let x1, x2 be set ; for C being Category
for c being Object of C
for p1, p2 being Morphism of C st x1 <> x2 holds
( c is_a_product_wrt p1,p2 iff c is_a_product_wrt (x1,x2) --> (p1,p2) )
let C be Category; for c being Object of C
for p1, p2 being Morphism of C st x1 <> x2 holds
( c is_a_product_wrt p1,p2 iff c is_a_product_wrt (x1,x2) --> (p1,p2) )
let c be Object of C; for p1, p2 being Morphism of C st x1 <> x2 holds
( c is_a_product_wrt p1,p2 iff c is_a_product_wrt (x1,x2) --> (p1,p2) )
let p1, p2 be Morphism of C; ( x1 <> x2 implies ( c is_a_product_wrt p1,p2 iff c is_a_product_wrt (x1,x2) --> (p1,p2) ) )
set F = (x1,x2) --> (p1,p2);
set I = {x1,x2};
assume A1:
x1 <> x2
; ( c is_a_product_wrt p1,p2 iff c is_a_product_wrt (x1,x2) --> (p1,p2) )
thus
( c is_a_product_wrt p1,p2 implies c is_a_product_wrt (x1,x2) --> (p1,p2) )
( c is_a_product_wrt (x1,x2) --> (p1,p2) implies c is_a_product_wrt p1,p2 )proof
assume A2:
c is_a_product_wrt p1,
p2
;
c is_a_product_wrt (x1,x2) --> (p1,p2)
then
(
dom p1 = c &
dom p2 = c )
;
hence
(
x1,
x2)
--> (
p1,
p2) is
Projections_family of
c,
{x1,x2}
by Th44;
CAT_3:def 14 for b being Object of C
for F9 being Projections_family of b,{x1,x2} st cods ((x1,x2) --> (p1,p2)) = cods F9 holds
ex h being Morphism of C st
( h in Hom (b,c) & ( for k being Morphism of C st k in Hom (b,c) holds
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k ) ) )
let b be
Object of
C;
for F9 being Projections_family of b,{x1,x2} st cods ((x1,x2) --> (p1,p2)) = cods F9 holds
ex h being Morphism of C st
( h in Hom (b,c) & ( for k being Morphism of C st k in Hom (b,c) holds
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k ) ) )let F9 be
Projections_family of
b,
{x1,x2};
( cods ((x1,x2) --> (p1,p2)) = cods F9 implies ex h being Morphism of C st
( h in Hom (b,c) & ( for k being Morphism of C st k in Hom (b,c) holds
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k ) ) ) )
assume A3:
cods ((x1,x2) --> (p1,p2)) = cods F9
;
ex h being Morphism of C st
( h in Hom (b,c) & ( for k being Morphism of C st k in Hom (b,c) holds
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k ) ) )
set f =
F9 /. x1;
set g =
F9 /. x2;
A4:
x1 in {x1,x2}
by TARSKI:def 2;
then
(cods ((x1,x2) --> (p1,p2))) /. x1 = cod (((x1,x2) --> (p1,p2)) /. x1)
by Def2;
then
cod (F9 /. x1) = cod (((x1,x2) --> (p1,p2)) /. x1)
by A3, A4, Def2;
then A5:
cod (F9 /. x1) = cod p1
by A1, Th3;
A6:
x2 in {x1,x2}
by TARSKI:def 2;
then
(cods ((x1,x2) --> (p1,p2))) /. x2 = cod (((x1,x2) --> (p1,p2)) /. x2)
by Def2;
then
cod (F9 /. x2) = cod (((x1,x2) --> (p1,p2)) /. x2)
by A3, A6, Def2;
then A7:
cod (F9 /. x2) = cod p2
by A1, Th3;
dom (F9 /. x2) = b
by A6, Th41;
then A8:
F9 /. x2 in Hom (
b,
(cod p2))
by A7;
dom (F9 /. x1) = b
by A4, Th41;
then
F9 /. x1 in Hom (
b,
(cod p1))
by A5;
then consider h being
Morphism of
C such that A9:
h in Hom (
b,
c)
and A10:
for
k being
Morphism of
C st
k in Hom (
b,
c) holds
( (
p1 (*) k = F9 /. x1 &
p2 (*) k = F9 /. x2 ) iff
h = k )
by A2, A8;
take
h
;
( h in Hom (b,c) & ( for k being Morphism of C st k in Hom (b,c) holds
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k ) ) )
thus
h in Hom (
b,
c)
by A9;
for k being Morphism of C st k in Hom (b,c) holds
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k )
let k be
Morphism of
C;
( k in Hom (b,c) implies ( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k ) )
assume A11:
k in Hom (
b,
c)
;
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k )
thus
( ( for
x being
set st
x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) implies
h = k )
( h = k implies for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x )proof
assume A12:
for
x being
set st
x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x
;
h = k
then
(((x1,x2) --> (p1,p2)) /. x2) (*) k = F9 /. x2
by A6;
then A13:
p2 (*) k = F9 /. x2
by A1, Th3;
(((x1,x2) --> (p1,p2)) /. x1) (*) k = F9 /. x1
by A4, A12;
then
p1 (*) k = F9 /. x1
by A1, Th3;
hence
h = k
by A10, A11, A13;
verum
end;
assume
h = k
;
for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x
then A14:
(
p1 (*) k = F9 /. x1 &
p2 (*) k = F9 /. x2 )
by A10, A11;
let x be
set ;
( x in {x1,x2} implies (((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x )
assume
x in {x1,x2}
;
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x
then
(
x = x1 or
x = x2 )
by TARSKI:def 2;
hence
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x
by A1, A14, Th3;
verum
end;
assume A15:
c is_a_product_wrt (x1,x2) --> (p1,p2)
; c is_a_product_wrt p1,p2
then A16:
(x1,x2) --> (p1,p2) is Projections_family of c,{x1,x2}
;
x2 in {x1,x2}
by TARSKI:def 2;
then A17:
dom (((x1,x2) --> (p1,p2)) /. x2) = c
by A16, Th41;
x1 in {x1,x2}
by TARSKI:def 2;
then
dom (((x1,x2) --> (p1,p2)) /. x1) = c
by A16, Th41;
hence
( dom p1 = c & dom p2 = c )
by A1, A17, Th3; CAT_3:def 15 for d being Object of C
for f, g being Morphism of C st f in Hom (d,(cod p1)) & g in Hom (d,(cod p2)) holds
ex h being Morphism of C st
( h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) ) )
let d be Object of C; for f, g being Morphism of C st f in Hom (d,(cod p1)) & g in Hom (d,(cod p2)) holds
ex h being Morphism of C st
( h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) ) )
let f, g be Morphism of C; ( f in Hom (d,(cod p1)) & g in Hom (d,(cod p2)) implies ex h being Morphism of C st
( h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) ) ) )
assume that
A18:
f in Hom (d,(cod p1))
and
A19:
g in Hom (d,(cod p2))
; ex h being Morphism of C st
( h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) ) )
( dom f = d & dom g = d )
by A18, A19, CAT_1:1;
then reconsider F9 = (x1,x2) --> (f,g) as Projections_family of d,{x1,x2} by Th44;
cods ((x1,x2) --> (p1,p2)) =
(x1,x2) --> ((cod p1),(cod p2))
by Th7
.=
(x1,x2) --> ((cod f),(cod p2))
by A18, CAT_1:1
.=
(x1,x2) --> ((cod f),(cod g))
by A19, CAT_1:1
.=
cods F9
by Th7
;
then consider h being Morphism of C such that
A20:
h in Hom (d,c)
and
A21:
for k being Morphism of C st k in Hom (d,c) holds
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k )
by A15;
take
h
; ( h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) ) )
thus
h in Hom (d,c)
by A20; for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k )
let k be Morphism of C; ( k in Hom (d,c) implies ( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) )
assume A22:
k in Hom (d,c)
; ( ( p1 (*) k = f & p2 (*) k = g ) iff h = k )
thus
( p1 (*) k = f & p2 (*) k = g implies h = k )
( h = k implies ( p1 (*) k = f & p2 (*) k = g ) )proof
assume A23:
(
p1 (*) k = f &
p2 (*) k = g )
;
h = k
now for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. xlet x be
set ;
( x in {x1,x2} implies (((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x )assume
x in {x1,x2}
;
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. xthen
(
x = x1 or
x = x2 )
by TARSKI:def 2;
then
( (
((x1,x2) --> (p1,p2)) /. x = p1 &
F9 /. x = f ) or (
((x1,x2) --> (p1,p2)) /. x = p2 &
F9 /. x = g ) )
by A1, Th3;
hence
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x
by A23;
verum end;
hence
h = k
by A21, A22;
verum
end;
assume A24:
h = k
; ( p1 (*) k = f & p2 (*) k = g )
x2 in {x1,x2}
by TARSKI:def 2;
then
(((x1,x2) --> (p1,p2)) /. x2) (*) k = F9 /. x2
by A21, A22, A24;
then A25:
(((x1,x2) --> (p1,p2)) /. x2) (*) k = g
by A1, Th3;
x1 in {x1,x2}
by TARSKI:def 2;
then
(((x1,x2) --> (p1,p2)) /. x1) (*) k = F9 /. x1
by A21, A22, A24;
then
(((x1,x2) --> (p1,p2)) /. x1) (*) k = f
by A1, Th3;
hence
( p1 (*) k = f & p2 (*) k = g )
by A1, A25, Th3; verum