Nuprl Lemma : fg-hom_wf
∀[X:Type]. ∀[G:Group{i}]. ∀[f:X ⟶ |G|]. ∀[w:free-word(X)]. (fg-hom(G;f;w) ∈ |G|)
Proof
Definitions occuring in Statement :
fg-hom: fg-hom(G;f;w),
free-word: free-word(X),
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
grp: Group{i},
grp_car: |g|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
grp: Group{i},
mon: Mon,
all: ∀x:A. B[x],
implies: P ⇒ Q,
transitive-reflexive-closure: R^*,
or: P ∨ Q,
prop: ℙ,
fg-hom: fg-hom(G;f;w),
squash: ↓T,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
true: True,
so_apply: x[s1;s2],
guard: {T},
rel_implies: R1 => R2,
word-rel: word-rel(X;w1;w2),
exists: ∃x:A. B[x],
and: P ∧ Q,
label: ...$L... t,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
inverse-letters: a = -b,
top: Top,
imon: IMonoid,
trans: Trans(T;x,y.E[x; y]),
free-word: free-word(X),
cand: A c∧ B,
word-equiv: word-equiv(X;w1;w2),
quotient: x,y:A//B[x; y],
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
free-word_wf,
grp_car_wf,
grp_wf,
transitive-reflexive-closure_wf,
list_wf,
word-rel_wf,
list_accum_wf,
squash_wf,
true_wf,
grp_id_wf,
grp_op_wf,
grp_inv_wf,
equal_wf,
transitive-closure-minimal,
member_wf,
and_wf,
append_wf,
cons_wf,
nil_wf,
subtype_rel_self,
iff_weakening_equal,
fg-hom-append,
list_accum_cons_lemma,
list_accum_nil_lemma,
grp_sig_wf,
monoid_p_wf,
inverse_wf,
grp_subtype_igrp,
mon_assoc,
grp_inverse,
mon_ident,
word-equiv_wf,
word-equiv-equiv,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
functionEquality,
setElimination,
rename,
universeEquality,
lambdaFormation,
unionElimination,
applyEquality,
unionEquality,
lambdaEquality,
imageElimination,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
instantiate,
independent_isectElimination,
voidElimination,
voidEquality,
setEquality,
cumulativity,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
productEquality
Latex:
\mforall{}[X:Type]. \mforall{}[G:Group\{i\}]. \mforall{}[f:X {}\mrightarrow{} |G|]. \mforall{}[w:free-word(X)]. (fg-hom(G;f;w) \mmember{} |G|)
Date html generated:
2019_10_31-AM-07_23_53
Last ObjectModification:
2018_08_21-PM-02_02_49
Theory : free!groups
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