Nuprl Lemma : functor-arrow-comp
∀[C,D:SmallCategory]. ∀[F:Functor(C;D)]. ∀[x,y,z:cat-ob(C)]. ∀[f:cat-arrow(C) x y]. ∀[g:cat-arrow(C) y z].
((functor-arrow(F) x z (cat-comp(C) x y z f g))
= (cat-comp(D) (functor-ob(F) x) (functor-ob(F) y) (functor-ob(F) z) (functor-arrow(F) x y f)
(functor-arrow(F) y z g))
∈ (cat-arrow(D) (functor-ob(F) x) (functor-ob(F) z)))
Proof
Definitions occuring in Statement :
functor-arrow: functor-arrow(F),
functor-ob: functor-ob(F),
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
top: Top,
all: ∀x:A. B[x],
mk-functor: mk-functor(ob;arrow),
and: P ∧ Q,
cat-functor: Functor(C1;C2),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-functor_wf,
cat-ob_wf,
cat-arrow_wf,
functor_arrow_pair_lemma,
functor_ob_pair_lemma
Rules used in proof :
because_Cache,
axiomEquality,
isectElimination,
applyEquality,
hypothesisEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
sqequalRule,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F:Functor(C;D)]. \mforall{}[x,y,z:cat-ob(C)]. \mforall{}[f:cat-arrow(C) x y].
\mforall{}[g:cat-arrow(C) y z].
((functor-arrow(F) x z (cat-comp(C) x y z f g))
= (cat-comp(D) (functor-ob(F) x) (functor-ob(F) y) (functor-ob(F) z) (functor-arrow(F) x y f)
(functor-arrow(F) y z g)))
Date html generated:
2017_01_11-AM-09_17_59
Last ObjectModification:
2017_01_10-PM-00_33_29
Theory : small!categories
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