:: MSUALG_9 semantic presentation
theorem Th1: :: MSUALG_9:1
theorem Th2: :: MSUALG_9:2
theorem Th3: :: MSUALG_9:3
theorem Th4: :: MSUALG_9:4
theorem Th5: :: MSUALG_9:5
theorem Th6: :: MSUALG_9:6
theorem Th7: :: MSUALG_9:7
theorem Th8: :: MSUALG_9:8
theorem Th9: :: MSUALG_9:9
theorem Th10: :: MSUALG_9:10
theorem Th11: :: MSUALG_9:11
theorem Th12: :: MSUALG_9:12
theorem Th13: :: MSUALG_9:13
theorem Th14: :: MSUALG_9:14
theorem Th15: :: MSUALG_9:15
theorem Th16: :: MSUALG_9:16
theorem Th17: :: MSUALG_9:17
theorem Th18: :: MSUALG_9:18
theorem Th19: :: MSUALG_9:19
theorem Th20: :: MSUALG_9:20
definition
let c
1 be
set ;
let c
2 be
ManySortedSet of c
1;
let c
3, c
4 be
V5 ManySortedSet of c
1;
let c
5 be
ManySortedFunction of c
2,
[|c3,c4|];
func Mpr1 c
5 -> ManySortedFunction of a
2,a
3 means :
Def1:
:: MSUALG_9:def 1
for b
1 being
set holds
( b
1 in a
1 implies a
6 . b
1 = pr1 (a5 . b1) );
existence
ex b1 being ManySortedFunction of c2,c3 st
for b2 being set holds
( b2 in c1 implies b1 . b2 = pr1 (c5 . b2) )
uniqueness
for b1, b2 being ManySortedFunction of c2,c3 holds
( ( for b3 being set holds
( b3 in c1 implies b1 . b3 = pr1 (c5 . b3) ) ) & ( for b3 being set holds
( b3 in c1 implies b2 . b3 = pr1 (c5 . b3) ) ) implies b1 = b2 )
func Mpr2 c
5 -> ManySortedFunction of a
2,a
4 means :
Def2:
:: MSUALG_9:def 2
for b
1 being
set holds
( b
1 in a
1 implies a
6 . b
1 = pr2 (a5 . b1) );
existence
ex b1 being ManySortedFunction of c2,c4 st
for b2 being set holds
( b2 in c1 implies b1 . b2 = pr2 (c5 . b2) )
uniqueness
for b1, b2 being ManySortedFunction of c2,c4 holds
( ( for b3 being set holds
( b3 in c1 implies b1 . b3 = pr2 (c5 . b3) ) ) & ( for b3 being set holds
( b3 in c1 implies b2 . b3 = pr2 (c5 . b3) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines Mpr1 MSUALG_9:def 1 :
:: deftheorem Def2 defines Mpr2 MSUALG_9:def 2 :
theorem Th21: :: MSUALG_9:21
theorem Th22: :: MSUALG_9:22
theorem Th23: :: MSUALG_9:23
theorem Th24: :: MSUALG_9:24
theorem Th25: :: MSUALG_9:25
theorem Th26: :: MSUALG_9:26
theorem Th27: :: MSUALG_9:27
theorem Th28: :: MSUALG_9:28
theorem Th29: :: MSUALG_9:29
theorem Th30: :: MSUALG_9:30
theorem Th31: :: MSUALG_9:31
theorem Th32: :: MSUALG_9:32
theorem Th33: :: MSUALG_9:33
theorem Th34: :: MSUALG_9:34
E17:
now
let c
1 be non
empty non
void ManySortedSign ;
let c
2 be
non-empty MSAlgebra of c
1;
let c
3, c
4 be
MSCongruence of c
2;
let c
5 be
ManySortedFunction of
(QuotMSAlg c2,c3),
(QuotMSAlg c2,c4);
assume E18:
for b
1 being
Element of c
1for b
2 being
Element of the
Sorts of
(QuotMSAlg c2,c3) . b
1for b
3 being
Element of the
Sorts of c
2 . b
1 holds
( b
2 = Class c
3,b
3 implies
(c5 . b1) . b
2 = Class c
4,b
3 )
;
thus
c
5 is
"onto"
proof
set c
6 = the
Sorts of
(QuotMSAlg c2,c3);
set c
7 = the
Sorts of
(QuotMSAlg c2,c4);
let c
8 be
set ;
:: according to MSUALG_3:def 3
assume
c
8 in the
carrier of c
1
;
then reconsider c
9 = c
8 as
SortSymbol of c
1 ;
rng (c5 . c9) c= the
Sorts of
(QuotMSAlg c2,c4) . c
9
;
hence
rng (c5 . c8) c= the
Sorts of
(QuotMSAlg c2,c4) . c
8
;
:: according to XBOOLE_0:def 10
let c
10 be
set ;
:: according to TARSKI:def 3
assume E19:
c
10 in the
Sorts of
(QuotMSAlg c2,c4) . c
8
;
c
10 in Class (c4 . c9)
by E19, MSUALG_4:def 8;
then consider c
11 being
set such that E20:
c
11 in the
Sorts of c
2 . c
9
and E21:
c
10 = Class (c4 . c9),c
11
by EQREL_1:def 5;
reconsider c
12 = c
11 as
Element of the
Sorts of c
2 . c
9 by E20;
E22:
dom (c5 . c9) = the
Sorts of
(QuotMSAlg c2,c3) . c
9
by FUNCT_2:def 1;
Class (c3 . c9),c
12 in Class (c3 . c9)
by EQREL_1:def 5;
then
Class c
3,c
12 in Class (c3 . c9)
;
then reconsider c
13 =
Class c
3,c
12 as
Element of the
Sorts of
(QuotMSAlg c2,c3) . c
9 by MSUALG_4:def 8;
(c5 . c9) . c
13 =
Class c
4,c
12
by E18
.=
Class (c4 . c9),c
12
;
hence
c
10 in rng (c5 . c8)
by E21, E22, FUNCT_1:def 5;
end;
end;
theorem Th35: :: MSUALG_9:35
theorem Th36: :: MSUALG_9:36
for b
1 being non
empty non
void ManySortedSign for b
2 being
non-empty MSAlgebra of b
1for b
3, b
4 being
MSCongruence of b
2for b
5 being
ManySortedFunction of
(QuotMSAlg b2,b3),
(QuotMSAlg b2,b4) holds
( ( for b
6 being
Element of b
1for b
7 being
Element of the
Sorts of
(QuotMSAlg b2,b3) . b
6for b
8 being
Element of the
Sorts of b
2 . b
6 holds
( b
7 = Class b
3,b
8 implies
(b5 . b6) . b
7 = Class b
4,b
8 ) ) implies b
5 is_epimorphism QuotMSAlg b
2,b
3,
QuotMSAlg b
2,b
4 )
theorem Th37: :: MSUALG_9:37