:: YELLOW20 semantic presentation
theorem Th1: :: YELLOW20:1
theorem Th2: :: YELLOW20:2
theorem Th3: :: YELLOW20:3
theorem Th4: :: YELLOW20:4
theorem Th5: :: YELLOW20:5
theorem Th6: :: YELLOW20:6
theorem Th7: :: YELLOW20:7
theorem Th8: :: YELLOW20:8
theorem Th9: :: YELLOW20:9
definition
let c
1, c
2 be
AltCatStr ;
pred c
1,c
2 have_the_same_composition means :
Def1:
:: YELLOW20:def 1
for b
1, b
2, b
3 being
set holds the
Comp of a
1 . [b1,b2,b3] tolerates the
Comp of a
2 . [b1,b2,b3];
symmetry
for b1, b2 being AltCatStr holds
( ( for b3, b4, b5 being set holds the Comp of b1 . [b3,b4,b5] tolerates the Comp of b2 . [b3,b4,b5] ) implies for b3, b4, b5 being set holds the Comp of b2 . [b3,b4,b5] tolerates the Comp of b1 . [b3,b4,b5] )
;
end;
:: deftheorem Def1 defines have_the_same_composition YELLOW20:def 1 :
theorem Th10: :: YELLOW20:10
for b
1, b
2 being
AltCatStr holds
( b
1,b
2 have_the_same_composition iff for b
3, b
4, b
5, b
6 being
set holds
( b
6 in dom (the Comp of b1 . [b3,b4,b5]) & b
6 in dom (the Comp of b2 . [b3,b4,b5]) implies
(the Comp of b1 . [b3,b4,b5]) . b
6 = (the Comp of b2 . [b3,b4,b5]) . b
6 ) )
theorem Th11: :: YELLOW20:11
for b
1, b
2 being non
empty transitive AltCatStr holds
( b
1,b
2 have_the_same_composition iff for b
3, b
4, b
5 being
object of b
1 holds
(
<^b3,b4^> <> {} &
<^b4,b5^> <> {} implies for b
6, b
7, b
8 being
object of b
2 holds
(
<^b6,b7^> <> {} &
<^b7,b8^> <> {} & b
6 = b
3 & b
7 = b
4 & b
8 = b
5 implies for b
9 being
Morphism of b
3,b
4for b
10 being
Morphism of b
6,b
7 holds
( b
10 = b
9 implies for b
11 being
Morphism of b
4,b
5for b
12 being
Morphism of b
7,b
8 holds
( b
12 = b
11 implies b
11 * b
9 = b
12 * b
10 ) ) ) ) )
theorem Th12: :: YELLOW20:12
:: deftheorem Def2 defines Intersect YELLOW20:def 2 :
theorem Th13: :: YELLOW20:13
theorem Th14: :: YELLOW20:14
theorem Th15: :: YELLOW20:15
theorem Th16: :: YELLOW20:16
theorem Th17: :: YELLOW20:17
for b
1, b
2 being
set for b
3, b
4 being
ManySortedSet of b
1for b
5, b
6 being
ManySortedSet of b
2for b
7, b
8 being
ManySortedSet of b
1 /\ b
2 holds
( b
7 = Intersect b
3,b
5 & b
8 = Intersect b
4,b
6 implies for b
9 being
ManySortedFunction of b
3,b
4for b
10 being
ManySortedFunction of b
5,b
6 holds
( ( for b
11 being
set holds
( b
11 in dom b
9 & b
11 in dom b
10 implies b
9 . b
11 tolerates b
10 . b
11 ) ) implies
Intersect b
9,b
10 is
ManySortedFunction of b
7,b
8 ) )
theorem Th18: :: YELLOW20:18
theorem Th19: :: YELLOW20:19
for b
1, b
2 being
set for b
3, b
4 being
ManySortedSet of
[:b1,b1:]for b
5, b
6 being
ManySortedSet of
[:b2,b2:] holds
ex b
7, b
8 being
ManySortedSet of
[:(b1 /\ b2),(b1 /\ b2):] st
( b
7 = Intersect b
3,b
5 & b
8 = Intersect b
4,b
6 &
Intersect {|b3,b4|},
{|b5,b6|} = {|b7,b8|} )
definition
let c
1, c
2 be
AltCatStr ;
assume E16:
c
1,c
2 have_the_same_composition
;
func Intersect c
1,c
2 -> strict AltCatStr means :
Def3:
:: YELLOW20:def 3
( the
carrier of a
3 = the
carrier of a
1 /\ the
carrier of a
2 & the
Arrows of a
3 = Intersect the
Arrows of a
1,the
Arrows of a
2 & the
Comp of a
3 = Intersect the
Comp of a
1,the
Comp of a
2 );
existence
ex b1 being strict AltCatStr st
( the carrier of b1 = the carrier of c1 /\ the carrier of c2 & the Arrows of b1 = Intersect the Arrows of c1,the Arrows of c2 & the Comp of b1 = Intersect the Comp of c1,the Comp of c2 )
uniqueness
for b1, b2 being strict AltCatStr holds
( the carrier of b1 = the carrier of c1 /\ the carrier of c2 & the Arrows of b1 = Intersect the Arrows of c1,the Arrows of c2 & the Comp of b1 = Intersect the Comp of c1,the Comp of c2 & the carrier of b2 = the carrier of c1 /\ the carrier of c2 & the Arrows of b2 = Intersect the Arrows of c1,the Arrows of c2 & the Comp of b2 = Intersect the Comp of c1,the Comp of c2 implies b1 = b2 )
;
end;
:: deftheorem Def3 defines Intersect YELLOW20:def 3 :
theorem Th20: :: YELLOW20:20
theorem Th21: :: YELLOW20:21
theorem Th22: :: YELLOW20:22
theorem Th23: :: YELLOW20:23
theorem Th24: :: YELLOW20:24
for b
1, b
2 being
AltCatStr holds
( b
1,b
2 have_the_same_composition implies for b
3, b
4 being
object of b
1for b
5, b
6 being
object of b
2for b
7, b
8 being
object of
(Intersect b1,b2) holds
( b
7 = b
3 & b
7 = b
5 & b
8 = b
4 & b
8 = b
6 &
<^b3,b4^> <> {} &
<^b5,b6^> <> {} implies for b
9 being
Morphism of b
3,b
4for b
10 being
Morphism of b
5,b
6 holds
( b
9 = b
10 implies b
9 in <^b7,b8^> ) ) )
theorem Th25: :: YELLOW20:25
theorem Th26: :: YELLOW20:26
scheme :: YELLOW20:sch 1
s1{ F
1()
-> category, F
2()
-> non
empty subcategory of F
1(), F
3()
-> non
empty subcategory of F
1(), P
1[
set ], P
2[
set ,
set ,
set ] } :
provided
E21:
for b
1 being
object of F
1() holds
( b
1 is
object of F
2() iff P
1[b
1] )
and
E22:
for b
1, b
2 being
object of F
1()
for b
3, b
4 being
object of F
2() holds
( b
3 = b
1 & b
4 = b
2 &
<^b1,b2^> <> {} implies for b
5 being
Morphism of b
1,b
2 holds
( b
5 in <^b3,b4^> iff P
2[b
1,b
2,b
5] ) )
and
E23:
for b
1 being
object of F
1() holds
( b
1 is
object of F
3() iff P
1[b
1] )
and
E24:
for b
1, b
2 being
object of F
1()
for b
3, b
4 being
object of F
3() holds
( b
3 = b
1 & b
4 = b
2 &
<^b1,b2^> <> {} implies for b
5 being
Morphism of b
1,b
2 holds
( b
5 in <^b3,b4^> iff P
2[b
1,b
2,b
5] ) )
theorem Th27: :: YELLOW20:27
theorem Th28: :: YELLOW20:28
theorem Th29: :: YELLOW20:29
theorem Th30: :: YELLOW20:30
theorem Th31: :: YELLOW20:31
theorem Th32: :: YELLOW20:32
theorem Th33: :: YELLOW20:33
theorem Th34: :: YELLOW20:34
theorem Th35: :: YELLOW20:35
theorem Th36: :: YELLOW20:36
theorem Th37: :: YELLOW20:37
definition
let c
1, c
2 be
category;
let c
3 be
FunctorStr of c
1,c
2;
let c
4, c
5 be
category;
pred c
4,c
5 are_isomorphic_under c
3 means :: YELLOW20:def 4
( a
4 is
subcategory of a
1 & a
5 is
subcategory of a
2 & ex b
1 being
covariant Functor of a
4,a
5 st
( b
1 is
bijective & ( for b
2 being
object of a
4for b
3 being
object of a
1 holds
( b
2 = b
3 implies b
1 . b
2 = a
3 . b
3 ) ) & ( for b
2, b
3 being
object of a
4for b
4, b
5 being
object of a
1 holds
(
<^b2,b3^> <> {} & b
2 = b
4 & b
3 = b
5 implies for b
6 being
Morphism of b
2,b
3for b
7 being
Morphism of b
4,b
5 holds
( b
6 = b
7 implies b
1 . b
6 = (Morph-Map a3,b4,b5) . b
7 ) ) ) ) );
pred c
4,c
5 are_anti-isomorphic_under c
3 means :: YELLOW20:def 5
( a
4 is
subcategory of a
1 & a
5 is
subcategory of a
2 & ex b
1 being
contravariant Functor of a
4,a
5 st
( b
1 is
bijective & ( for b
2 being
object of a
4for b
3 being
object of a
1 holds
( b
2 = b
3 implies b
1 . b
2 = a
3 . b
3 ) ) & ( for b
2, b
3 being
object of a
4for b
4, b
5 being
object of a
1 holds
(
<^b2,b3^> <> {} & b
2 = b
4 & b
3 = b
5 implies for b
6 being
Morphism of b
2,b
3for b
7 being
Morphism of b
4,b
5 holds
( b
6 = b
7 implies b
1 . b
6 = (Morph-Map a3,b4,b5) . b
7 ) ) ) ) );
end;
:: deftheorem Def4 defines are_isomorphic_under YELLOW20:def 4 :
for b
1, b
2 being
categoryfor b
3 being
FunctorStr of b
1,b
2for b
4, b
5 being
category holds
( b
4,b
5 are_isomorphic_under b
3 iff ( b
4 is
subcategory of b
1 & b
5 is
subcategory of b
2 & ex b
6 being
covariant Functor of b
4,b
5 st
( b
6 is
bijective & ( for b
7 being
object of b
4for b
8 being
object of b
1 holds
( b
7 = b
8 implies b
6 . b
7 = b
3 . b
8 ) ) & ( for b
7, b
8 being
object of b
4for b
9, b
10 being
object of b
1 holds
(
<^b7,b8^> <> {} & b
7 = b
9 & b
8 = b
10 implies for b
11 being
Morphism of b
7,b
8for b
12 being
Morphism of b
9,b
10 holds
( b
11 = b
12 implies b
6 . b
11 = (Morph-Map b3,b9,b10) . b
12 ) ) ) ) ) );
:: deftheorem Def5 defines are_anti-isomorphic_under YELLOW20:def 5 :
for b
1, b
2 being
categoryfor b
3 being
FunctorStr of b
1,b
2for b
4, b
5 being
category holds
( b
4,b
5 are_anti-isomorphic_under b
3 iff ( b
4 is
subcategory of b
1 & b
5 is
subcategory of b
2 & ex b
6 being
contravariant Functor of b
4,b
5 st
( b
6 is
bijective & ( for b
7 being
object of b
4for b
8 being
object of b
1 holds
( b
7 = b
8 implies b
6 . b
7 = b
3 . b
8 ) ) & ( for b
7, b
8 being
object of b
4for b
9, b
10 being
object of b
1 holds
(
<^b7,b8^> <> {} & b
7 = b
9 & b
8 = b
10 implies for b
11 being
Morphism of b
7,b
8for b
12 being
Morphism of b
9,b
10 holds
( b
11 = b
12 implies b
6 . b
11 = (Morph-Map b3,b9,b10) . b
12 ) ) ) ) ) );
theorem Th38: :: YELLOW20:38
theorem Th39: :: YELLOW20:39
theorem Th40: :: YELLOW20:40
theorem Th41: :: YELLOW20:41
theorem Th42: :: YELLOW20:42
theorem Th43: :: YELLOW20:43
scheme :: YELLOW20:sch 4
s4{ F
1()
-> non
empty category, F
2()
-> non
empty category, F
3()
-> covariant Functor of F
1(),F
2(), F
4()
-> non
empty subcategory of F
1(), F
5()
-> non
empty subcategory of F
2() } :
provided
E32:
F
3() is
bijective
and
E33:
for b
1 being
object of F
1() holds
( b
1 is
object of F
4() iff F
3()
. b
1 is
object of F
5() )
and
E34:
for b
1, b
2 being
object of F
1() holds
(
<^b1,b2^> <> {} implies for b
3, b
4 being
object of F
4() holds
( b
3 = b
1 & b
4 = b
2 implies for b
5, b
6 being
object of F
5() holds
( b
5 = F
3()
. b
1 & b
6 = F
3()
. b
2 implies for b
7 being
Morphism of b
1,b
2 holds
( b
7 in <^b3,b4^> iff F
3()
. b
7 in <^b5,b6^> ) ) ) )
scheme :: YELLOW20:sch 5
s5{ F
1()
-> non
empty category, F
2()
-> non
empty category, F
3()
-> contravariant Functor of F
1(),F
2(), F
4()
-> non
empty subcategory of F
1(), F
5()
-> non
empty subcategory of F
2() } :
provided
E32:
F
3() is
bijective
and
E33:
for b
1 being
object of F
1() holds
( b
1 is
object of F
4() iff F
3()
. b
1 is
object of F
5() )
and
E34:
for b
1, b
2 being
object of F
1() holds
(
<^b1,b2^> <> {} implies for b
3, b
4 being
object of F
4() holds
( b
3 = b
1 & b
4 = b
2 implies for b
5, b
6 being
object of F
5() holds
( b
5 = F
3()
. b
1 & b
6 = F
3()
. b
2 implies for b
7 being
Morphism of b
1,b
2 holds
( b
7 in <^b3,b4^> iff F
3()
. b
7 in <^b6,b5^> ) ) ) )
theorem Th44: :: YELLOW20:44
theorem Th45: :: YELLOW20:45
theorem Th46: :: YELLOW20:46
theorem Th47: :: YELLOW20:47
theorem Th48: :: YELLOW20:48
theorem Th49: :: YELLOW20:49
theorem Th50: :: YELLOW20:50
theorem Th51: :: YELLOW20:51
theorem Th52: :: YELLOW20:52
theorem Th53: :: YELLOW20:53
theorem Th54: :: YELLOW20:54
theorem Th55: :: YELLOW20:55
theorem Th56: :: YELLOW20:56