Nuprl Lemma : free-group-hom
∀[X:Type]. ∀G:Group{i}. ∀f:X ⟶ |G|. ∃F:MonHom(free-group(X),G). ∀x:X. ((F free-letter(x)) = (f x) ∈ |G|)
Proof
Definitions occuring in Statement :
free-letter: free-letter(x),
free-group: free-group(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
monoid_hom: MonHom(M1,M2),
grp: Group{i},
grp_car: |g|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
grp: Group{i},
mon: Mon,
so_lambda: λ2x.t[x],
monoid_hom: MonHom(M1,M2),
free-group: free-group(X),
grp_car: |g|,
pi1: fst(t),
so_apply: x[s],
prop: ℙ,
free-letter: free-letter(x),
fg-lift: fg-lift(G;f),
fg-hom: fg-hom(G;f;w),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
imon: IMonoid,
and: P ∧ Q
Lemmas referenced :
fg-lift_wf,
monoid_hom_wf,
free-group_wf,
all_wf,
equal_wf,
grp_car_wf,
free-letter_wf,
free-word_wf,
grp_wf,
list_accum_cons_lemma,
list_accum_nil_lemma,
mon_ident,
grp_sig_wf,
monoid_p_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
dependent_functionElimination,
functionExtensionality,
applyEquality,
hypothesis,
lambdaEquality,
setElimination,
rename,
setEquality,
because_Cache,
sqequalRule,
functionEquality,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination
Latex:
\mforall{}[X:Type]. \mforall{}G:Group\{i\}. \mforall{}f:X {}\mrightarrow{} |G|. \mexists{}F:MonHom(free-group(X),G). \mforall{}x:X. ((F free-letter(x)) = (f x))
Date html generated:
2020_05_20-AM-08_23_13
Last ObjectModification:
2017_01_16-AM-00_26_54
Theory : free!groups
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