:: ANPROJ_1 semantic presentation
:: deftheorem Def1 ANPROJ_1:def 1 :
canceled;
:: deftheorem Def2 defines are_Prop ANPROJ_1:def 2 :
theorem Th1: :: ANPROJ_1:1
canceled;
theorem Th2: :: ANPROJ_1:2
canceled;
theorem Th3: :: ANPROJ_1:3
canceled;
theorem Th4: :: ANPROJ_1:4
canceled;
theorem Th5: :: ANPROJ_1:5
theorem Th6: :: ANPROJ_1:6
theorem Th7: :: ANPROJ_1:7
:: deftheorem Def3 defines are_LinDep ANPROJ_1:def 3 :
theorem Th8: :: ANPROJ_1:8
canceled;
theorem Th9: :: ANPROJ_1:9
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
are_Prop b
2,b
3 &
are_Prop b
4,b
5 &
are_Prop b
6,b
7 & b
2,b
4,b
6 are_LinDep implies b
3,b
5,b
7 are_LinDep )
theorem Th10: :: ANPROJ_1:10
for b
1 being
RealLinearSpacefor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3,b
4 are_LinDep implies ( b
2,b
4,b
3 are_LinDep & b
3,b
2,b
4 are_LinDep & b
4,b
3,b
2 are_LinDep & b
4,b
2,b
3 are_LinDep & b
3,b
4,b
2 are_LinDep ) )
Lemma8:
for b1 being RealLinearSpace
for b2, b3 being Element of b1
for b4, b5 being Real holds
( (b4 * b2) + (b5 * b3) = 0. b1 implies b4 * b2 = (- b5) * b3 )
Lemma9:
for b1 being RealLinearSpace
for b2, b3, b4 being Element of b1
for b5, b6, b7 being Real holds
( ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1 & b5 <> 0 implies b2 = ((- ((b5 " ) * b6)) * b3) + ((- ((b5 " ) * b7)) * b4) )
theorem Th11: :: ANPROJ_1:11
Lemma11:
for b1 being RealLinearSpace
for b2 being Element of b1
for b3, b4, b5 being Real holds ((b3 + b4) + b5) * b2 = ((b3 * b2) + (b4 * b2)) + (b5 * b2)
Lemma12:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds ((b2 + b3) + b4) + ((b5 + b6) + b7) = ((b2 + b5) + (b3 + b6)) + (b4 + b7)
Lemma13:
for b1 being RealLinearSpace
for b2, b3 being Element of b1
for b4, b5, b6 being Real holds ((b4 * b5) * b2) + ((b4 * b6) * b3) = b4 * ((b5 * b2) + (b6 * b3))
theorem Th12: :: ANPROJ_1:12
Lemma14:
for b1 being RealLinearSpace
for b2, b3, b4 being Element of b1
for b5, b6 being Real holds
( b2 + (b5 * b3) = b4 + (b6 * b3) implies ((b5 - b6) * b3) + b2 = b4 )
theorem Th13: :: ANPROJ_1:13
for b
1 being
RealLinearSpacefor b
2, b
3, b
4 being
Element of b
1for b
5, b
6, b
7, b
8, b
9, b
10 being
Real holds
( not b
2,b
3,b
4 are_LinDep &
((b5 * b2) + (b6 * b3)) + (b7 * b4) = ((b8 * b2) + (b9 * b3)) + (b10 * b4) implies ( b
5 = b
8 & b
6 = b
9 & b
7 = b
10 ) )
theorem Th14: :: ANPROJ_1:14
theorem Th15: :: ANPROJ_1:15
theorem Th16: :: ANPROJ_1:16
theorem Th17: :: ANPROJ_1:17
theorem Th18: :: ANPROJ_1:18
theorem Th19: :: ANPROJ_1:19
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
are_Prop b
2,b
3 & b
2,b
3,b
4 are_LinDep & b
2,b
3,b
5 are_LinDep & b
2,b
3,b
6 are_LinDep & b
2 is_Prop_Vect & b
3 is_Prop_Vect implies b
4,b
5,b
6 are_LinDep )
Lemma21:
for b1 being RealLinearSpace
for b2, b3 being Element of b1
for b4, b5, b6 being Real holds b4 * ((b5 * b2) + (b6 * b3)) = ((b4 * b5) * b2) + ((b4 * b6) * b3)
theorem Th20: :: ANPROJ_1:20
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( not b
2,b
3,b
4 are_LinDep & b
2,b
3,b
5 are_LinDep & b
3,b
4,b
6 are_LinDep & ( for b
7 being
Element of b
1 holds
not ( b
2,b
4,b
7 are_LinDep & b
5,b
6,b
7 are_LinDep & b
7 is_Prop_Vect ) ) )
theorem Th21: :: ANPROJ_1:21
Lemma22:
for b1 being RealLinearSpace
for b2, b3, b4 being Element of b1 holds
( not b2,b3,b4 are_LinDep implies for b5, b6 being Element of b1 holds
not ( b5 is_Prop_Vect & b6 is_Prop_Vect & not are_Prop b5,b6 & ( for b7 being Element of b1 holds b5,b6,b7 are_LinDep ) ) )
theorem Th22: :: ANPROJ_1:22
Lemma23:
for b1 being RealLinearSpace
for b2, b3, b4 being Element of b1
for b5, b6, b7, b8, b9, b10, b11, b12, b13 being Real holds ((b11 * ((b5 * b2) + (b6 * b3))) + (b12 * ((b7 * b3) + (b8 * b4)))) + (b13 * ((b9 * b2) + (b10 * b4))) = ((((b11 * b5) + (b13 * b9)) * b2) + (((b11 * b6) + (b12 * b7)) * b3)) + (((b12 * b8) + (b13 * b10)) * b4)
theorem Th23: :: ANPROJ_1:23
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
not ( b
2,b
3,b
4 are_LinDep & b
5,b
6,b
4 are_LinDep & b
2,b
5,b
7 are_LinDep & b
3,b
6,b
7 are_LinDep & b
2,b
6,b
8 are_LinDep & b
3,b
5,b
8 are_LinDep & b
7,b
4,b
8 are_LinDep & b
7 is_Prop_Vect & b
4 is_Prop_Vect & b
8 is_Prop_Vect & not b
2,b
3,b
6 are_LinDep & not b
2,b
3,b
5 are_LinDep & not b
2,b
5,b
6 are_LinDep & not b
3,b
5,b
6 are_LinDep )
:: deftheorem Def4 defines Proper_Vectors_of ANPROJ_1:def 4 :
theorem Th24: :: ANPROJ_1:24
canceled;
theorem Th25: :: ANPROJ_1:25
canceled;
theorem Th26: :: ANPROJ_1:26
definition
let c
1 be
RealLinearSpace;
func Proportionality_as_EqRel_of c
1 -> Equivalence_Relation of
Proper_Vectors_of a
1 means :
Def5:
:: ANPROJ_1:def 5
for b
1, b
2 being
set holds
(
[b1,b2] in a
2 iff ( b
1 in Proper_Vectors_of a
1 & b
2 in Proper_Vectors_of a
1 & ex b
3, b
4 being
Element of a
1 st
( b
1 = b
3 & b
2 = b
4 &
are_Prop b
3,b
4 ) ) );
existence
ex b1 being Equivalence_Relation of Proper_Vectors_of c1 st
for b2, b3 being set holds
( [b2,b3] in b1 iff ( b2 in Proper_Vectors_of c1 & b3 in Proper_Vectors_of c1 & ex b4, b5 being Element of c1 st
( b2 = b4 & b3 = b5 & are_Prop b4,b5 ) ) )
uniqueness
for b1, b2 being Equivalence_Relation of Proper_Vectors_of c1 holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ( b3 in Proper_Vectors_of c1 & b4 in Proper_Vectors_of c1 & ex b5, b6 being Element of c1 st
( b3 = b5 & b4 = b6 & are_Prop b5,b6 ) ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ( b3 in Proper_Vectors_of c1 & b4 in Proper_Vectors_of c1 & ex b5, b6 being Element of c1 st
( b3 = b5 & b4 = b6 & are_Prop b5,b6 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines Proportionality_as_EqRel_of ANPROJ_1:def 5 :
theorem Th27: :: ANPROJ_1:27
canceled;
theorem Th28: :: ANPROJ_1:28
theorem Th29: :: ANPROJ_1:29
:: deftheorem Def6 defines Dir ANPROJ_1:def 6 :
:: deftheorem Def7 defines ProjectivePoints ANPROJ_1:def 7 :
:: deftheorem Def8 defines trivial ANPROJ_1:def 8 :
theorem Th30: :: ANPROJ_1:30
canceled;
theorem Th31: :: ANPROJ_1:31
canceled;
theorem Th32: :: ANPROJ_1:32
canceled;
theorem Th33: :: ANPROJ_1:33
theorem Th34: :: ANPROJ_1:34
theorem Th35: :: ANPROJ_1:35
definition
let c
1 be non
trivial RealLinearSpace;
func ProjectiveCollinearity c
1 -> Relation3 of
ProjectivePoints a
1 means :
Def9:
:: ANPROJ_1:def 9
for b
1, b
2, b
3 being
set holds
(
[b1,b2,b3] in a
2 iff ex b
4, b
5, b
6 being
Element of a
1 st
( b
1 = Dir b
4 & b
2 = Dir b
5 & b
3 = Dir b
6 & b
4 is_Prop_Vect & b
5 is_Prop_Vect & b
6 is_Prop_Vect & b
4,b
5,b
6 are_LinDep ) );
existence
ex b1 being Relation3 of ProjectivePoints c1 st
for b2, b3, b4 being set holds
( [b2,b3,b4] in b1 iff ex b5, b6, b7 being Element of c1 st
( b2 = Dir b5 & b3 = Dir b6 & b4 = Dir b7 & b5 is_Prop_Vect & b6 is_Prop_Vect & b7 is_Prop_Vect & b5,b6,b7 are_LinDep ) )
uniqueness
for b1, b2 being Relation3 of ProjectivePoints c1 holds
( ( for b3, b4, b5 being set holds
( [b3,b4,b5] in b1 iff ex b6, b7, b8 being Element of c1 st
( b3 = Dir b6 & b4 = Dir b7 & b5 = Dir b8 & b6 is_Prop_Vect & b7 is_Prop_Vect & b8 is_Prop_Vect & b6,b7,b8 are_LinDep ) ) ) & ( for b3, b4, b5 being set holds
( [b3,b4,b5] in b2 iff ex b6, b7, b8 being Element of c1 st
( b3 = Dir b6 & b4 = Dir b7 & b5 = Dir b8 & b6 is_Prop_Vect & b7 is_Prop_Vect & b8 is_Prop_Vect & b6,b7,b8 are_LinDep ) ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines ProjectiveCollinearity ANPROJ_1:def 9 :
:: deftheorem Def10 defines ProjectiveSpace ANPROJ_1:def 10 :
theorem Th36: :: ANPROJ_1:36
canceled;
theorem Th37: :: ANPROJ_1:37
canceled;
theorem Th38: :: ANPROJ_1:38
canceled;
theorem Th39: :: ANPROJ_1:39
theorem Th40: :: ANPROJ_1:40
theorem Th41: :: ANPROJ_1:41
theorem Th42: :: ANPROJ_1:42