:: Desargues Theorem In Projective 3-Space
:: by Eugeniusz Kusak
::
:: Received August 13, 1990
:: Copyright (c) 1990 Association of Mizar Users

theorem Th1: :: PROJDES1:1

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c being Element of FCPS st a,b,c is_collinear holds
( b,c,a is_collinear & c,a,b is_collinear & b,a,c is_collinear & a,c,b is_collinear & c,b,a is_collinear )

proof end;

theorem :: PROJDES1:2

canceled;

theorem :: PROJDES1:3

canceled;

theorem :: PROJDES1:4

canceled;

theorem :: PROJDES1:5

for FCPS being up-3-dimensional CollProjectiveSpace
for p being Element of FCPS holds
not for q, r being Element of FCPS holds p,q,r is_collinear

proof end;

theorem :: PROJDES1:6

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, b' being Element of FCPS st not a,b,c is_collinear & a,b,b' is_collinear & a <> b' holds
not a,b',c is_collinear

proof end;

theorem :: PROJDES1:7

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, d being Element of FCPS st not a,b,c is_collinear & a,b,d is_collinear & a,c,d is_collinear holds
a = d

proof end;

theorem Th8: :: PROJDES1:8

for FCPS being up-3-dimensional CollProjectiveSpace
for o, a, d, d', s, a' being Element of FCPS st not o,a,d is_collinear & o,d,d' is_collinear & a,d,s is_collinear & d <> d' & a',d',s is_collinear & o,a,a' is_collinear & o <> a' holds
s <> d

proof end;

Lm1: for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, b', c' being Element of FCPS st not a,b,c is_collinear & a,b,b' is_collinear & a,c,c' is_collinear & a <> b' holds
b' <> c'

proof end;

definition

let FCPS be up-3-dimensional CollProjectiveSpace;
let a, b, c, d be Element of FCPS;
pred a,b,c,d are_coplanar means :Def1: :: PROJDES1:def 1
ex x being Element of FCPS st
( a,b,x is_collinear & c,d,x is_collinear );

end;

:: deftheorem Def1 defines are_coplanar PROJDES1:def 1 :

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, d being Element of FCPS holds
( a,b,c,d are_coplanar iff ex x being Element of FCPS st
( a,b,x is_collinear & c,d,x is_collinear ) );

theorem :: PROJDES1:9

canceled;

theorem Th10: :: PROJDES1:10

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, d being Element of FCPS st ( a,b,c is_collinear or b,c,d is_collinear or c,d,a is_collinear or d,a,b is_collinear ) holds
a,b,c,d are_coplanar

proof end;

Lm2: for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, d being Element of FCPS st a,b,c,d are_coplanar holds
b,a,c,d are_coplanar

proof end;

Lm3: for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, d being Element of FCPS st a,b,c,d are_coplanar holds
b,c,d,a are_coplanar

proof end;

theorem Th11: :: PROJDES1:11

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, d being Element of FCPS st a,b,c,d are_coplanar holds
( b,c,d,a are_coplanar & c,d,a,b are_coplanar & d,a,b,c are_coplanar & b,a,c,d are_coplanar & c,b,d,a are_coplanar & d,c,a,b are_coplanar & a,d,b,c are_coplanar & a,c,d,b are_coplanar & b,d,a,c are_coplanar & c,a,b,d are_coplanar & d,b,c,a are_coplanar & c,a,d,b are_coplanar & d,b,a,c are_coplanar & a,c,b,d are_coplanar & b,d,c,a are_coplanar & a,b,d,c are_coplanar & a,d,c,b are_coplanar & b,c,a,d are_coplanar & b,a,d,c are_coplanar & c,b,a,d are_coplanar & c,d,b,a are_coplanar & d,a,c,b are_coplanar & d,c,b,a are_coplanar )

proof end;

Lm4: for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, p, q being Element of FCPS st not a,b,c is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar holds
a,b,p,q are_coplanar

proof end;

theorem Th12: :: PROJDES1:12

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, p, q, r, s being Element of FCPS st not a,b,c is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar & a,b,c,r are_coplanar & a,b,c,s are_coplanar holds
p,q,r,s are_coplanar

proof end;

Lm5: for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, p, q, r being Element of FCPS st not a,b,c is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar & a,b,c,r are_coplanar holds
a,p,q,r are_coplanar

proof end;

theorem :: PROJDES1:13

for FCPS being up-3-dimensional CollProjectiveSpace
for p, q, r, a, b, c, s being Element of FCPS st not p,q,r is_collinear & a,b,c,p are_coplanar & a,b,c,r are_coplanar & a,b,c,q are_coplanar & p,q,r,s are_coplanar holds
a,b,c,s are_coplanar

proof end;

Lm6: for FCPS being up-3-dimensional CollProjectiveSpace
for p, q, r, a, b being Element of FCPS st p <> q & p,q,r is_collinear & a,b,p,q are_coplanar holds
a,b,p,r are_coplanar

proof end;

theorem Th14: :: PROJDES1:14

for FCPS being up-3-dimensional CollProjectiveSpace
for p, q, r, a, b, c being Element of FCPS st p <> q & p,q,r is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar holds
a,b,c,r are_coplanar

proof end;

theorem :: PROJDES1:15

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, p, q, r, s being Element of FCPS st not a,b,c is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar & a,b,c,r are_coplanar & a,b,c,s are_coplanar holds
ex x being Element of FCPS st
( p,q,x is_collinear & r,s,x is_collinear )

proof end;

theorem Th16: :: PROJDES1:16

for FCPS being up-3-dimensional CollProjectiveSpace holds
not for a, b, c, d being Element of FCPS holds a,b,c,d are_coplanar

proof end;

theorem Th17: :: PROJDES1:17

for FCPS being up-3-dimensional CollProjectiveSpace
for p, q, r being Element of FCPS st not p,q,r is_collinear holds
ex s being Element of FCPS st not p,q,r,s are_coplanar

proof end;

theorem Th18: :: PROJDES1:18

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, d being Element of FCPS st ( a = b or a = c or b = c or a = d or b = d or d = c ) holds
a,b,c,d are_coplanar

proof end;

theorem Th19: :: PROJDES1:19

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, o, a' being Element of FCPS st not a,b,c,o are_coplanar & o,a,a' is_collinear & a <> a' holds
not a,b,c,a' are_coplanar

proof end;

theorem Th20: :: PROJDES1:20

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, a', b', c', p, q, r being Element of FCPS st not a,b,c is_collinear & not a',b',c' is_collinear & a,b,c,p are_coplanar & a,b,c,q are_coplanar & a,b,c,r are_coplanar & a',b',c',p are_coplanar & a',b',c',q are_coplanar & a',b',c',r are_coplanar & not a,b,c,a' are_coplanar holds
p,q,r is_collinear

proof end;

theorem Th21: :: PROJDES1:21

for FCPS being up-3-dimensional CollProjectiveSpace
for a, a', o, b, c, b', c', p, q, r being Element of FCPS st a <> a' & o,a,a' is_collinear & not a,b,c,o are_coplanar & not a',b',c' is_collinear & a,b,p is_collinear & a',b',p is_collinear & b,c,q is_collinear & b',c',q is_collinear & a,c,r is_collinear & a',c',r is_collinear holds
p,q,r is_collinear

proof end;

theorem Th22: :: PROJDES1:22

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, d, o being Element of FCPS st not a,b,c,d are_coplanar & a,b,c,o are_coplanar & not a,b,o is_collinear holds
not a,b,d,o are_coplanar

proof end;

theorem Th23: :: PROJDES1:23

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, o, a', b', c' being Element of FCPS st not a,b,c,o are_coplanar & o,a,a' is_collinear & o,b,b' is_collinear & o,c,c' is_collinear & o <> a' & o <> b' & o <> c' holds
( not a',b',c' is_collinear & not a',b',c',o are_coplanar )

proof end;

theorem Th24: :: PROJDES1:24

for FCPS being up-3-dimensional CollProjectiveSpace
for a, b, c, o, d, d', a', b', c', s, t, u being Element of FCPS st a,b,c,o are_coplanar & not a,b,c,d are_coplanar & not a,b,d,o are_coplanar & not b,c,d,o are_coplanar & not a,c,d,o are_coplanar & o,d,d' is_collinear & o,a,a' is_collinear & o,b,b' is_collinear & o,c,c' is_collinear & a,d,s is_collinear & a',d',s is_collinear & b,d,t is_collinear & b',d',t is_collinear & c,d,u is_collinear & o <> a' & o <> d' & d <> d' & o <> b' holds
not s,t,u is_collinear

proof end;

definition

let FCPS be up-3-dimensional CollProjectiveSpace;
let o, a, b, c be Element of FCPS;
pred o,a,b,c constitute_a_quadrangle means :Def2: :: PROJDES1:def 2
( not a,b,c is_collinear & not o,a,b is_collinear & not o,b,c is_collinear & not o,c,a is_collinear );

end;

:: deftheorem Def2 defines constitute_a_quadrangle PROJDES1:def 2 :

for FCPS being up-3-dimensional CollProjectiveSpace
for o, a, b, c being Element of FCPS holds
( o,a,b,c constitute_a_quadrangle iff ( not a,b,c is_collinear & not o,a,b is_collinear & not o,b,c is_collinear & not o,c,a is_collinear ) );

Lm7: for FCPS being up-3-dimensional CollProjectiveSpace
for o, a', b', c', a, b, c, p, q, r being Element of FCPS st o <> a' & o <> b' & o <> c' & a <> a' & b <> b' & o,a,b,c constitute_a_quadrangle & o,a,a' is_collinear & o,b,b' is_collinear & o,c,c' is_collinear & a,b,p is_collinear & a',b',p is_collinear & b,c,q is_collinear & b',c',q is_collinear & a,c,r is_collinear & a',c',r is_collinear holds
p,q,r is_collinear

proof end;

theorem :: PROJDES1:25

canceled;

theorem Th26: :: PROJDES1:26

for FCPS being up-3-dimensional CollProjectiveSpace
for o, a, b, c, a', b', c', p, r, q being Element of FCPS st not o,a,b is_collinear & not o,b,c is_collinear & not o,a,c is_collinear & o,a,a' is_collinear & o,b,b' is_collinear & o,c,c' is_collinear & a,b,p is_collinear & a',b',p is_collinear & a <> a' & b,c,r is_collinear & b',c',r is_collinear & a,c,q is_collinear & b <> b' & a',c',q is_collinear & o <> a' & o <> b' & o <> c' holds
r,q,p is_collinear

proof end;

theorem Th27: :: PROJDES1:27

for CS being up-3-dimensional CollProjectiveSpace holds CS is Desarguesian

proof end;

registration

cluster up-3-dimensional -> Desarguesian up-3-dimensional L13();
correctness
coherence
for b₁ being up-3-dimensional CollProjectiveSpace holds b₁ is Desarguesian;
by Th27;

end;