Section Def.
Variable G G' : GROUP.
Variable f : Hom G G'.
Definition surj_group_quo_ker :
Hom G (group_quo G (Ker f) (kernel_normal (G:=G) (G':=G') (f:=f))) :=
group_quo_surj (kernel_normal (f:=f)).
red in |- *.
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
intros y; exists y; try assumption.
simpl in |- *.
apply Trans with (Ap (sgroup_map (monoid_sgroup_hom f)) (monoid_unit G));
auto with algebra.
Qed.
Definition inj_coKer_group : Hom (coKer f:GROUP) G'.
apply (BUILD_HOM_GROUP (G:=coKer f) (G':=G') (ff:=inj_part (coKer f))).
simpl in |- *.
auto with algebra.
simpl in |- *.
auto with algebra.
simpl in |- *.
auto with algebra.
Defined.
exact (inj_part_injective (E:=G') (A:=coKer f)).
Qed.
Definition bij_group_quo_ker_coKer :
Hom (group_quo G (Ker f) (kernel_normal (G:=G) (G':=G') (f:=f)):GROUP)
(coKer f).
apply
(BUILD_HOM_GROUP
(G:=group_quo G (Ker f) (kernel_normal (G:=G) (G':=G') (f:=f)))
(G':=coKer f)
(ff:=fun x : G =>
Build_subtype (P:=image (sgroup_map (monoid_sgroup_hom f)) (full G))
(subtype_elt:=f x) (coKer_prop f x))).
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
simpl in |- *.
intros x y H'; try assumption.
apply
GROUP_reg_right
with (group_inverse _ (Ap (sgroup_map (monoid_sgroup_hom f)) y));
auto with algebra.
apply
Trans
with
(sgroup_law G' (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(Ap (sgroup_map (monoid_sgroup_hom f)) (group_inverse G y)));
auto with algebra.
apply
Trans
with
(Ap (sgroup_map (monoid_sgroup_hom f))
(sgroup_law G x (group_inverse G y))); auto with algebra.
apply Trans with (monoid_unit G'); auto with algebra.
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
simpl in |- *.
intros x y; try assumption.
exact (SGROUP_hom_prop f x y).
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
simpl in |- *.
auto with algebra.
Defined.
red in |- *.
split; [ try assumption | idtac ].
red in |- *.
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
simpl in |- *.
intros x y H'; try assumption.
apply
Trans
with
(sgroup_law G' (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(Ap (sgroup_map (monoid_sgroup_hom f)) (group_inverse G y)));
auto with algebra.
apply
Trans
with
(sgroup_law G' (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(group_inverse _ (Ap (sgroup_map (monoid_sgroup_hom f)) y)));
auto with algebra.
apply
Trans
with
(sgroup_law G' (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(group_inverse _ (Ap (sgroup_map (monoid_sgroup_hom f)) x)));
auto with algebra.
red in |- *.
intros y; try assumption.
elim y.
intros y' subtype_prf; try assumption.
elim subtype_prf.
intros x H'; try assumption.
elim H'; intros H'0 H'1; try exact H'0; clear H'.
exists x; try assumption.
Qed.

Theorem factor_group_hom :
Equal f
(comp_hom inj_coKer_group
(comp_hom bij_group_quo_ker_coKer surj_group_quo_ker)).