:: PENCIL_4 semantic presentation
theorem Th1: :: PENCIL_4:1
for b
1, b
2 being
Nat holds
not ( 1
<= b
1 & b
1 < b
2 & 3
<= b
2 & not b
1 + 1
< b
2 & not 2
<= b
1 )
theorem Th2: :: PENCIL_4:2
theorem Th3: :: PENCIL_4:3
theorem Th4: :: PENCIL_4:4
theorem Th5: :: PENCIL_4:5
theorem Th6: :: PENCIL_4:6
:: deftheorem Def1 defines segment PENCIL_4:def 1 :
theorem Th7: :: PENCIL_4:7
:: deftheorem Def2 defines pencil PENCIL_4:def 2 :
theorem Th8: :: PENCIL_4:8
:: deftheorem Def3 defines pencil PENCIL_4:def 3 :
theorem Th9: :: PENCIL_4:9
theorem Th10: :: PENCIL_4:10
definition
let c
1 be
Field;
let c
2 be
finite-dimensional VectSp of c
1;
let c
3 be
Nat;
func c
3 Pencils_of c
2 -> Subset-Family of
(a3 Subspaces_of a2) means :
Def4:
:: PENCIL_4:def 4
for b
1 being
set holds
( b
1 in a
4 iff ex b
2, b
3 being
Subspace of a
2 st
( b
2 is
Subspace of b
3 &
(dim b2) + 1
= a
3 &
dim b
3 = a
3 + 1 & b
1 = pencil b
2,b
3,a
3 ) );
existence
ex b1 being Subset-Family of (c3 Subspaces_of c2) st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being Subspace of c2 st
( b3 is Subspace of b4 & (dim b3) + 1 = c3 & dim b4 = c3 + 1 & b2 = pencil b3,b4,c3 ) )
uniqueness
for b1, b2 being Subset-Family of (c3 Subspaces_of c2) holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being Subspace of c2 st
( b4 is Subspace of b5 & (dim b4) + 1 = c3 & dim b5 = c3 + 1 & b3 = pencil b4,b5,c3 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being Subspace of c2 st
( b4 is Subspace of b5 & (dim b4) + 1 = c3 & dim b5 = c3 + 1 & b3 = pencil b4,b5,c3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines Pencils_of PENCIL_4:def 4 :
theorem Th11: :: PENCIL_4:11
theorem Th12: :: PENCIL_4:12
theorem Th13: :: PENCIL_4:13
theorem Th14: :: PENCIL_4:14
theorem Th15: :: PENCIL_4:15
theorem Th16: :: PENCIL_4:16
theorem Th17: :: PENCIL_4:17
:: deftheorem Def5 defines PencilSpace PENCIL_4:def 5 :
theorem Th18: :: PENCIL_4:18
theorem Th19: :: PENCIL_4:19
theorem Th20: :: PENCIL_4:20
theorem Th21: :: PENCIL_4:21
theorem Th22: :: PENCIL_4:22
theorem Th23: :: PENCIL_4:23
:: deftheorem Def6 defines SubspaceSet PENCIL_4:def 6 :
theorem Th24: :: PENCIL_4:24
theorem Th25: :: PENCIL_4:25
theorem Th26: :: PENCIL_4:26
:: deftheorem Def7 defines GrassmannSpace PENCIL_4:def 7 :
theorem Th27: :: PENCIL_4:27
theorem Th28: :: PENCIL_4:28
theorem Th29: :: PENCIL_4:29
theorem Th30: :: PENCIL_4:30
theorem Th31: :: PENCIL_4:31
theorem Th32: :: PENCIL_4:32
:: deftheorem Def8 defines PairSet PENCIL_4:def 8 :
for b
1 being
set for b
2 being
set holds
( b
2 = PairSet b
1 iff for b
3 being
set holds
( b
3 in b
2 iff ex b
4, b
5 being
set st
( b
4 in b
1 & b
5 in b
1 & b
3 = {b4,b5} ) ) );
:: deftheorem Def9 defines PairSet PENCIL_4:def 9 :
for b
1, b
2 being
set for b
3 being
set holds
( b
3 = PairSet b
1,b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5 being
set st
( b
5 in b
2 & b
4 = {b1,b5} ) ) );
:: deftheorem Def10 defines PairSetFamily PENCIL_4:def 10 :
:: deftheorem Def11 defines VeroneseSpace PENCIL_4:def 11 :