Nuprl Lemma : fg-hom-append
∀[X:Type]. ∀[G:Group{i}]. ∀[f:X ⟶ |G|]. ∀[a,b:(X + X) List].
(fg-hom(G;f;a @ b) = (fg-hom(G;f;a) * fg-hom(G;f;b)) ∈ |G|)
Proof
Definitions occuring in Statement :
fg-hom: fg-hom(G;f;w),
append: as @ bs,
list: T List,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T,
grp: Group{i},
grp_op: *,
grp_car: |g|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fg-hom: fg-hom(G;f;w),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
grp: Group{i},
mon: Mon,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
imon: IMonoid,
and: P ∧ Q,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_accum_append,
subtype_rel_list,
top_wf,
list_accum_wf,
grp_car_wf,
grp_id_wf,
grp_op_wf,
grp_inv_wf,
equal_wf,
list_wf,
grp_wf,
list_induction,
all_wf,
list_accum_nil_lemma,
mon_ident,
grp_sig_wf,
monoid_p_wf,
inverse_wf,
list_accum_cons_lemma,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
mon_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
unionEquality,
hypothesis,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
functionEquality,
universeEquality,
setEquality,
cumulativity,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate
Latex:
\mforall{}[X:Type]. \mforall{}[G:Group\{i\}]. \mforall{}[f:X {}\mrightarrow{} |G|]. \mforall{}[a,b:(X + X) List].
(fg-hom(G;f;a @ b) = (fg-hom(G;f;a) * fg-hom(G;f;b)))
Date html generated:
2019_10_31-AM-07_23_49
Last ObjectModification:
2018_08_21-PM-02_02_33
Theory : free!groups
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