:: Axioms of Incidence
:: by Wojciech A. Trybulec
::
:: Received April 14, 1989
:: Copyright (c) 1990 Association of Mizar Users
:: deftheorem Def1 defines on INCSP_1:def 1 :
:: deftheorem Def2 defines on INCSP_1:def 2 :
:: deftheorem Def3 defines on INCSP_1:def 3 :
:: deftheorem Def4 defines on INCSP_1:def 4 :
:: deftheorem Def5 defines on INCSP_1:def 5 :
:: deftheorem Def6 defines linear INCSP_1:def 6 :
:: deftheorem Def7 defines planar INCSP_1:def 7 :
theorem :: INCSP_1:1
canceled;
theorem :: INCSP_1:2
canceled;
theorem :: INCSP_1:3
canceled;
theorem :: INCSP_1:4
canceled;
theorem :: INCSP_1:5
canceled;
theorem :: INCSP_1:6
canceled;
theorem :: INCSP_1:7
canceled;
theorem :: INCSP_1:8
canceled;
theorem :: INCSP_1:9
canceled;
theorem :: INCSP_1:10
canceled;
theorem Th11: :: INCSP_1:11
theorem Th12: :: INCSP_1:12
theorem Th13: :: INCSP_1:13
theorem Th14: :: INCSP_1:14
theorem Th15: :: INCSP_1:15
theorem Th16: :: INCSP_1:16
theorem Th17: :: INCSP_1:17
theorem Th18: :: INCSP_1:18
theorem Th19: :: INCSP_1:19
theorem Th20: :: INCSP_1:20
theorem Th21: :: INCSP_1:21
theorem :: INCSP_1:22
theorem :: INCSP_1:23
:: deftheorem Def8 defines with_non-trivial_lines INCSP_1:def 8 :
:: deftheorem Def9 defines linear INCSP_1:def 9 :
:: deftheorem Def10 defines up-2-rank INCSP_1:def 10 :
definition
let S be
IncStruct ;
attr S is
with_non-empty_planes means :
Def11:
:: INCSP_1:def 11
for
P being
PLANE of
S ex
A being
POINT of
S st
A on P;
attr S is
planar means :
Def12:
:: INCSP_1:def 12
for
A,
B,
C being
POINT of
S ex
P being
PLANE of
S st
{A,B,C} on P;
attr S is
with_<=1_plane_per_3_pts means :
Def13:
:: INCSP_1:def 13
for
A,
B,
C being
POINT of
S for
P,
Q being
PLANE of
S st not
{A,B,C} is
linear &
{A,B,C} on P &
{A,B,C} on Q holds
P = Q;
attr S is
with_lines_inside_planes means :
Def14:
:: INCSP_1:def 14
for
L being
LINE of
S for
P being
PLANE of
S st ex
A,
B being
POINT of
S st
(
A <> B &
{A,B} on L &
{A,B} on P ) holds
L on P;
attr S is
with_planes_intersecting_in_2_pts means :
Def15:
:: INCSP_1:def 15
for
A being
POINT of
S for
P,
Q being
PLANE of
S st
A on P &
A on Q holds
ex
B being
POINT of
S st
(
A <> B &
B on P &
B on Q );
attr S is
up-3-dimensional means :
Def16:
:: INCSP_1:def 16
not for
A,
B,
C,
D being
POINT of
S holds
{A,B,C,D} is
planar;
attr S is
inc-compatible means :
Def17:
:: INCSP_1:def 17
for
A being
POINT of
S for
L being
LINE of
S for
P being
PLANE of
S st
A on L &
L on P holds
A on P;
end;
:: deftheorem Def11 defines with_non-empty_planes INCSP_1:def 11 :
:: deftheorem Def12 defines planar INCSP_1:def 12 :
:: deftheorem Def13 defines with_<=1_plane_per_3_pts INCSP_1:def 13 :
:: deftheorem Def14 defines with_lines_inside_planes INCSP_1:def 14 :
:: deftheorem Def15 defines with_planes_intersecting_in_2_pts INCSP_1:def 15 :
:: deftheorem Def16 defines up-3-dimensional INCSP_1:def 16 :
:: deftheorem Def17 defines inc-compatible INCSP_1:def 17 :
:: deftheorem Def18 defines IncSpace-like INCSP_1:def 18 :
theorem :: INCSP_1:24
canceled;
theorem :: INCSP_1:25
canceled;
theorem :: INCSP_1:26
canceled;
theorem :: INCSP_1:27
canceled;
theorem :: INCSP_1:28
canceled;
theorem :: INCSP_1:29
canceled;
theorem :: INCSP_1:30
canceled;
theorem :: INCSP_1:31
canceled;
theorem :: INCSP_1:32
canceled;
theorem :: INCSP_1:33
canceled;
theorem :: INCSP_1:34
canceled;
theorem Th35: :: INCSP_1:35
theorem Th36: :: INCSP_1:36
theorem Th37: :: INCSP_1:37
theorem Th38: :: INCSP_1:38
theorem Th39: :: INCSP_1:39
theorem Th40: :: INCSP_1:40
for
S being
IncSpace for
A,
B,
C,
D being
POINT of
S for
P being
PLANE of
S st not
{A,B,C} is
linear &
{A,B,C} on P & not
D on P holds
not
{A,B,C,D} is
planar
theorem :: INCSP_1:41
for
S being
IncSpace for
K,
L being
LINE of
S st ( for
P being
PLANE of
S holds
( not
K on P or not
L on P ) ) holds
K <> L
Lm1:
for S being IncSpace
for A being POINT of S
for L being LINE of S ex B being POINT of S st
( A <> B & B on L )
theorem :: INCSP_1:42
for
S being
IncSpace for
L,
L1,
L2 being
LINE of
S st ( for
P being
PLANE of
S holds
( not
L on P or not
L1 on P or not
L2 on P ) ) & ex
A being
POINT of
S st
(
A on L &
A on L1 &
A on L2 ) holds
L <> L1
theorem :: INCSP_1:43
for
S being
IncSpace for
L1,
L2,
L being
LINE of
S for
P being
PLANE of
S st
L1 on P &
L2 on P & not
L on P &
L1 <> L2 holds
for
Q being
PLANE of
S holds
( not
L on Q or not
L1 on Q or not
L2 on Q )
theorem Th44: :: INCSP_1:44
theorem Th45: :: INCSP_1:45
theorem :: INCSP_1:46
theorem :: INCSP_1:47
theorem Th48: :: INCSP_1:48
for
S being
IncSpace for
A being
POINT of
S for
L being
LINE of
S st not
A on L holds
ex
P being
PLANE of
S st
for
Q being
PLANE of
S holds
( (
A on Q &
L on Q ) iff
P = Q )
theorem Th49: :: INCSP_1:49
for
S being
IncSpace for
K,
L being
LINE of
S st
K <> L & ex
A being
POINT of
S st
(
A on K &
A on L ) holds
ex
P being
PLANE of
S st
for
Q being
PLANE of
S holds
( (
K on Q &
L on Q ) iff
P = Q )
:: deftheorem Def19 defines Line INCSP_1:def 19 :
definition
let S be
IncSpace;
let A,
B,
C be
POINT of
S;
assume A1:
not
{A,B,C} is
linear
;
func Plane A,
B,
C -> PLANE of
S means :
Def20:
:: INCSP_1:def 20
{A,B,C} on it;
correctness
existence
ex b1 being PLANE of S st {A,B,C} on b1;
uniqueness
for b1, b2 being PLANE of S st {A,B,C} on b1 & {A,B,C} on b2 holds
b1 = b2;
by A1, Def12, Def13;
end;
:: deftheorem Def20 defines Plane INCSP_1:def 20 :
:: deftheorem Def21 defines Plane INCSP_1:def 21 :
:: deftheorem Def22 defines Plane INCSP_1:def 22 :
theorem :: INCSP_1:50
canceled;
theorem :: INCSP_1:51
canceled;
theorem :: INCSP_1:52
canceled;
theorem :: INCSP_1:53
canceled;
theorem :: INCSP_1:54
canceled;
theorem :: INCSP_1:55
canceled;
theorem :: INCSP_1:56
canceled;
theorem :: INCSP_1:57
theorem Th58: :: INCSP_1:58
theorem Th59: :: INCSP_1:59
theorem :: INCSP_1:60
theorem Th61: :: INCSP_1:61
theorem :: INCSP_1:62
theorem :: INCSP_1:63
canceled;
theorem :: INCSP_1:64
theorem Th65: :: INCSP_1:65
theorem :: INCSP_1:66
theorem :: INCSP_1:67
theorem :: INCSP_1:68
for
S being
IncSpace for
A,
B,
C,
D being
POINT of
S st not
{A,B,C} is
linear &
D on Plane A,
B,
C holds
{A,B,C,D} is
planar
theorem :: INCSP_1:69
theorem :: INCSP_1:70
Lm2:
for S being IncSpace
for P being PLANE of S ex A, B, C, D being POINT of S st
( A on P & not {A,B,C,D} is planar )
theorem Th71: :: INCSP_1:71
theorem :: INCSP_1:72
theorem :: INCSP_1:73
theorem Th74: :: INCSP_1:74
theorem Th75: :: INCSP_1:75
theorem Th76: :: INCSP_1:76
theorem Th77: :: INCSP_1:77
theorem :: INCSP_1:78
theorem :: INCSP_1:79
theorem Th80: :: INCSP_1:80
theorem :: INCSP_1:81
for
S being
IncSpace for
A being
POINT of
S ex
L,
L1,
L2 being
LINE of
S st
(
A on L &
A on L1 &
A on L2 & ( for
P being
PLANE of
S holds
( not
L on P or not
L1 on P or not
L2 on P ) ) )
theorem :: INCSP_1:82
theorem :: INCSP_1:83
theorem :: INCSP_1:84
theorem :: INCSP_1:85
theorem :: INCSP_1:86
canceled;
theorem :: INCSP_1:87
theorem :: INCSP_1:88
for
S being
IncSpace for
P,
Q being
PLANE of
S holds
( not
P <> Q or for
A being
POINT of
S holds
( not
A on P or not
A on Q ) or ex
L being
LINE of
S st
for
B being
POINT of
S holds
( (
B on P &
B on Q ) iff
B on L ) )