Nuprl Lemma : functor-arrow-prod-id
∀[A,B,C:SmallCategory]. ∀[F:Functor(A × B;C)]. ∀[a:cat-ob(A)]. ∀[b:cat-ob(B)].
((arrow(F) <a, b> <a, b> <cat-id(A) a, cat-id(B) b>) = (cat-id(C) (ob(F) <a, b>)) ∈ (cat-arrow(C) (ob(F) <a, b>) (ob(F\000C) <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,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
top: Top,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
cat-arrow_wf,
functor-ob_wf,
product-cat_wf,
ob_product_lemma,
functor-arrow-id,
cat-id_wf,
iff_weakening_equal,
cat-ob_wf,
cat-functor_wf,
small-category_wf,
id_prod_cat_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[A,B,C:SmallCategory]. \mforall{}[F:Functor(A \mtimes{} B;C)]. \mforall{}[a:cat-ob(A)]. \mforall{}[b:cat-ob(B)].
((arrow(F) <a, b> <a, b> <cat-id(A) a, cat-id(B) b>) = (cat-id(C) (ob(F) <a, b>)))
Date html generated:
2017_10_05-AM-00_47_57
Last ObjectModification:
2017_07_28-AM-09_19_51
Theory : small!categories
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