Nuprl Lemma : functor-curry_wf
∀[A,B,C:SmallCategory]. (functor-curry(A;B) ∈ Functor(FUN(A × B;C);FUN(A;FUN(B;C))))
Proof
Definitions occuring in Statement :
functor-curry: functor-curry(A;B),
product-cat: A × B,
functor-cat: FUN(C1;C2),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
functor-curry: functor-curry(A;B),
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
top: Top,
so_apply: x[s],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
subtype_rel: A ⊆r B,
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
product-cat: A × B,
cat-ob: cat-ob(C),
so_apply: x[s1;s2;s3],
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
implies: P ⇒ Q,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
nat-trans: nat-trans(C;D;F;G),
trans-comp: t1 o t2,
identity-trans: identity-trans(C;D;F)
Lemmas referenced :
mk-functor_wf,
functor-cat_wf,
product-cat_wf,
functor_cat_ob_lemma,
istype-void,
functor-ob_wf,
ob_product_lemma,
cat-ob_wf,
functor-arrow_wf,
cat-id_wf,
subtype_rel_self,
cat-arrow_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cat-comp_wf,
functor-arrow-prod-comp,
iff_weakening_equal,
cat-comp-ident1,
functor-arrow-prod-id,
functor_cat_arrow_lemma,
mk-nat-trans_wf,
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
functor_cat_comp_lemma,
functor_cat_id_lemma,
trans_comp_ap_lemma,
ident_trans_ap_lemma,
small-category_wf,
cat-comp-ident2,
cat-functor_wf,
ap_mk_nat_trans_lemma,
nat-trans-equation,
nat-trans-assoc-equation,
cat-comp-assoc,
nat-trans-comp-equation,
nat-trans-assoc-comp-equation
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
lambdaEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
applyEquality,
independent_pairEquality,
universeIsType,
independent_isectElimination,
lambdaFormation_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
instantiate,
universeEquality,
productElimination,
equalityIstype,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination,
rename,
axiomEquality,
isectIsTypeImplies,
functionEquality,
functionIsType
Latex:
\mforall{}[A,B,C:SmallCategory]. (functor-curry(A;B) \mmember{} Functor(FUN(A \mtimes{} B;C);FUN(A;FUN(B;C))))
Date html generated:
2019_10_31-AM-07_24_35
Last ObjectModification:
2018_12_13-PM-03_03_44
Theory : small!categories
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