:: Transformations in Affine Spaces
:: by Henryk Oryszczyszyn and Krzysztof Pra\.zmowski
::
:: Received May 31, 1990
:: Copyright (c) 1990 Association of Mizar Users
theorem :: TRANSGEO:1
canceled;
theorem Th2: :: TRANSGEO:2
theorem :: TRANSGEO:3
canceled;
theorem Th4: :: TRANSGEO:4
:: deftheorem defines \ TRANSGEO:def 1 :
scheme :: TRANSGEO:sch 1
EXPermutation{
F1()
-> non
empty set ,
P1[
set ,
set ] } :
provided
A1:
for
x being
Element of
F1() ex
y being
Element of
F1() st
P1[
x,
y]
and A2:
for
y being
Element of
F1() ex
x being
Element of
F1() st
P1[
x,
y]
and A3:
for
x,
y,
x' being
Element of
F1() st
P1[
x,
y] &
P1[
x',
y] holds
x = x'
and A4:
for
x,
y,
y' being
Element of
F1() st
P1[
x,
y] &
P1[
x,
y'] holds
y = y'
theorem :: TRANSGEO:5
canceled;
theorem :: TRANSGEO:6
canceled;
theorem :: TRANSGEO:7
canceled;
theorem :: TRANSGEO:8
canceled;
theorem :: TRANSGEO:9
theorem :: TRANSGEO:10
canceled;
theorem :: TRANSGEO:11
Lm1:
for A being non empty set
for f, g, h being Permutation of A st f * g = f * h holds
g = h
Lm2:
for A being non empty set
for g, f, h being Permutation of A st g * f = h * f holds
g = h
theorem :: TRANSGEO:12
canceled;
theorem :: TRANSGEO:13
theorem :: TRANSGEO:14
canceled;
theorem :: TRANSGEO:15
canceled;
theorem :: TRANSGEO:16
theorem :: TRANSGEO:17
theorem :: TRANSGEO:18
theorem :: TRANSGEO:19
theorem :: TRANSGEO:20
theorem :: TRANSGEO:21
:: deftheorem Def2 defines is_FormalIz_of TRANSGEO:def 2 :
theorem :: TRANSGEO:22
canceled;
theorem Th23: :: TRANSGEO:23
theorem Th24: :: TRANSGEO:24
theorem :: TRANSGEO:25
theorem Th26: :: TRANSGEO:26
for
A being non
empty set for
f,
g being
Permutation of
A for
R being
Relation of
[:A,A:] st ( for
a,
b,
x,
y,
z,
t being
Element of
A st
[[x,y],[a,b]] in R &
[[a,b],[z,t]] in R &
a <> b holds
[[x,y],[z,t]] in R ) & ( for
x,
y,
z being
Element of
A holds
[[x,x],[y,z]] in R ) &
f is_FormalIz_of R &
g is_FormalIz_of R holds
f * g is_FormalIz_of R
definition
let A be non
empty set ;
let f be
Permutation of
A;
let R be
Relation of
[:A,A:];
pred f is_automorphism_of R means :
Def3:
:: TRANSGEO:def 3
for
x,
y,
z,
t being
Element of
A holds
(
[[x,y],[z,t]] in R iff
[[(f . x),(f . y)],[(f . z),(f . t)]] in R );
end;
:: deftheorem Def3 defines is_automorphism_of TRANSGEO:def 3 :
for
A being non
empty set for
f being
Permutation of
A for
R being
Relation of
[:A,A:] holds
(
f is_automorphism_of R iff for
x,
y,
z,
t being
Element of
A holds
(
[[x,y],[z,t]] in R iff
[[(f . x),(f . y)],[(f . z),(f . t)]] in R ) );
theorem :: TRANSGEO:27
canceled;
theorem Th28: :: TRANSGEO:28
theorem Th29: :: TRANSGEO:29
theorem Th30: :: TRANSGEO:30
theorem :: TRANSGEO:31
theorem :: TRANSGEO:32
for
A being non
empty set for
f being
Permutation of
A for
R being
Relation of
[:A,A:] st ( for
a,
b,
x,
y,
z,
t being
Element of
A st
[[x,y],[a,b]] in R &
[[a,b],[z,t]] in R &
a <> b holds
[[x,y],[z,t]] in R ) & ( for
x,
y,
z being
Element of
A holds
[[x,x],[y,z]] in R ) &
R is_symmetric_in [:A,A:] &
f is_FormalIz_of R holds
f is_automorphism_of R
theorem :: TRANSGEO:33
:: deftheorem Def4 defines is_DIL_of TRANSGEO:def 4 :
theorem :: TRANSGEO:34
canceled;
theorem Th35: :: TRANSGEO:35
definition
let IT be non
empty AffinStruct ;
attr IT is
CongrSpace-like means :
Def5:
:: TRANSGEO:def 5
( ( for
x,
y,
z,
t,
a,
b being
Element of
IT st
x,
y // a,
b &
a,
b // z,
t &
a <> b holds
x,
y // z,
t ) & ( for
x,
y,
z being
Element of
IT holds
x,
x // y,
z ) & ( for
x,
y,
z,
t being
Element of
IT st
x,
y // z,
t holds
z,
t // x,
y ) & ( for
x,
y being
Element of
IT holds
x,
y // x,
y ) );
end;
:: deftheorem Def5 defines CongrSpace-like TRANSGEO:def 5 :
for
IT being non
empty AffinStruct holds
(
IT is
CongrSpace-like iff ( ( for
x,
y,
z,
t,
a,
b being
Element of
IT st
x,
y // a,
b &
a,
b // z,
t &
a <> b holds
x,
y // z,
t ) & ( for
x,
y,
z being
Element of
IT holds
x,
x // y,
z ) & ( for
x,
y,
z,
t being
Element of
IT st
x,
y // z,
t holds
z,
t // x,
y ) & ( for
x,
y being
Element of
IT holds
x,
y // x,
y ) ) );
Lm3:
for CS being CongrSpace holds the CONGR of CS is_reflexive_in [:the carrier of CS,the carrier of CS:]
Lm4:
for CS being CongrSpace holds the CONGR of CS is_symmetric_in [:the carrier of CS,the carrier of CS:]
theorem :: TRANSGEO:36
canceled;
theorem Th37: :: TRANSGEO:37
theorem Th38: :: TRANSGEO:38
theorem Th39: :: TRANSGEO:39
theorem Th40: :: TRANSGEO:40
:: deftheorem Def6 defines positive_dilatation TRANSGEO:def 6 :
theorem :: TRANSGEO:41
canceled;
theorem Th42: :: TRANSGEO:42
:: deftheorem Def7 defines negative_dilatation TRANSGEO:def 7 :
theorem :: TRANSGEO:43
canceled;
theorem Th44: :: TRANSGEO:44
theorem :: TRANSGEO:45
theorem :: TRANSGEO:46
theorem :: TRANSGEO:47
theorem :: TRANSGEO:48
theorem :: TRANSGEO:49
:: deftheorem Def8 defines dilatation TRANSGEO:def 8 :
theorem :: TRANSGEO:50
canceled;
theorem Th51: :: TRANSGEO:51
theorem :: TRANSGEO:52
theorem :: TRANSGEO:53
theorem :: TRANSGEO:54
theorem :: TRANSGEO:55
theorem Th56: :: TRANSGEO:56
theorem Th57: :: TRANSGEO:57
theorem Th58: :: TRANSGEO:58
theorem Th59: :: TRANSGEO:59
theorem Th60: :: TRANSGEO:60
theorem Th61: :: TRANSGEO:61
for
OAS being
OAffinSpace for
a,
b,
c,
d being
Element of
OAS holds
( not
a,
b '||' c,
d or
a,
c '||' b,
d or ex
x being
Element of
OAS st
(
LIN a,
c,
x &
LIN b,
d,
x ) )
theorem Th62: :: TRANSGEO:62
theorem Th63: :: TRANSGEO:63
theorem Th64: :: TRANSGEO:64
theorem :: TRANSGEO:65
:: deftheorem Def9 defines translation TRANSGEO:def 9 :
theorem :: TRANSGEO:66
canceled;
theorem Th67: :: TRANSGEO:67
theorem :: TRANSGEO:68
canceled;
theorem Th69: :: TRANSGEO:69
theorem Th70: :: TRANSGEO:70
theorem Th71: :: TRANSGEO:71
theorem :: TRANSGEO:72
Lm5:
for OAS being OAffinSpace
for a, b being Element of OAS
for f being Permutation of the carrier of OAS st f is translation & not LIN a,f . a,b holds
( a,b // f . a,f . b & a,f . a // b,f . b )
Lm6:
for OAS being OAffinSpace
for a, b being Element of OAS
for f being Permutation of the carrier of OAS st f is translation & a <> f . a & LIN a,f . a,b holds
a,f . a // b,f . b
Lm7:
for OAS being OAffinSpace
for a, b being Element of OAS
for f being Permutation of the carrier of OAS st f is translation & Mid a,f . a,b & a <> f . a holds
a,b // f . a,f . b
Lm8:
for OAS being OAffinSpace
for a, b being Element of OAS
for f being Permutation of the carrier of OAS st f is translation & a <> f . a & b <> f . a & Mid a,b,f . a holds
a,b // f . a,f . b
Lm9:
for OAS being OAffinSpace
for a, b being Element of OAS
for f being Permutation of the carrier of OAS st f is translation & a <> f . a & LIN a,f . a,b holds
a,b // f . a,f . b
theorem Th73: :: TRANSGEO:73
theorem Th74: :: TRANSGEO:74
theorem Th75: :: TRANSGEO:75
theorem Th76: :: TRANSGEO:76
theorem Th77: :: TRANSGEO:77
theorem Th78: :: TRANSGEO:78
theorem Th79: :: TRANSGEO:79
theorem :: TRANSGEO:80
theorem :: TRANSGEO:81
canceled;
theorem Th82: :: TRANSGEO:82
theorem :: TRANSGEO:83
:: deftheorem Def10 defines dilatation TRANSGEO:def 10 :
theorem :: TRANSGEO:84
canceled;
theorem Th85: :: TRANSGEO:85
theorem Th86: :: TRANSGEO:86
theorem Th87: :: TRANSGEO:87
theorem Th88: :: TRANSGEO:88
theorem Th89: :: TRANSGEO:89
theorem :: TRANSGEO:90
theorem Th91: :: TRANSGEO:91
theorem Th92: :: TRANSGEO:92
for
AFS being
AffinSpace for
a,
b,
c,
d being
Element of
AFS holds
( not
a,
b // c,
d or
a,
c // b,
d or ex
x being
Element of
AFS st
(
LIN a,
c,
x &
LIN b,
d,
x ) )
theorem Th93: :: TRANSGEO:93
theorem Th94: :: TRANSGEO:94
theorem Th95: :: TRANSGEO:95
theorem :: TRANSGEO:96
theorem Th97: :: TRANSGEO:97
for
AFS being
AffinSpace for
a,
b,
c,
d1,
d2 being
Element of
AFS st not
LIN a,
b,
c &
a,
b // c,
d1 &
a,
b // c,
d2 &
a,
c // b,
d1 &
a,
c // b,
d2 holds
d1 = d2
:: deftheorem Def11 defines translation TRANSGEO:def 11 :
theorem :: TRANSGEO:98
canceled;
theorem :: TRANSGEO:99
theorem Th100: :: TRANSGEO:100
theorem :: TRANSGEO:101
canceled;
theorem Th102: :: TRANSGEO:102
theorem Th103: :: TRANSGEO:103
theorem Th104: :: TRANSGEO:104
theorem :: TRANSGEO:105
:: deftheorem defines collineation TRANSGEO:def 12 :
theorem :: TRANSGEO:106
canceled;
theorem Th107: :: TRANSGEO:107
theorem Th108: :: TRANSGEO:108
theorem :: TRANSGEO:109
theorem Th110: :: TRANSGEO:110
theorem Th111: :: TRANSGEO:111
theorem Th112: :: TRANSGEO:112
theorem Th113: :: TRANSGEO:113
theorem :: TRANSGEO:114
theorem Th115: :: TRANSGEO:115
theorem :: TRANSGEO:116
theorem :: TRANSGEO:117