Nuprl Lemma : nerve_box_label_same
∀[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;[])].
  ∀f:I-face(cubical-nerve(C);I)
    ((ob(cube(f)) L) = nerve_box_label(box;L) ∈ cat-ob(C)) supposing 
       ((f ∈ box) and 
       (direction(f) = (L dimension(f)) ∈ ℕ2)) 
  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)
, 
face-cube: cube(f)
, 
face-direction: direction(f)
, 
face-dimension: dimension(f)
, 
I-face: I-face(X;I)
, 
name-morph: name-morph(I;J)
, 
nameset: nameset(L)
, 
coordinate_name: Cname
, 
functor-ob: ob(F)
, 
cat-ob: cat-ob(C)
, 
small-category: SmallCategory
, 
l_member: (x ∈ l)
, 
null: null(as)
, 
nil: []
, 
list: T List
, 
int_seg: {i..j-}
, 
assert: ↑b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
or: P ∨ Q
, 
apply: f a
, 
natural_number: $n
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
nerve_box_label: nerve_box_label(box;L)
, 
subtype_rel: A ⊆r B
, 
name-morph: name-morph(I;J)
, 
cat-ob: cat-ob(C)
, 
pi1: fst(t)
, 
poset-cat: poset-cat(J)
, 
so_lambda: λ2x.t[x]
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
so_apply: x[s]
, 
and: P ∧ Q
, 
open_box: open_box(X;I;J;x;i)
, 
top: Top
, 
nameset: nameset(L)
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
not: ¬A
, 
I-face: I-face(X;I)
, 
face-dimension: dimension(f)
, 
face-direction: direction(f)
, 
pi2: snd(t)
, 
face-name: face-name(f)
, 
coordinate_name: Cname
, 
int_upper: {i...}
, 
sq_type: SQType(T)
, 
guard: {T}
, 
int_seg: {i..j-}
, 
false: False
, 
squash: ↓T
, 
true: True
, 
uiff: uiff(P;Q)
, 
cubical-nerve: cubical-nerve(X)
, 
cube-set-restriction: f(s)
, 
label: ...$L... t
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
poset-functor: poset-functor(J;K;f)
, 
functor-comp: functor-comp(F;G)
, 
so_lambda: so_lambda(x,y,z.t[x; y; z])
, 
so_apply: x[s1;s2;s3]
, 
cand: A c∧ B
, 
isname: isname(z)
, 
le_int: i ≤z j
, 
lt_int: i <z j
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
assert: ↑b
, 
bfalse: ff
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
lelt: i ≤ j < k
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
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, 
l_member_wf, 
int_seg_wf, 
face-direction_wf, 
face-dimension_wf, 
extd-nameset-nil, 
or_wf, 
not_wf, 
null_wf3, 
subtype_rel_list, 
top_wf, 
name-morph_wf, 
open_box_wf, 
list_wf, 
small-category_wf, 
decidable__equal-coordinate_name, 
pairwise-implies, 
face-name_wf, 
subtype_base_sq, 
set_subtype_base, 
le_wf, 
int_subtype_base, 
lelt_wf, 
face-cube_wf, 
squash_wf, 
true_wf, 
cubical-set_wf, 
cubical-nerve-I-cube, 
name-morph_subtype, 
list-diff_wf, 
cname_deq_wf, 
cons_wf, 
nameset_subtype, 
list-diff-subset, 
cat-functor_wf, 
poset-cat_wf, 
functor-ob_wf, 
cat-ob_wf, 
adjacent-compatible-iff, 
I-cube_wf, 
functor-comp_wf, 
poset-functor_wf, 
member_wf, 
iff_weakening_equal, 
face-map_wf2, 
subtype_rel_wf, 
list-diff2, 
list-diff2-sym, 
ob_mk_functor_lemma, 
ob_pair_lemma, 
cat_ob_pair_lemma, 
face-map-comp-id, 
decidable__equal_int, 
int_seg_properties, 
false_wf, 
int_seg_subtype, 
int_seg_cases, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformless_wf, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_wf, 
member-nameset-diff, 
intformeq_wf, 
intformnot_wf, 
int_formula_prop_eq_lemma, 
int_formula_prop_not_lemma, 
decidable__le, 
member_singleton
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lambdaFormation, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
because_Cache, 
applyEquality, 
sqequalRule, 
setEquality, 
functionEquality, 
hypothesis, 
lambdaEquality, 
functionExtensionality, 
independent_isectElimination, 
productEquality, 
setElimination, 
rename, 
natural_numberEquality, 
equalityTransitivity, 
equalitySymmetry, 
dependent_functionElimination, 
independent_functionElimination, 
isect_memberEquality, 
axiomEquality, 
voidElimination, 
voidEquality, 
productElimination, 
unionElimination, 
cumulativity, 
promote_hyp, 
instantiate, 
intEquality, 
independent_pairEquality, 
imageElimination, 
imageMemberEquality, 
baseClosed, 
hyp_replacement, 
applyLambdaEquality, 
universeEquality, 
independent_pairFormation, 
hypothesis_subsumption, 
addEquality, 
dependent_pairFormation, 
int_eqEquality, 
computeAll, 
dependent_set_memberEquality, 
addLevel, 
impliesFunctionality
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;[])].
    \mforall{}f:I-face(cubical-nerve(C);I)
        ((ob(cube(f))  L)  =  nerve\_box\_label(box;L))  supposing 
              ((f  \mmember{}  box)  and 
              (direction(f)  =  (L  dimension(f)))) 
    supposing  ((L  x)  =  i)  \mvee{}  (\mneg{}\muparrow{}null(J))
Date html generated:
2017_10_05-PM-03_37_07
Last ObjectModification:
2017_07_28-AM-11_25_29
Theory : cubical!sets
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