:: MSUALG_3 semantic presentation
:: deftheorem Def1 defines id MSUALG_3:def 1 :
:: deftheorem Def2 defines "1-1" MSUALG_3:def 2 :
theorem Th1: :: MSUALG_3:1
:: deftheorem Def3 defines "onto" MSUALG_3:def 3 :
theorem Th2: :: MSUALG_3:2
theorem Th3: :: MSUALG_3:3
theorem Th4: :: MSUALG_3:4
:: deftheorem Def4 MSUALG_3:def 4 :
canceled;
:: deftheorem Def5 defines "" MSUALG_3:def 5 :
theorem Th5: :: MSUALG_3:5
theorem Th6: :: MSUALG_3:6
:: deftheorem Def6 MSUALG_3:def 6 :
canceled;
:: deftheorem Def7 defines # MSUALG_3:def 7 :
E11:
now
let c
1 be non
empty non
void ManySortedSign ;
let c
2, c
3 be
MSAlgebra of c
1;
let c
4 be
OperSymbol of c
1;
let c
5 be
ManySortedFunction of c
2,c
3;
let c
6 be
Element of
Args c
4,c
2;
let c
7, c
8 be
Function;
assume E12:
( c
6 = c
7 & c
6 in Args c
4,c
2 & c
8 in Args c
4,c
3 )
;
E13:
rng (the_arity_of c4) c= the
carrier of c
1
by FINSEQ_1:def 4;
then E14:
rng (the_arity_of c4) c= dom the
Sorts of c
2
by PBOOLE:def 3;
E15:
(
Args c
4,c
2 = product (the Sorts of c2 * (the_arity_of c4)) &
Args c
4,c
3 = product (the Sorts of c3 * (the_arity_of c4)) )
by PRALG_2:10;
then E16:
dom c
7 =
dom (the Sorts of c2 * (the_arity_of c4))
by E12, CARD_3:18
.=
dom (the_arity_of c4)
by E14, RELAT_1:46
;
E17:
rng (the_arity_of c4) c= dom the
Sorts of c
3
by E13, PBOOLE:def 3;
E18:
dom c
8 = dom (the Sorts of c3 * (the_arity_of c4))
by E12, E15, CARD_3:18;
then E19:
dom c
8 = dom (the_arity_of c4)
by E17, RELAT_1:46;
E20:
dom (the Sorts of c2 * (the_arity_of c4)) = (c5 * (the_arity_of c4)) " (SubFuncs (rng (c5 * (the_arity_of c4))))
then E21:
the
Sorts of c
2 * (the_arity_of c4) = doms (c5 * (the_arity_of c4))
by E20, FUNCT_6:def 2;
hereby
assume
c
8 = c
5 # c
6
;
then E22:
c
8 = (Frege (c5 * (the_arity_of c4))) . c
6
by E12, Def7;
let c
9 be
Nat;
assume E23:
c
9 in dom c
7
;
then
(the_arity_of c4) . c
9 in the
carrier of c
1
by E16, FINSEQ_2:13;
then
(the_arity_of c4) . c
9 in dom c
5
by PBOOLE:def 3;
then E24:
c
9 in dom (c5 * (the_arity_of c4))
by E16, E23, FUNCT_1:21;
then E25:
(c5 * (the_arity_of c4)) . c
9 =
c
5 . ((the_arity_of c4) . c9)
by FUNCT_1:22
.=
c
5 . ((the_arity_of c4) /. c9)
by E16, E23, FINSEQ_4:def 4
;
thus c
8 . c
9 =
((c5 * (the_arity_of c4)) .. c7) . c
9
by E12, E15, E21, E22, PRALG_2:def 8
.=
(c5 . ((the_arity_of c4) /. c9)) . (c7 . c9)
by E24, E25, PRALG_1:def 17
;
end;
assume E22:
for b
1 being
Nat holds
( b
1 in dom c
7 implies c
8 . b
1 = (c5 . ((the_arity_of c4) /. b1)) . (c7 . b1) )
;
E23:
rng (the_arity_of c4) c= dom c
5
by E13, PBOOLE:def 3;
c
5 # c
6 is
Element of
product (the Sorts of c3 * (the_arity_of c4))
by PRALG_2:10;
then reconsider c
9 = c
5 # c
6 as
Function ;
E24:
c
5 # c
6 = (Frege (c5 * (the_arity_of c4))) . c
6
by E12, Def7;
then
c
5 # c
6 = (c5 * (the_arity_of c4)) .. c
7
by E12, E15, E21, PRALG_2:def 8;
then E25:
dom c
9 =
dom (c5 * (the_arity_of c4))
by PRALG_1:def 17
.=
dom c
7
by E16, E23, RELAT_1:46
;
now
let c
10 be
set ;
assume E26:
c
10 in dom c
7
;
then reconsider c
11 = c
10 as
Nat by E16;
(the_arity_of c4) . c
11 in the
carrier of c
1
by E16, E26, FINSEQ_2:13;
then
(the_arity_of c4) . c
11 in dom c
5
by PBOOLE:def 3;
then E27:
c
11 in dom (c5 * (the_arity_of c4))
by E16, E26, FUNCT_1:21;
then E28:
(c5 * (the_arity_of c4)) . c
11 =
c
5 . ((the_arity_of c4) . c11)
by FUNCT_1:22
.=
c
5 . ((the_arity_of c4) /. c11)
by E16, E26, FINSEQ_4:def 4
;
thus c
8 . c
10 =
(c5 . ((the_arity_of c4) /. c11)) . (c7 . c11)
by E22, E26
.=
((c5 * (the_arity_of c4)) .. c7) . c
11
by E27, E28, PRALG_1:def 17
.=
c
9 . c
10
by E12, E15, E21, E24, PRALG_2:def 8
;
end;
hence
c
8 = c
5 # c
6
by E16, E19, E25, FUNCT_1:9;
end;
:: deftheorem Def8 defines # MSUALG_3:def 8 :
theorem Th7: :: MSUALG_3:7
theorem Th8: :: MSUALG_3:8
:: deftheorem Def9 defines is_homomorphism MSUALG_3:def 9 :
theorem Th9: :: MSUALG_3:9
theorem Th10: :: MSUALG_3:10
:: deftheorem Def10 defines is_epimorphism MSUALG_3:def 10 :
theorem Th11: :: MSUALG_3:11
:: deftheorem Def11 defines is_monomorphism MSUALG_3:def 11 :
theorem Th12: :: MSUALG_3:12
:: deftheorem Def12 defines is_isomorphism MSUALG_3:def 12 :
theorem Th13: :: MSUALG_3:13
Lemma24:
for b1 being non empty non void ManySortedSign
for b2, b3 being non-empty MSAlgebra of b1
for b4 being ManySortedFunction of b2,b3 holds
( b4 is_isomorphism b2,b3 implies b4 "" is_homomorphism b3,b2 )
theorem Th14: :: MSUALG_3:14
theorem Th15: :: MSUALG_3:15
:: deftheorem Def13 defines are_isomorphic MSUALG_3:def 13 :
theorem Th16: :: MSUALG_3:16
theorem Th17: :: MSUALG_3:17
theorem Th18: :: MSUALG_3:18
:: deftheorem Def14 defines Image MSUALG_3:def 14 :
theorem Th19: :: MSUALG_3:19
Lemma30:
for b1 being non empty non void ManySortedSign
for b2, b3 being non-empty MSAlgebra of b1
for b4 being ManySortedFunction of b2,b3 holds
( b4 is_homomorphism b2,b3 implies b4 is ManySortedFunction of b2,(Image b4) )
theorem Th20: :: MSUALG_3:20
theorem Th21: :: MSUALG_3:21
Lemma32:
for b1 being non empty non void ManySortedSign
for b2, b3 being non-empty MSAlgebra of b1
for b4 being non-empty MSSubAlgebra of b3
for b5 being ManySortedFunction of b2,b3
for b6 being ManySortedFunction of b2,b4 holds
( b5 = b6 & b6 is_homomorphism b2,b4 implies b5 is_homomorphism b2,b3 )
theorem Th22: :: MSUALG_3:22
theorem Th23: :: MSUALG_3:23
theorem Th24: :: MSUALG_3:24