:: The Well Ordering Relations
:: by Grzegorz Bancerek
::
:: Received April 4, 1989
:: Copyright (c) 1990 Association of Mizar Users
Lm1:
for R being Relation holds
( R is reflexive iff for x being set st x in field R holds
[x,x] in R )
Lm2:
for R being Relation holds
( R is transitive iff for x, y, z being set st [x,y] in R & [y,z] in R holds
[x,z] in R )
Lm3:
for R being Relation holds
( R is antisymmetric iff for x, y being set st [x,y] in R & [y,x] in R holds
x = y )
Lm4:
for R being Relation holds
( R is connected iff for x, y being set st x in field R & y in field R & x <> y & not [x,y] in R holds
[y,x] in R )
:: deftheorem Def1 defines -Seg WELLORD1:def 1 :
theorem :: WELLORD1:1
canceled;
theorem Th2: :: WELLORD1:2
:: deftheorem Def2 defines well_founded WELLORD1:def 2 :
:: deftheorem Def3 defines is_well_founded_in WELLORD1:def 3 :
theorem :: WELLORD1:3
canceled;
theorem :: WELLORD1:4
canceled;
theorem Th5: :: WELLORD1:5
:: deftheorem Def4 defines well-ordering WELLORD1:def 4 :
:: deftheorem Def5 defines well_orders WELLORD1:def 5 :
theorem :: WELLORD1:6
canceled;
theorem :: WELLORD1:7
canceled;
theorem :: WELLORD1:8
theorem :: WELLORD1:9
theorem Th10: :: WELLORD1:10
theorem :: WELLORD1:11
theorem :: WELLORD1:12
theorem Th13: :: WELLORD1:13
:: deftheorem defines |_2 WELLORD1:def 6 :
theorem :: WELLORD1:14
canceled;
theorem :: WELLORD1:15
canceled;
theorem :: WELLORD1:16
canceled;
theorem Th17: :: WELLORD1:17
theorem Th18: :: WELLORD1:18
Lm5:
for X being set
for R being Relation holds dom (X | R) c= dom R
theorem Th19: :: WELLORD1:19
theorem Th20: :: WELLORD1:20
theorem Th21: :: WELLORD1:21
theorem Th22: :: WELLORD1:22
theorem Th23: :: WELLORD1:23
theorem Th24: :: WELLORD1:24
theorem Th25: :: WELLORD1:25
theorem Th26: :: WELLORD1:26
theorem :: WELLORD1:27
theorem :: WELLORD1:28
theorem Th29: :: WELLORD1:29
theorem Th30: :: WELLORD1:30
theorem Th31: :: WELLORD1:31
theorem Th32: :: WELLORD1:32
theorem Th33: :: WELLORD1:33
theorem :: WELLORD1:34
canceled;
theorem Th35: :: WELLORD1:35
theorem Th36: :: WELLORD1:36
theorem Th37: :: WELLORD1:37
theorem Th38: :: WELLORD1:38
theorem Th39: :: WELLORD1:39
theorem Th40: :: WELLORD1:40
theorem Th41: :: WELLORD1:41
theorem Th42: :: WELLORD1:42
theorem Th43: :: WELLORD1:43
:: deftheorem Def7 defines is_isomorphism_of WELLORD1:def 7 :
theorem :: WELLORD1:44
canceled;
theorem Th45: :: WELLORD1:45
:: deftheorem Def8 defines are_isomorphic WELLORD1:def 8 :
theorem :: WELLORD1:46
canceled;
theorem Th47: :: WELLORD1:47
theorem :: WELLORD1:48
theorem Th49: :: WELLORD1:49
theorem Th50: :: WELLORD1:50
theorem Th51: :: WELLORD1:51
theorem Th52: :: WELLORD1:52
theorem Th53: :: WELLORD1:53
theorem Th54: :: WELLORD1:54
theorem Th55: :: WELLORD1:55
definition
let R,
S be
Relation;
assume A1:
(
R is
well-ordering &
R,
S are_isomorphic )
;
func canonical_isomorphism_of R,
S -> Function means :
Def9:
:: WELLORD1:def 9
it is_isomorphism_of R,
S;
existence
ex b1 being Function st b1 is_isomorphism_of R,S
by A1, Def8;
uniqueness
for b1, b2 being Function st b1 is_isomorphism_of R,S & b2 is_isomorphism_of R,S holds
b1 = b2
by A1, Th55;
end;
:: deftheorem Def9 defines canonical_isomorphism_of WELLORD1:def 9 :
theorem :: WELLORD1:56
canceled;
theorem Th57: :: WELLORD1:57
theorem Th58: :: WELLORD1:58
theorem Th59: :: WELLORD1:59
theorem Th60: :: WELLORD1:60
theorem Th61: :: WELLORD1:61
theorem Th62: :: WELLORD1:62
theorem Th63: :: WELLORD1:63
theorem :: WELLORD1:64