Nuprl Lemma : nerve_box_label_wf
∀[C:SmallCategory]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(C);I;J;x;i)]. ∀[L:name-morph(I;[])].
nerve_box_label(box;L) ∈ cat-ob(C) supposing ((L x) = i ∈ ℕ2) ∨ (¬↑null(J))
Proof
Definitions occuring in Statement :
nerve_box_label: nerve_box_label(box;L),
cubical-nerve: cubical-nerve(X),
open_box: open_box(X;I;J;x;i),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
cat-ob: cat-ob(C),
small-category: SmallCategory,
null: null(as),
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
or: P ∨ Q,
member: t ∈ T,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nerve_box_label: nerve_box_label(box;L),
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
open_box: open_box(X;I;J;x;i),
and: P ∧ Q,
I-face: I-face(X;I),
face-cube: cube(f),
pi2: snd(t),
top: Top,
nameset: nameset(L)
Lemmas referenced :
nerve-box-face_wf,
subtype_rel_self,
nameset_wf,
extd-nameset_wf,
nil_wf,
coordinate_name_wf,
all_wf,
assert_wf,
isname_wf,
equal_wf,
set_wf,
I-face_wf,
cubical-nerve_wf,
and_wf,
l_member_wf,
int_seg_wf,
face-direction_wf,
face-dimension_wf,
cubical-nerve-I-cube,
or_wf,
extd-nameset-nil,
not_wf,
null_wf3,
subtype_rel_list,
top_wf,
name-morph_wf,
open_box_wf,
list_wf,
small-category_wf,
functor-ob_wf,
poset-cat_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
name-morph_subtype,
nameset_subtype,
list-diff-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
setEquality,
functionEquality,
hypothesis,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
because_Cache,
lambdaFormation,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(C);I;J;x;i)]. \mforall{}[L:name-morph(I;[])].
nerve\_box\_label(box;L) \mmember{} cat-ob(C) supposing ((L x) = i) \mvee{} (\mneg{}\muparrow{}null(J))
Date html generated:
2016_06_16-PM-07_03_02
Last ObjectModification:
2015_12_28-PM-04_18_02
Theory : cubical!sets
Home
Index