Nuprl Lemma : functor-comp-id
∀[A,B:SmallCategory]. ∀[F:Functor(A;B)].
((functor-comp(F;1) = F ∈ Functor(A;B)) ∧ (functor-comp(1;F) = F ∈ Functor(A;B)))
Proof
Definitions occuring in Statement :
id_functor: 1,
functor-comp: functor-comp(F;G),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
id_functor: 1,
functor-comp: functor-comp(F;G),
all: ∀x:A. B[x],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal-functors,
functor-comp_wf,
id_functor_wf,
ob_mk_functor_lemma,
functor-ob_wf,
cat-ob_wf,
arrow_mk_functor_lemma,
functor-arrow_wf,
cat-arrow_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
applyEquality,
because_Cache,
independent_pairFormation,
productElimination,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}[A,B:SmallCategory]. \mforall{}[F:Functor(A;B)]. ((functor-comp(F;1) = F) \mwedge{} (functor-comp(1;F) = F))
Date html generated:
2017_01_19-PM-02_53_40
Last ObjectModification:
2017_01_13-PM-01_28_24
Theory : small!categories
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