let I be set ; :: thesis: for C being Category
for c, d being Object of C
for F being Projections_family of c,I
for F9 being Projections_family of d,I st c is_a_product_wrt F & d is_a_product_wrt F9 & cods F = cods F9 holds
c,d are_isomorphic
let C be Category; :: thesis: for c, d being Object of C
for F being Projections_family of c,I
for F9 being Projections_family of d,I st c is_a_product_wrt F & d is_a_product_wrt F9 & cods F = cods F9 holds
c,d are_isomorphic
let c, d be Object of C; :: thesis: for F being Projections_family of c,I
for F9 being Projections_family of d,I st c is_a_product_wrt F & d is_a_product_wrt F9 & cods F = cods F9 holds
c,d are_isomorphic
let F be Projections_family of c,I; :: thesis: for F9 being Projections_family of d,I st c is_a_product_wrt F & d is_a_product_wrt F9 & cods F = cods F9 holds
c,d are_isomorphic
let F9 be Projections_family of d,I; :: thesis: ( c is_a_product_wrt F & d is_a_product_wrt F9 & cods F = cods F9 implies c,d are_isomorphic )
assume that
A1: c is_a_product_wrt F and
A2: d is_a_product_wrt F9 and
A3: cods F = cods F9 ; :: thesis: c,d are_isomorphic
consider gf being Morphism of C such that
gf in Hom (d,d) and
A4: for k being Morphism of C st k in Hom (d,d) holds
( ( for x being set st x in I holds
(F9 /. x) (*) k = F9 /. x ) iff gf = k ) by A2;
consider f being Morphism of C such that
A5: f in Hom (d,c) and
A6: 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 f = k ) by A1, A3;
reconsider f = f as Morphism of d,c by A5, CAT_1:def 5;
consider fg being Morphism of C such that
fg in Hom (c,c) and
A7: for k being Morphism of C st k in Hom (c,c) holds
( ( for x being set st x in I holds
(F /. x) (*) k = F /. x ) iff fg = k ) by A1;
consider g being Morphism of C such that
A8: g in Hom (c,d) and
A9: for k being Morphism of C st k in Hom (c,d) holds
( ( for x being set st x in I holds
(F9 /. x) (*) k = F /. x ) iff g = k ) by A2, A3;
reconsider g = g as Morphism of c,d by A8, CAT_1:def 5;
A10: cod g = d by A8, CAT_1:1;
take g ; :: according to CAT_1:def 20 :: thesis: g is invertible
thus ( Hom (c,d) <> {} & Hom (d,c) <> {} ) by A5, A8; :: according to CAT_1:def 16 :: thesis: ex b₁ being Morphism of d,c st
( g * b₁ = id d & b₁ * g = id c )
take f ; :: thesis: ( g * f = id d & f * g = id c )
A13: dom g = c by A8, CAT_1:1;
A14: cod f = c by A5, CAT_1:1;
A15: dom f = d by A5, CAT_1:1;
thus g * f = g (*) f by A5, A8, CAT_1:def 13
.= gf by A4, A16
.= id d by A4, A12 ; :: thesis: f * g = id c
thus f * g = f (*) g by A5, A8, CAT_1:def 13
.= fg by A7, A18
.= id c by A11, A7 ; :: thesis: verum