Nuprl Lemma : poset-functor-comp
∀[I,J,K:Cname List]. ∀[f:name-morph(I;J)]. ∀[g:name-morph(J;K)].
(poset-functor(I;K;(f o g))
= functor-comp(poset-functor(J;K;g);poset-functor(I;J;f))
∈ Functor(poset-cat(K);poset-cat(I)))
Proof
Definitions occuring in Statement :
poset-functor: poset-functor(J;K;f),
poset-cat: poset-cat(J),
name-comp: (f o g),
name-morph: name-morph(I;J),
coordinate_name: Cname,
functor-comp: functor-comp(F;G),
cat-functor: Functor(C1;C2),
list: T List,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
functor-arrow: arrow(F),
pi2: snd(t),
functor-comp: functor-comp(F;G),
mk-functor: mk-functor,
poset-functor: poset-functor(J;K;f),
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],
poset-cat: poset-cat(J),
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 :
equal-functors,
poset-cat_wf,
poset-functor_wf,
name-comp_wf,
functor-comp_wf,
cat-ob_wf,
functor-arrow_wf,
cat-arrow_wf,
name-morph_wf,
list_wf,
coordinate_name_wf,
ob_pair_lemma,
ob_mk_functor_lemma,
cat_ob_pair_lemma,
equal_wf,
squash_wf,
true_wf,
nil_wf,
name-comp-assoc,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
lambdaFormation,
sqequalRule,
applyEquality,
because_Cache,
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[I,J,K:Cname List]. \mforall{}[f:name-morph(I;J)]. \mforall{}[g:name-morph(J;K)].
(poset-functor(I;K;(f o g)) = functor-comp(poset-functor(J;K;g);poset-functor(I;J;f)))
Date html generated:
2017_10_05-AM-10_29_06
Last ObjectModification:
2017_07_28-AM-11_24_03
Theory : cubical!sets
Home
Index