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finished some lemma

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This commit is contained in:
Egbert Rijke 2016-12-08 14:16:40 -05:00
parent 4ea75446ba
commit 8d586d587b
2 changed files with 14 additions and 11 deletions
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20
algebra/quotient_group.hlean
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@ -142,21 +142,19 @@ namespace group
definition qg_map [constructor] : G →g quotient_group N := definition qg_map [constructor] : G →g quotient_group N :=
homomorphism.mk class_of (λ g h, idp) homomorphism.mk class_of (λ g h, idp)
definition ab_gq_map {G : AbGroup} (N : subgroup_rel G) : G →g quotient_ab_group N := definition ab_qg_map {G : AbGroup} (N : subgroup_rel G) : G →g quotient_ab_group N :=
begin begin
fapply homomorphism.mk, fapply homomorphism.mk,
exact class_of, exact class_of,
exact λ g h, idp exact λ g h, idp
end end
definition surjective_ab_gq_map {A : AbGroup} (N : subgroup_rel A) : is_surjective (ab_gq_map N) := definition surjective_ab_qg_map {A : AbGroup} (N : subgroup_rel A) : is_surjective (ab_qg_map N) :=
begin begin
intro x, intro x, induction x,
induction x,
fapply image.mk, fapply image.mk,
exact a, exact a, reflexivity,
reflexivity, apply is_prop.elimo
sorry --Floris please help
end end
namespace quotient namespace quotient
@ -176,7 +174,7 @@ namespace group
unfold quotient_rel, rewrite e, exact H unfold quotient_rel, rewrite e, exact H
end end
definition ab_gq_map_eq_one {K : subgroup_rel A} (g :A) (H : K g) : ab_gq_map K g = 1 := definition ab_qg_map_eq_one {K : subgroup_rel A} (g :A) (H : K g) : ab_qg_map K g = 1 :=
begin begin
apply eq_of_rel, apply eq_of_rel,
have e : (g * 1⁻¹ = g), have e : (g * 1⁻¹ = g),
@ -257,7 +255,7 @@ definition kernel_quotient_extension {A B : AbGroup} (f : A →g B) : quotient_a
end end
definition kernel_quotient_extension_triangle {A B : AbGroup} (f : A →g B) : definition kernel_quotient_extension_triangle {A B : AbGroup} (f : A →g B) :
kernel_quotient_extension f ∘g ab_gq_map (kernel_subgroup f) ~ f := kernel_quotient_extension f ∘g ab_qg_map (kernel_subgroup f) ~ f :=
begin begin
intro a, intro a,
apply quotient_group_compute apply quotient_group_compute
@ -273,10 +271,10 @@ definition ab_group_quotient_homomorphism (A B : AbGroup)(K : subgroup_rel A)(L
(p : Π(a:A), K(a) → L(f a)) : quotient_ab_group K →g quotient_ab_group L := (p : Π(a:A), K(a) → L(f a)) : quotient_ab_group K →g quotient_ab_group L :=
begin begin
fapply quotient_group_elim, fapply quotient_group_elim,
exact (ab_gq_map L) ∘g f, exact (ab_qg_map L) ∘g f,
intro a, intro a,
intro k, intro k,
exact @ab_gq_map_eq_one B L (f a) (p a k), exact @ab_qg_map_eq_one B L (f a) (p a k),
end end
definition ab_group_kernel_factor {A B C: AbGroup} (f : A →g B)(g : A →g C){i : C →g B}(H : f = i ∘g g ) definition ab_group_kernel_factor {A B C: AbGroup} (f : A →g B)(g : A →g C){i : C →g B}(H : f = i ∘g g )

5
move_to_lib.hlean
View file

@ -130,3 +130,8 @@ namespace sphere
-- end -- end
end sphere end sphere
definition image_pathover {A B : Type} (f : A → B) {x y : B} (p : x = y) (u : image f x) (v : image f y) : u =[p] v :=
begin
apply is_prop.elimo
end