:: WEDDWITT semantic presentation
theorem Th1: :: WEDDWITT:1
for b
1 being
Natfor b
2 being
Real holds
( 1
< b
2 & b
2 |^ b
1 = 1 implies b
1 = 0 )
theorem Th2: :: WEDDWITT:2
for b
1, b
2, b
3 being
Natfor b
4 being
Real holds
( 1
< b
4 & 0
< b
2 implies b
4 |^ ((b1 * b2) + b3) = (b4 |^ b1) * (b4 |^ ((b1 * (b2 -' 1)) + b3)) )
theorem Th3: :: WEDDWITT:3
theorem Th4: :: WEDDWITT:4
theorem Th5: :: WEDDWITT:5
theorem Th6: :: WEDDWITT:6
:: deftheorem Def1 defines Centralizer WEDDWITT:def 1 :
theorem Th7: :: WEDDWITT:7
theorem Th8: :: WEDDWITT:8
:: deftheorem Def2 defines -con_map WEDDWITT:def 2 :
theorem Th9: :: WEDDWITT:9
theorem Th10: :: WEDDWITT:10
theorem Th11: :: WEDDWITT:11
theorem Th12: :: WEDDWITT:12
theorem Th13: :: WEDDWITT:13
:: deftheorem Def3 defines conjugate_Classes WEDDWITT:def 3 :
theorem Th14: :: WEDDWITT:14
theorem Th15: :: WEDDWITT:15
theorem Th16: :: WEDDWITT:16
:: deftheorem Def4 defines center WEDDWITT:def 4 :
theorem Th17: :: WEDDWITT:17
Lemma22:
for b1 being Skew-Field holds 1. (center b1) = 1. b1
theorem Th18: :: WEDDWITT:18
theorem Th19: :: WEDDWITT:19
theorem Th20: :: WEDDWITT:20
theorem Th21: :: WEDDWITT:21
theorem Th22: :: WEDDWITT:22
theorem Th23: :: WEDDWITT:23
:: deftheorem Def5 defines centralizer WEDDWITT:def 5 :
theorem Th24: :: WEDDWITT:24
theorem Th25: :: WEDDWITT:25
theorem Th26: :: WEDDWITT:26
theorem Th27: :: WEDDWITT:27
theorem Th28: :: WEDDWITT:28
theorem Th29: :: WEDDWITT:29
theorem Th30: :: WEDDWITT:30
theorem Th31: :: WEDDWITT:31
definition
let c
1 be
Skew-Field;
func VectSp_over_center c
1 -> strict VectSp of
center a
1 means :
Def6:
:: WEDDWITT:def 6
(
LoopStr(# the
carrier of a
2,the
add of a
2,the
Zero of a
2 #)
= LoopStr(# the
carrier of a
1,the
add of a
1,the
Zero of a
1 #) & the
lmult of a
2 = the
mult of a
1 | [:the carrier of (center a1),the carrier of a1:] );
existence
ex b1 being strict VectSp of center c1 st
( LoopStr(# the carrier of b1,the add of b1,the Zero of b1 #) = LoopStr(# the carrier of c1,the add of c1,the Zero of c1 #) & the lmult of b1 = the mult of c1 | [:the carrier of (center c1),the carrier of c1:] )
uniqueness
for b1, b2 being strict VectSp of center c1 holds
( LoopStr(# the carrier of b1,the add of b1,the Zero of b1 #) = LoopStr(# the carrier of c1,the add of c1,the Zero of c1 #) & the lmult of b1 = the mult of c1 | [:the carrier of (center c1),the carrier of c1:] & LoopStr(# the carrier of b2,the add of b2,the Zero of b2 #) = LoopStr(# the carrier of c1,the add of c1,the Zero of c1 #) & the lmult of b2 = the mult of c1 | [:the carrier of (center c1),the carrier of c1:] implies b1 = b2 )
;
end;
:: deftheorem Def6 defines VectSp_over_center WEDDWITT:def 6 :
theorem Th32: :: WEDDWITT:32
theorem Th33: :: WEDDWITT:33
definition
let c
1 be
Skew-Field;
let c
2 be
Element of c
1;
func VectSp_over_center c
2 -> strict VectSp of
center a
1 means :
Def7:
:: WEDDWITT:def 7
(
LoopStr(# the
carrier of a
3,the
add of a
3,the
Zero of a
3 #)
= LoopStr(# the
carrier of
(centralizer a2),the
add of
(centralizer a2),the
Zero of
(centralizer a2) #) & the
lmult of a
3 = the
mult of a
1 | [:the carrier of (center a1),the carrier of (centralizer a2):] );
existence
ex b1 being strict VectSp of center c1 st
( LoopStr(# the carrier of b1,the add of b1,the Zero of b1 #) = LoopStr(# the carrier of (centralizer c2),the add of (centralizer c2),the Zero of (centralizer c2) #) & the lmult of b1 = the mult of c1 | [:the carrier of (center c1),the carrier of (centralizer c2):] )
uniqueness
for b1, b2 being strict VectSp of center c1 holds
( LoopStr(# the carrier of b1,the add of b1,the Zero of b1 #) = LoopStr(# the carrier of (centralizer c2),the add of (centralizer c2),the Zero of (centralizer c2) #) & the lmult of b1 = the mult of c1 | [:the carrier of (center c1),the carrier of (centralizer c2):] & LoopStr(# the carrier of b2,the add of b2,the Zero of b2 #) = LoopStr(# the carrier of (centralizer c2),the add of (centralizer c2),the Zero of (centralizer c2) #) & the lmult of b2 = the mult of c1 | [:the carrier of (center c1),the carrier of (centralizer c2):] implies b1 = b2 )
;
end;
:: deftheorem Def7 defines VectSp_over_center WEDDWITT:def 7 :
theorem Th34: :: WEDDWITT:34
theorem Th35: :: WEDDWITT:35
theorem Th36: :: WEDDWITT:36
theorem Th37: :: WEDDWITT:37
theorem Th38: :: WEDDWITT:38
theorem Th39: :: WEDDWITT:39
theorem Th40: :: WEDDWITT:40
E46:
now
let c
1 be
Skew-Field;
let c
2 be
Element of c
1;
set c
3 =
centralizer c
2;
let c
4, c
5 be
Element of
(centralizer c2);
assume E47:
c
5 = 1. c
1
;
E48:
[c4,c5] in [:the carrier of (centralizer c2),the carrier of (centralizer c2):]
by ZFMISC_1:106;
E49:
[c5,c4] in [:the carrier of (centralizer c2),the carrier of (centralizer c2):]
by ZFMISC_1:106;
E50:
the
carrier of
(centralizer c2) c= the
carrier of c
1
by Th24;
c
4 in the
carrier of
(centralizer c2)
;
then reconsider c
6 = c
4 as
Element of c
1 by E50;
E51:
the
mult of
(centralizer c2) = the
mult of c
1 || the
carrier of
(centralizer c2)
by Def5;
thus c
4 * c
5 =
the
mult of
(centralizer c2) . c
4,c
5
.=
the
mult of
(centralizer c2) . [c4,c5]
.=
the
mult of c
1 . [c4,c5]
by E51, E48, FUNCT_1:72
.=
the
mult of c
1 . c
4,
(1. c1)
by E47
.=
c
6 * (1. c1)
.=
c
4
by GROUP_1:def 5
;
thus c
5 * c
4 =
the
mult of
(centralizer c2) . c
5,c
4
.=
the
mult of
(centralizer c2) . [c5,c4]
.=
the
mult of c
1 . [c5,c4]
by E51, E49, FUNCT_1:72
.=
the
mult of c
1 . (1. c1),c
4
by E47
.=
(1. c1) * c
6
.=
c
4
by GROUP_1:def 5
;
end;
theorem Th41: :: WEDDWITT:41