Nuprl Lemma : functor-uncurry_wf
∀[A,B,C:SmallCategory]. (functor-uncurry(C) ∈ Functor(FUN(A;FUN(B;C));FUN(A × B;C)))
Proof
Definitions occuring in Statement :
functor-uncurry: functor-uncurry(C),
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,
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
pi1: fst(t),
subtype_rel: A ⊆r B,
cat-ob: cat-ob(C),
functor-cat: FUN(C1;C2),
cat-functor: Functor(C1;C2),
pi2: snd(t),
so_apply: x[s],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
nat-trans: nat-trans(C;D;F;G),
uimplies: b supposing a,
so_apply: x[s1;s2;s3],
functor-uncurry: functor-uncurry(C),
trans-comp: t1 o t2,
squash: ↓T,
prop: ℙ,
cat-arrow: cat-arrow(C),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
identity-trans: identity-trans(C;D;F)
Lemmas referenced :
functor_cat_ob_lemma,
istype-void,
mk-functor_wf,
product-cat_wf,
functor-ob_wf,
functor-cat_wf,
ob_product_lemma,
subtype_rel_self,
cat-functor_wf,
cat-ob_wf,
cat-comp_wf,
functor-arrow_wf,
arrow_prod_lemma,
functor_cat_arrow_lemma,
cat-arrow_wf,
pi1_wf_top,
subtype_rel_product,
top_wf,
pi2_wf,
mk-nat-trans_wf,
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
functor_cat_comp_lemma,
trans_comp_ap_lemma,
squash_wf,
all_wf,
equal_wf,
comp_product_cat_lemma,
id_prod_cat_lemma,
ap_mk_nat_trans_lemma,
functor_cat_id_lemma,
ident_trans_ap_lemma,
small-category_wf,
nat-trans_wf,
cat-id_wf,
true_wf,
istype-universe,
functor-arrow-comp,
cat-comp-assoc,
nat-trans-equation,
nat-trans-comp-equation,
nat-trans-assoc-comp-equation,
nat-trans-assoc-equation,
iff_weakening_equal,
functor-arrow-id,
cat-comp-ident2,
cat-comp-ident1,
trans-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
isectElimination,
hypothesisEquality,
lambdaEquality_alt,
applyEquality,
because_Cache,
productElimination,
universeIsType,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productIsType,
independent_isectElimination,
imageElimination,
functionIsType,
imageMemberEquality,
baseClosed,
axiomEquality,
isectIsTypeImplies,
natural_numberEquality,
instantiate,
universeEquality,
independent_functionElimination,
applyLambdaEquality
Latex:
\mforall{}[A,B,C:SmallCategory]. (functor-uncurry(C) \mmember{} Functor(FUN(A;FUN(B;C));FUN(A \mtimes{} B;C)))
Date html generated:
2019_10_31-AM-07_24_44
Last ObjectModification:
2018_12_13-AM-10_03_58
Theory : small!categories
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