Nuprl Lemma : groupoids_wf
Groupoids ∈ SmallCategory'
Proof
Definitions occuring in Statement :
groupoids: Groupoids,
small-category: SmallCategory,
member: t ∈ T
Definitions unfolded in proof :
groupoids: Groupoids,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: so_lambda(x,y,z,w,v.t[x; y; z; w; v]),
so_apply: x[s1;s2;s3;s4;s5],
uimplies: b supposing a,
groupoid-map: groupoid-map(G;H),
all: ∀x:A. B[x],
id_functor: 1,
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
functor-comp: functor-comp(F;G),
true: True,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
mk-cat_wf,
groupoid_wf,
groupoid-map_wf,
id_functor_wf,
groupoid-cat_wf,
ob_mk_functor_lemma,
istype-void,
arrow_mk_functor_lemma,
groupoid-inv_wf,
cat-arrow_wf,
cat-ob_wf,
functor-ob_wf,
functor-arrow_wf,
functor-comp_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
functor-comp-id,
functor-comp-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality_alt,
cumulativity,
hypothesisEquality,
inhabitedIsType,
universeIsType,
independent_isectElimination,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
applyEquality,
functionIsType,
because_Cache,
equalityIstype,
setElimination,
rename,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
independent_pairFormation
Latex:
Groupoids \mmember{} SmallCategory'
Date html generated:
2019_10_31-AM-07_25_00
Last ObjectModification:
2019_05_07-PM-10_26_46
Theory : small!categories
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