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].
((F x z (cat-comp(C) x y z f g)) = (cat-comp(D) (F x) (F y) (F z) (F x y f) (F y z g)) ∈ (cat-arrow(D) (F x) (F z)))
Proof
Definitions occuring in Statement :
functor-arrow: arrow(F),
functor-ob: 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 :
uall: ∀[x:A]. B[x],
cat-functor: Functor(C1;C2),
and: P ∧ Q,
all: ∀x:A. B[x],
member: t ∈ T
Lemmas referenced :
ob_pair_lemma,
arrow_pair_lemma,
cat-arrow_wf,
cat-ob_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
hypothesisEquality,
universeIsType,
applyEquality,
isectElimination,
because_Cache,
inhabitedIsType
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].
((F x z (cat-comp(C) x y z f g)) = (cat-comp(D) (F x) (F y) (F z) (F x y f) (F y z g)))
Date html generated:
2020_05_20-AM-07_51_05
Last ObjectModification:
2019_12_30-PM-02_10_27
Theory : small!categories
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