Nuprl Lemma : functor-arrow-prod-id
∀[A,B,C:SmallCategory]. ∀[F:Functor(A × B;C)]. ∀[a:cat-ob(A)]. ∀[b:cat-ob(B)].
((F <a, b> <a, b> <cat-id(A) a, cat-id(B) b>) = (cat-id(C) (F <a, b>)) ∈ (cat-arrow(C) (F <a, b>) (F <a, b>)))
Proof
Definitions occuring in Statement :
product-cat: A × B,
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
apply: f a,
pair: <a, b>,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
cat-ob: cat-ob(C),
pi1: fst(t),
product-cat: A × B,
pi2: snd(t)
Lemmas referenced :
functor-arrow-id,
product-cat_wf,
cat-ob_wf,
cat-functor_wf,
small-category_wf,
id_prod_cat_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
independent_pairEquality,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
Error :memTop
Latex:
\mforall{}[A,B,C:SmallCategory]. \mforall{}[F:Functor(A \mtimes{} B;C)]. \mforall{}[a:cat-ob(A)]. \mforall{}[b:cat-ob(B)].
((F <a, b> <a, b> <cat-id(A) a, cat-id(B) b>) = (cat-id(C) (F <a, b>)))
Date html generated:
2020_05_20-AM-07_54_15
Last ObjectModification:
2019_12_30-PM-03_22_13
Theory : small!categories
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