Nuprl Lemma : fg-lift_wf
∀[X:Type]
∀G:Group{i}. ∀f:X ⟶ |G|. (fg-lift(G;f) ∈ {F:MonHom(free-group(X),G)| ∀x:X. ((F free-letter(x)) = (f x) ∈ |G|)} )
Proof
Definitions occuring in Statement :
fg-lift: fg-lift(G;f),
free-letter: free-letter(x),
free-group: free-group(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
set: {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],
member: t ∈ T,
all: ∀x:A. B[x],
grp: Group{i},
mon: Mon,
fg-lift: fg-lift(G;f),
monoid_hom: MonHom(M1,M2),
free-group: free-group(X),
grp_car: |g|,
pi1: fst(t),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
grp_op: *,
pi2: snd(t),
infix_ap: x f y,
prop: ℙ,
free-word: free-word(X),
implies: P ⇒ Q,
cand: A c∧ B,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
squash: ↓T,
true: True,
guard: {T},
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
free-append: w + w',
free-letter: free-letter(x),
fg-hom: fg-hom(G;f;w),
top: Top,
imon: IMonoid,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
grp_car_wf,
grp_wf,
fg-hom_wf,
free-group_wf,
grp_hom_formation,
grp_subtype_igrp,
monoid_hom_p_wf,
word-equiv-equiv,
list_wf,
word-equiv_wf,
equal-wf-base,
equal_wf,
squash_wf,
true_wf,
free-append_wf,
subtype_quotient,
fg-hom-append,
iff_weakening_equal,
free-word_wf,
quotient-member-eq,
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,
all_wf,
free-letter_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
dependent_set_memberEquality,
sqequalHypSubstitution,
hypothesis,
functionEquality,
cumulativity,
hypothesisEquality,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
universeEquality,
functionExtensionality,
applyEquality,
independent_isectElimination,
unionEquality,
promote_hyp,
independent_pairFormation,
pointwiseFunctionality,
pertypeElimination,
productElimination,
independent_functionElimination,
productEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality
Latex:
\mforall{}[X:Type]
\mforall{}G:Group\{i\}. \mforall{}f:X {}\mrightarrow{} |G|.
(fg-lift(G;f) \mmember{} \{F:MonHom(free-group(X),G)| \mforall{}x:X. ((F free-letter(x)) = (f x))\} )
Date html generated:
2020_05_20-AM-08_23_11
Last ObjectModification:
2017_07_28-AM-09_18_54
Theory : free!groups
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