Nuprl Lemma : functor-comp_wf
∀[A,B,C:SmallCategory]. ∀[F:Functor(A;B)]. ∀[G:Functor(B;C)]. (functor-comp(F;G) ∈ Functor(A;C))
Proof
Definitions occuring in Statement :
functor-comp: functor-comp(F;G),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
functor-comp: functor-comp(F;G),
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
mk-functor_wf,
functor-ob_wf,
cat-ob_wf,
functor-arrow_wf,
cat-arrow_wf,
equal_wf,
squash_wf,
true_wf,
functor-arrow-comp,
cat-comp_wf,
iff_weakening_equal,
functor-arrow-id,
cat-id_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
because_Cache,
hypothesis,
independent_isectElimination,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[A,B,C:SmallCategory]. \mforall{}[F:Functor(A;B)]. \mforall{}[G:Functor(B;C)]. (functor-comp(F;G) \mmember{} Functor(A;C))
Date html generated:
2017_10_05-AM-00_47_23
Last ObjectModification:
2017_07_28-AM-09_19_40
Theory : small!categories
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