:: 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 :
E85:
now
let S be non
empty non
void ManySortedSign ;
let U1 be
MSAlgebra of
S,
U2 be
MSAlgebra of
S;
let o be
OperSymbol of
S;
let F be
ManySortedFunction of
U1,
U2;
let x be
Element of
Args o,
U1;
let f be
Function,
u be
Function;
assume E30:
(
x = f &
x in Args o,
U1 &
u in Args o,
U2 )
;
E33:
rng (the_arity_of o) c= the
carrier of
S
by FINSEQ_1:def 4;
then E34:
rng (the_arity_of o) c= dom the
Sorts of
U1
by PBOOLE:def 3;
E35:
(
Args o,
U1 = product (the Sorts of U1 * (the_arity_of o)) &
Args o,
U2 = product (the Sorts of U2 * (the_arity_of o)) )
by PRALG_2:10;
then E49:
dom f =
dom (the Sorts of U1 * (the_arity_of o))
by , CARD_3:18
.=
dom (the_arity_of o)
by , RELAT_1:46
;
E50:
rng (the_arity_of o) c= dom the
Sorts of
U2
by , PBOOLE:def 3;
E51:
dom u = dom (the Sorts of U2 * (the_arity_of o))
by , , CARD_3:18;
then E54:
dom u = dom (the_arity_of o)
by , RELAT_1:46;
E55:
dom (the Sorts of U1 * (the_arity_of o)) = (F * (the_arity_of o)) " (SubFuncs (rng (F * (the_arity_of o))))
then E70:
the
Sorts of
U1 * (the_arity_of o) = doms (F * (the_arity_of o))
by , FUNCT_6:def 2;
hereby
assume
u = F # x
;
then E71:
u = (Frege (F * (the_arity_of o))) . x
by , ;
let n be
Nat;
assume E72:
n in dom f
;
then
(the_arity_of o) . n in the
carrier of
S
by , FINSEQ_2:13;
then
(the_arity_of o) . n in dom F
by PBOOLE:def 3;
then E73:
n in dom (F * (the_arity_of o))
by , , FUNCT_1:21;
then E74:
(F * (the_arity_of o)) . n =
F . ((the_arity_of o) . n)
by FUNCT_1:22
.=
F . ((the_arity_of o) /. n)
by , , FINSEQ_4:def 4
;
thus u . n =
((F * (the_arity_of o)) .. f) . n
by , , , , PRALG_2:def 8
.=
(F . ((the_arity_of o) /. n)) . (f . n)
by , , PRALG_1:def 17
;
end;
assume E75:
for
n being
Nat st
n in dom f holds
u . n = (F . ((the_arity_of o) /. n)) . (f . n)
;
E76:
rng (the_arity_of o) c= dom F
by , PBOOLE:def 3;
F # x is
Element of
product (the Sorts of U2 * (the_arity_of o))
by PRALG_2:10;
then reconsider g =
F # x as
Function ;
E77:
F # x = (Frege (F * (the_arity_of o))) . x
by , ;
then
F # x = (F * (the_arity_of o)) .. f
by , , , PRALG_2:def 8;
then E78:
dom g =
dom (F * (the_arity_of o))
by PRALG_1:def 17
.=
dom f
by , , RELAT_1:46
;
now
let e be
set ;
assume E79:
e in dom f
;
then reconsider n =
e as
Nat by , ORDINAL1:def 13;
(the_arity_of o) . n in the
carrier of
S
by , , FINSEQ_2:13;
then
(the_arity_of o) . n in dom F
by PBOOLE:def 3;
then E80:
n in dom (F * (the_arity_of o))
by , , FUNCT_1:21;
then E81:
(F * (the_arity_of o)) . n =
F . ((the_arity_of o) . n)
by FUNCT_1:22
.=
F . ((the_arity_of o) /. n)
by , , FINSEQ_4:def 4
;
thus u . e =
(F . ((the_arity_of o) /. n)) . (f . n)
by ,
.=
((F * (the_arity_of o)) .. f) . n
by , Def1, PRALG_1:def 17
.=
g . e
by , , , , PRALG_2:def 8
;
end;
hence
u = F # x
by , , , 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
Lemma103:
for S being non empty non void ManySortedSign
for U1, U2 being non-empty MSAlgebra of S
for H being ManySortedFunction of U1,U2 st H is_isomorphism U1,U2 holds
H "" is_homomorphism U2,U1
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
Lemma127:
for S being non empty non void ManySortedSign
for U1, U2 being non-empty MSAlgebra of S
for F being ManySortedFunction of U1,U2 st F is_homomorphism U1,U2 holds
F is ManySortedFunction of U1,(Image F)
theorem Th20: :: MSUALG_3:20
theorem Th21: :: MSUALG_3:21
Lemma135:
for S being non empty non void ManySortedSign
for U1, U2 being non-empty MSAlgebra of S
for U3 being non-empty MSSubAlgebra of U2
for F being ManySortedFunction of U1,U2
for G being ManySortedFunction of U1,U3 st F = G & G is_homomorphism U1,U3 holds
F is_homomorphism U1,U2
theorem Th22: :: MSUALG_3:22
theorem Th23: :: MSUALG_3:23
theorem Th24: :: MSUALG_3:24