let I be set ; :: thesis: for C being Category
for c being Object of C
for F being Function of I, the carrier' of C holds
( c is_a_product_wrt F iff c opp is_a_coproduct_wrt F opp )
let C be Category; :: thesis: for c being Object of C
for F being Function of I, the carrier' of C holds
( c is_a_product_wrt F iff c opp is_a_coproduct_wrt F opp )
let c be Object of C; :: thesis: for F being Function of I, the carrier' of C holds
( c is_a_product_wrt F iff c opp is_a_coproduct_wrt F opp )
let F be Function of I, the carrier' of C; :: thesis: ( c is_a_product_wrt F iff c opp is_a_coproduct_wrt F opp )
thus ( c is_a_product_wrt F implies c opp is_a_coproduct_wrt F opp ) :: thesis: ( c opp is_a_coproduct_wrt F opp implies c is_a_product_wrt F )
proof
assume A1: c is_a_product_wrt F ; :: thesis: c opp is_a_coproduct_wrt F opp
then F is Projections_family of c,I ;
hence F opp is Injections_family of c opp ,I by Th68; :: according to CAT_3:def 17 :: thesis: for d being Object of (C opp)
for F9 being Injections_family of d,I st doms (F opp) = doms F9 holds
ex h being Morphism of (C opp) st
( h in Hom ((c opp),d) & ( for k being Morphism of (C opp) st k in Hom ((c opp),d) holds
( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) iff h = k ) ) )
let d be Object of (C opp); :: thesis: for F9 being Injections_family of d,I st doms (F opp) = doms F9 holds
ex h being Morphism of (C opp) st
( h in Hom ((c opp),d) & ( for k being Morphism of (C opp) st k in Hom ((c opp),d) holds
( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) iff h = k ) ) )
let F9 be Injections_family of d,I; :: thesis: ( doms (F opp) = doms F9 implies ex h being Morphism of (C opp) st
( h in Hom ((c opp),d) & ( for k being Morphism of (C opp) st k in Hom ((c opp),d) holds
( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) iff h = k ) ) ) )
assume A2: doms (F opp) = doms F9 ; :: thesis: ex h being Morphism of (C opp) st
( h in Hom ((c opp),d) & ( for k being Morphism of (C opp) st k in Hom ((c opp),d) holds
( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) iff h = k ) ) )
reconsider oppF9 = opp F9 as Projections_family of opp d,I by Th69;
now :: thesis: for x being set st x in I holds
(cods F) /. x = (cods oppF9) /. x
let x be set ; :: thesis: ( x in I implies (cods F) /. x = (cods oppF9) /. x )
reconsider gg = F /. x as Morphism of dom (F /. x), cod (F /. x) by CAT_1:4;
A3: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A4: gg opp = (F /. x) opp by OPPCAT_1:def 6;
reconsider g9 = F9 /. x as Morphism of dom (F9 /. x), cod (F9 /. x) by CAT_1:4;
Hom ((dom g9),(cod g9)) <> {} by CAT_1:2;
then A5: g9 opp = (F9 /. x) opp by OPPCAT_1:def 6;
assume A6: x in I ; :: thesis: (cods F) /. x = (cods oppF9) /. x
hence (cods F) /. x = cod (F /. x) by Def2
.= dom (gg opp) by A3, OPPCAT_1:10
.= dom ((F opp) /. x) by A6, Def3, A4
.= (doms F9) /. x by A2, A6, Def1
.= dom (F9 /. x) by A6, Def1
.= cod (opp (F9 /. x)) by A5, OPPCAT_1:11
.= cod (oppF9 /. x) by A6, Def4
.= (cods oppF9) /. x by A6, Def2 ;
:: thesis: verum
end;
then consider h being Morphism of C such that
A7: h in Hom ((opp d),c) and
A8: for k being Morphism of C st k in Hom ((opp d),c) holds
( ( for x being set st x in I holds
(F /. x) (*) k = oppF9 /. x ) iff h = k ) by A1, Th1;
take h opp ; :: thesis: ( h opp in Hom ((c opp),d) & ( for k being Morphism of (C opp) st k in Hom ((c opp),d) holds
( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) iff h opp = k ) ) )
h in Hom ((c opp),((opp d) opp)) by A7, OPPCAT_1:5;
hence h opp in Hom ((c opp),d) ; :: thesis: for k being Morphism of (C opp) st k in Hom ((c opp),d) holds
( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) iff h opp = k )
let k be Morphism of (C opp); :: thesis: ( k in Hom ((c opp),d) implies ( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) iff h opp = k ) )
assume A9: k in Hom ((c opp),d) ; :: thesis: ( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) iff h opp = k )
then A10: opp k in Hom ((opp d),(opp (c opp))) by OPPCAT_1:6;
thus ( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ) implies h opp = k ) :: thesis: ( h opp = k implies for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x )
proof
assume A11: for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x ; :: thesis: h opp = k
now :: thesis: for x being set st x in I holds
oppF9 /. x = (F /. x) (*) (opp k)
let x be set ; :: thesis: ( x in I implies oppF9 /. x = (F /. x) (*) (opp k) )
assume A12: x in I ; :: thesis: oppF9 /. x = (F /. x) (*) (opp k)
reconsider gg = F /. x as Morphism of dom (F /. x), cod (F /. x) by CAT_1:4;
A13: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A14: gg opp = (F /. x) opp by OPPCAT_1:def 6;
F is Projections_family of c,I by A1;
then dom (F /. x) = c by A12, Th41;
then cod ((F /. x) opp) = c opp by A13, A14, OPPCAT_1:12;
then cod ((F opp) /. x) = c opp by A12, Def3;
then A15: dom k = cod ((F opp) /. x) by A9, CAT_1:1;
opp (k (*) ((F opp) /. x)) = opp (F9 /. x) by A11, A12;
hence oppF9 /. x = opp (k (*) ((F opp) /. x)) by A12, Def4
.= (opp ((F opp) /. x)) (*) (opp k) by A15, OPPCAT_1:18
.= (opp ((F /. x) opp)) (*) (opp k) by A12, Def3
.= (F /. x) (*) (opp k) ;
:: thesis: verum
end;
hence h opp = k by A8, A10; :: thesis: verum
end;
assume A16: h opp = k ; :: thesis: for x being set st x in I holds
k (*) ((F opp) /. x) = F9 /. x
let x be set ; :: thesis: ( x in I implies k (*) ((F opp) /. x) = F9 /. x )
assume A17: x in I ; :: thesis: k (*) ((F opp) /. x) = F9 /. x
F is Projections_family of c,I by A1;
then dom (F /. x) = c by A17, Th41;
then A18: cod (opp k) = dom (F /. x) by A10, CAT_1:1;
reconsider ff = opp k as Morphism of dom (opp k), cod (opp k) by CAT_1:4;
reconsider gg = F /. x as Morphism of cod (opp k), cod (F /. x) by A18, CAT_1:4;
A19: ( Hom ((dom (opp k)),(cod (opp k))) <> {} & Hom ((dom (F /. x)),(cod (F /. x))) <> {} ) by CAT_1:2;
then A20: ff opp = (opp k) opp by OPPCAT_1:def 6;
A21: gg opp = (F /. x) opp by A19, A18, OPPCAT_1:def 6;
(F /. x) (*) (opp k) = oppF9 /. x by A8, A10, A17, A16;
then ((opp k) opp) (*) ((F /. x) opp) = (oppF9 /. x) opp by A18, A20, A21, A19, OPPCAT_1:16;
hence k (*) ((F opp) /. x) = (oppF9 /. x) opp by A17, Def3
.= (opp (F9 /. x)) opp by A17, Def4
.= F9 /. x ;
:: thesis: verum
end;
assume A22: c opp is_a_coproduct_wrt F opp ; :: thesis: c is_a_product_wrt F
then F opp is Injections_family of c opp ,I ;
hence F is Projections_family of c,I by Th68; :: according to CAT_3:def 14 :: thesis: for b being Object of C
for F9 being Projections_family of b,I st cods F = 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 I holds
(F /. x) (*) k = F9 /. x ) iff h = k ) ) )
let d be Object of C; :: thesis: for F9 being Projections_family of d,I st cods F = cods F9 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
( ( for x being set st x in I holds
(F /. x) (*) k = F9 /. x ) iff h = k ) ) )
let F9 be Projections_family of d,I; :: thesis: ( cods F = cods F9 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
( ( for x being set st x in I holds
(F /. x) (*) k = F9 /. x ) iff h = k ) ) ) )
assume A23: cods F = cods F9 ; :: thesis: 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
( ( for x being set st x in I holds
(F /. x) (*) k = F9 /. x ) iff h = k ) ) )
A24: now :: thesis: for x being set st x in I holds
(doms (F opp)) /. x = (doms (F9 opp)) /. x
let x be set ; :: thesis: ( x in I implies (doms (F opp)) /. x = (doms (F9 opp)) /. x )
reconsider gg = F /. x as Morphism of dom (F /. x), cod (F /. x) by CAT_1:4;
A25: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A26: gg opp = (F /. x) opp by OPPCAT_1:def 6;
reconsider g9 = F9 /. x as Morphism of dom (F9 /. x), cod (F9 /. x) by CAT_1:4;
A27: Hom ((dom g9),(cod g9)) <> {} by CAT_1:2;
then A28: g9 opp = (F9 /. x) opp by OPPCAT_1:def 6;
assume A29: x in I ; :: thesis: (doms (F opp)) /. x = (doms (F9 opp)) /. x
hence (doms (F opp)) /. x = dom ((F opp) /. x) by Def1
.= dom (gg opp) by A29, Def3, A26
.= cod (F /. x) by A25, OPPCAT_1:10
.= (cods F9) /. x by A23, A29, Def2
.= cod (F9 /. x) by A29, Def2
.= dom (g9 opp) by A27, OPPCAT_1:10
.= dom ((F9 opp) /. x) by A29, Def3, A28
.= (doms (F9 opp)) /. x by A29, Def1 ;
:: thesis: verum
end;
reconsider F9opp = F9 opp as Injections_family of d opp ,I by Th68;
consider h being Morphism of (C opp) such that
A30: h in Hom ((c opp),(d opp)) and
A31: for k being Morphism of (C opp) st k in Hom ((c opp),(d opp)) holds
( ( for x being set st x in I holds
k (*) ((F opp) /. x) = F9opp /. x ) iff h = k ) by A22, A24, Th1;
take opp h ; :: thesis: ( opp h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( for x being set st x in I holds
(F /. x) (*) k = F9 /. x ) iff opp h = k ) ) )
opp h in Hom ((opp (d opp)),(opp (c opp))) by A30, OPPCAT_1:6;
hence opp h in Hom (d,c) ; :: thesis: for k being Morphism of C st k in Hom (d,c) holds
( ( for x being set st x in I holds
(F /. x) (*) k = F9 /. x ) iff opp h = k )
let k be Morphism of C; :: thesis: ( k in Hom (d,c) implies ( ( for x being set st x in I holds
(F /. x) (*) k = F9 /. x ) iff opp h = k ) )
assume A32: k in Hom (d,c) ; :: thesis: ( ( for x being set st x in I holds
(F /. x) (*) k = F9 /. x ) iff opp h = k )
then A33: k opp in Hom ((c opp),(d opp)) by OPPCAT_1:5;
thus ( ( for x being set st x in I holds
(F /. x) (*) k = F9 /. x ) implies opp h = k ) :: thesis: ( opp h = k implies for x being set st x in I holds
(F /. x) (*) k = F9 /. x )
proof
assume A34: for x being set st x in I holds
(F /. x) (*) k = F9 /. x ; :: thesis: opp h = k
now :: thesis: for x being set st x in I holds
F9opp /. x = (k opp) (*) ((F opp) /. x)
let x be set ; :: thesis: ( x in I implies F9opp /. x = (k opp) (*) ((F opp) /. x) )
assume A35: x in I ; :: thesis: F9opp /. x = (k opp) (*) ((F opp) /. x)
reconsider gg = F /. x as Morphism of dom (F /. x), cod (F /. x) by CAT_1:4;
A36: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A37: gg opp = (F /. x) opp by OPPCAT_1:def 6;
F opp is Injections_family of c opp ,I by A22;
then cod ((F opp) /. x) = c opp by A35, Th62;
then cod (gg opp) = c opp by A35, Def3, A37;
then dom (F /. x) = c by A36, OPPCAT_1:10;
then A38: cod k = dom (F /. x) by A32, CAT_1:1;
reconsider ff = k as Morphism of dom k, cod k by CAT_1:4;
reconsider gg = F /. x as Morphism of cod k, cod (F /. x) by A38, CAT_1:4;
A39: ( Hom ((dom k),(cod k)) <> {} & Hom ((dom (F /. x)),(cod (F /. x))) <> {} ) by CAT_1:2;
then A40: ff opp = k opp by OPPCAT_1:def 6;
A41: gg opp = (F /. x) opp by A39, A38, OPPCAT_1:def 6;
(F /. x) (*) k = F9 /. x by A34, A35;
then (k opp) (*) ((F /. x) opp) = (F9 /. x) opp by A38, A40, A41, A39, OPPCAT_1:16;
hence F9opp /. x = (k opp) (*) ((F /. x) opp) by A35, Def3
.= (k opp) (*) ((F opp) /. x) by A35, Def3 ;
:: thesis: verum
end;
hence opp h = k by A31, A33; :: thesis: verum
end;
assume A42: opp h = k ; :: thesis: for x being set st x in I holds
(F /. x) (*) k = F9 /. x
let x be set ; :: thesis: ( x in I implies (F /. x) (*) k = F9 /. x )
assume A43: x in I ; :: thesis: (F /. x) (*) k = F9 /. x
reconsider gg = F /. x as Morphism of dom (F /. x), cod (F /. x) by CAT_1:4;
A44: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A45: gg opp = (F /. x) opp by OPPCAT_1:def 6;
F opp is Injections_family of c opp ,I by A22;
then cod ((F opp) /. x) = c opp by A43, Th62;
then cod (gg opp) = c opp by A43, Def3, A45;
then dom (F /. x) = c by A44, OPPCAT_1:10;
then A46: cod k = dom (F /. x) by A32, CAT_1:1;
reconsider ff = k as Morphism of dom k, cod k by CAT_1:4;
reconsider gg = F /. x as Morphism of cod k, cod (F /. x) by A46, CAT_1:4;
A47: ( Hom ((dom k),(cod k)) <> {} & Hom ((dom (F /. x)),(cod (F /. x))) <> {} ) by CAT_1:2;
then A48: ff opp = k opp by OPPCAT_1:def 6;
A49: gg opp = (F /. x) opp by A47, A46, OPPCAT_1:def 6;
(k opp) (*) ((F opp) /. x) = F9opp /. x by A31, A33, A43, A42;
then (k opp) (*) ((F opp) /. x) = (F9 /. x) opp by A43, Def3;
hence F9 /. x = (k opp) (*) ((F /. x) opp) by A43, Def3
.= ((F /. x) (*) k) opp by A46, A48, A49, A47, OPPCAT_1:16
.= (F /. x) (*) k ;
:: thesis: verum