c1,c2],c1],[[c2,c1],c1],[[c2,c2],c2]} & ( for b2, b3 being set holds
not [b2,b3] = b1 ) )
for b1, b2, b3 being set holds
( [b1,b2] in {[[c1,c1],c1],[[c1,c2],c1],[[c2,c1],c1],[[c2,c2],c2]} & [b1,b3] in {[[c1,c1],c1],[[c1,c2],c1],[[c2,c1],c1],[[c2,c2],c2]} implies b2 = b3 )
then reconsider c16 = {[[c1,c1],c1],[[c1,c2],c1],[[c2,c1],c1],[[c2,c2],c2]} as Function by Lemma28, FUNCT_1:2;
for b1 being set holds
( b1 in [:c4,c4:] iff ex b2 being set st [b1,b2] in {[[c1,c1],c1],[[c1,c2],c1],[[c2,c1],c1],[[c2,c2],c2]} )
then Lemma29:
[:c4,c4:] = dom c16
by RELAT_1:def 4;
then Lemma30:
[c1,c1] in dom c16
by Lemma12;
[[c1,c1],c1] in c16
by ENUMSET1:def 2;
then Lemma31:
c16 . [c1,c1] = c1
by Lemma30, FUNCT_1:def 4;
Lemma32:
[c1,c2] in dom c16
by Lemma12, Lemma29;
[[c1,c2],c1] in c16
by ENUMSET1:def 2;
then Lemma33:
c16 . [c1,c2] = c1
by Lemma32, FUNCT_1:def 4;
Lemma34:
[c2,c1] in dom c16
by Lemma12, Lemma29;
[[c2,c1],c1] in c16
by ENUMSET1:def 2;
then Lemma35:
c16 . [c2,c1] = c1
by Lemma34, FUNCT_1:def 4;
Lemma36:
[c2,c2] in dom c16
by Lemma12, Lemma29;
[[c2,c2],c2] in c16
by ENUMSET1:def 2;
then Lemma37:
c16 . [c2,c2] = c2
by Lemma36, FUNCT_1:def 4;
then Lemma38:
for b1 being set holds
( b1 in [:c4,c4:] implies c16 . b1 in c4 )
by Lemma10, Lemma11, Lemma12, Lemma31, Lemma33, Lemma35;
then Lemma39:
c16 is BinOp of c4
by Lemma29, FUNCT_2:5;
Lemma40:
c4 \ {c1} = {c2}
by Lemma9, ZFMISC_1:23;
then Lemma41:
[:(c4 \ {c1}),(c4 \ {c1}):] = {[c2,c2]}
by ZFMISC_1:35;
Lemma42:
for b1 being set holds
( b1 in [:(c4 \ {c1}),(c4 \ {c1}):] implies c16 . b1 in c4 \ {c1} )
reconsider c17 = c2 as Element of c4 \ {c5} by Lemma40, TARSKI:def 1;
reconsider c18 = c11 as BinOp of c4 by Lemma15, Lemma27, FUNCT_2:5;
reconsider c19 = c16 as BinOp of c4 by Lemma29, Lemma38, FUNCT_2:5;
Lemma43:
for b1, b2, b3 being Element of c4 holds c18 . (c18 . b1,b2),b3 = c18 . b1,(c18 . b2,b3)
Lemma44:
for b1 being Element of c4 holds
( c18 . b1,c5 = b1 & c18 . c5,b1 = b1 )
Lemma45:
for b1 being Element of c4 holds
ex b2 being Element of c4 st
( c18 . b1,b2 = c5 & c18 . b2,b1 = c5 )
for b1, b2 being Element of c4 holds c18 . b1,b2 = c18 . b2,b1
then Lemma46:
LoopStr(# c4,c18,c5 #) is Group
by Def1, Def2, Def3, Def4, Lemma43, Lemma44, Lemma45;
reconsider c20 = c16 as DnT of c5,c4 by Lemma39, Lemma42, REALSET1:def 8;
Lemma47:
for b1 being non empty set
for b2 being BinOp of b1
for b3 being Element of b1 holds
( b1 = c4 \ {c5} & b3 = c17 & b2 = c20 ! c4,c5 implies LoopStr(# b1,b2,b3 #) is Group )
Lemma48:
for b1, b2, b3 being Element of c4 holds
( c19 . [b1,(c18 . [b2,b3])] = c18 . [(c19 . [b1,b2]),(c19 . [b1,b3])] & c19 . [(c18 . [b1,b2]),b3] = c18 . [(c19 . [b1,b3]),(c19 . [b2,b3])] )
definition
let c
21 be
doubleLoopStr ;
attr a
1 is
Field-like means :
Def8:
:: REALSET2:def 8
ex b
1 being non
trivial set ex b
2 being
BinOp of b
1ex b
3 being
Element of b
1ex b
4 being
DnT of b
3,b
1ex b
5 being
Element of b
1 \ {b3} st
( a
1 = field b
1,b
2,b
4,b
3,b
5 &
LoopStr(# b
1,b
2,b
3 #) is
Group & ( for b
6 being non
empty set for b
7 being
BinOp of b
6for b
8 being
Element of b
6 holds
( b
6 = b
1 \ {b3} & b
8 = b
5 & b
7 = b
4 ! b
1,b
3 implies
LoopStr(# b
6,b
7,b
8 #) is
Group ) ) & ( for b
6, b
7, b
8 being
Element of b
1 holds
( b
4 . [b6,(b2 . [b7,b8])] = b
2 . [(b4 . [b6,b7]),(b4 . [b6,b8])] & b
4 . [(b2 . [b6,b7]),b8] = b
2 . [(b4 . [b6,b8]),(b4 . [b7,b8])] ) ) );
end;
:: deftheorem Def8 defines Field-like REALSET2:def 8 :
for b
1 being
doubleLoopStr holds
( b
1 is
Field-like iff ex b
2 being non
trivial set ex b
3 being
BinOp of b
2ex b
4 being
Element of b
2ex b
5 being
DnT of b
4,b
2ex b
6 being
Element of b
2 \ {b4} st
( b
1 = field b
2,b
3,b
5,b
4,b
6 &
LoopStr(# b
2,b
3,b
4 #) is
Group & ( for b
7 being non
empty set for b
8 being
BinOp of b
7for b
9 being
Element of b
7 holds
( b
7 = b
2 \ {b4} & b
9 = b
6 & b
8 = b
5 ! b
2,b
4 implies
LoopStr(# b
7,b
8,b
9 #) is
Group ) ) & ( for b
7, b
8, b
9 being
Element of b
2 holds
( b
5 . [b7,(b3 . [b8,b9])] = b
3 . [(b5 . [b7,b8]),(b5 . [b7,b9])] & b
5 . [(b3 . [b7,b8]),b9] = b
3 . [(b5 . [b7,b9]),(b5 . [b8,b9])] ) ) ) );
definition
let c
21 be
Field;
func suppf c
1 -> non
trivial set means :
Def9:
:: REALSET2:def 9
ex b
1 being
BinOp of a
2ex b
2 being
Element of a
2ex b
3 being
DnT of b
2,a
2ex b
4 being
Element of a
2 \ {b2} st a
1 = field a
2,b
1,b
3,b
2,b
4;
existence
ex b1 being non trivial set ex b2 being BinOp of b1ex b3 being Element of b1ex b4 being DnT of b3,b1ex b5 being Element of b1 \ {b3} st c21 = field b1,b2,b4,b3,b5
uniqueness
for b1, b2 being non trivial set holds
( ex b3 being BinOp of b1ex b4 being Element of b1ex b5 being DnT of b4,b1ex b6 being Element of b1 \ {b4} st c21 = field b1,b3,b5,b4,b6 & ex b3 being BinOp of b2ex b4 being Element of b2ex b5 being DnT of b4,b2ex b6 being Element of b2 \ {b4} st c21 = field b2,b3,b5,b4,b6 implies b1 = b2 )
end;
:: deftheorem Def9 defines suppf REALSET2:def 9 :
definition
let c
21 be
Field;
func odf c
1 -> BinOp of
suppf a
1 means :
Def10:
:: REALSET2:def 10
ex b
1 being
Element of
suppf a
1ex b
2 being
DnT of b
1,
suppf a
1ex b
3 being
Element of
(suppf a1) \ {b1} st a
1 = field (suppf a1),a
2,b
2,b
1,b
3;
existence
ex b1 being BinOp of suppf c21ex b2 being Element of suppf c21ex b3 being DnT of b2, suppf c21ex b4 being Element of (suppf c21) \ {b2} st c21 = field (suppf c21),b1,b3,b2,b4
by Def9;
uniqueness
for b1, b2 being BinOp of suppf c21 holds
( ex b3 being Element of suppf c21ex b4 being DnT of b3, suppf c21ex b5 being Element of (suppf c21) \ {b3} st c21 = field (suppf c21),b1,b4,b3,b5 & ex b3 being Element of suppf c21ex b4 being DnT of b3, suppf c21ex b5 being Element of (suppf c21) \ {b3} st c21 = field (suppf c21),b2,b4,b3,b5 implies b1 = b2 )
end;
:: deftheorem Def10 defines odf REALSET2:def 10 :
definition
let c
21 be
Field;
func ndf c
1 -> Element of
suppf a
1 means :
Def11:
:: REALSET2:def 11
ex b
1 being
DnT of a
2,
suppf a
1ex b
2 being
Element of
(suppf a1) \ {a2} st a
1 = field (suppf a1),
(odf a1),b
1,a
2,b
2;
existence
ex b1 being Element of suppf c21ex b2 being DnT of b1, suppf c21ex b3 being Element of (suppf c21) \ {b1} st c21 = field (suppf c21),(odf c21),b2,b1,b3
by Def10;
uniqueness
for b1, b2 being Element of suppf c21 holds
( ex b3 being DnT of b1, suppf c21ex b4 being Element of (suppf c21) \ {b1} st c21 = field (suppf c21),(odf c21),b3,b1,b4 & ex b3 being DnT of b2, suppf c21ex b4 being Element of (suppf c21) \ {b2} st c21 = field (suppf c21),(odf c21),b3,b2,b4 implies b1 = b2 )
end;
:: deftheorem Def11 defines ndf REALSET2:def 11 :
definition
let c
21 be
Field;
func omf c
1 -> DnT of
ndf a
1,
suppf a
1 means :
Def12:
:: REALSET2:def 12
ex b
1 being
Element of
(suppf a1) \ {(ndf a1)} st a
1 = field (suppf a1),
(odf a1),a
2,
(ndf a1),b
1;
existence
ex b1 being DnT of ndf c21, suppf c21ex b2 being Element of (suppf c21) \ {(ndf c21)} st c21 = field (suppf c21),(odf c21),b1,(ndf c21),b2
by Def11;
uniqueness
for b1, b2 being DnT of ndf c21, suppf c21 holds
( ex b3 being Element of (suppf c21) \ {(ndf c21)} st c21 = field (suppf c21),(odf c21),b1,(ndf c21),b3 & ex b3 being Element of (suppf c21) \ {(ndf c21)} st c21 = field (suppf c21),(odf c21),b2,(ndf c21),b3 implies b1 = b2 )
end;
:: deftheorem Def12 defines omf REALSET2:def 12 :
definition
let c
21 be
Field;
func nmf c
1 -> Element of
(suppf a1) \ {(ndf a1)} means :
Def13:
:: REALSET2:def 13
a
1 = field (suppf a1),
(odf a1),
(omf a1),
(ndf a1),a
2;
existence
ex b1 being Element of (suppf c21) \ {(ndf c21)} st c21 = field (suppf c21),(odf c21),(omf c21),(ndf c21),b1
by Def12;
uniqueness
for b1, b2 being Element of (suppf c21) \ {(ndf c21)} holds
( c21 = field (suppf c21),(odf c21),(omf c21),(ndf c21),b1 & c21 = field (suppf c21),(odf c21),(omf c21),(ndf c21),b2 implies b1 = b2 )
end;
:: deftheorem Def13 defines nmf REALSET2:def 13 :
theorem Th1: :: REALSET2:1
theorem Th2: :: REALSET2:2
theorem Th3: :: REALSET2:3
for b
1 being
Fieldfor b
2, b
3, b
4 being
Element of
suppf b
1 holds
(
(omf b1) . [b2,((odf b1) . [b3,b4])] = (odf b1) . [((omf b1) . [b2,b3]),((omf b1) . [b2,b4])] &
(omf b1) . [((odf b1) . [b2,b3]),b4] = (odf b1) . [((omf b1) . [b2,b4]),((omf b1) . [b3,b4])] )
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