Nuprl Lemma : nerve-box-common-face_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:cat-ob(poset-cat(I))]. ∀[z:nameset(I)].
  nerve-box-common-face(box;L;z) ∈ {f:I-face(cubical-nerve(C);I)| 
                                    (f ∈ box)
                                    ∧ (direction(f) = (L dimension(f)) ∈ ℕ2)
                                    ∧ (direction(f) = (flip(L;z) dimension(f)) ∈ ℕ2)}  
  supposing (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname))) ∨ (((L x) = i ∈ ℕ2) ∧ (¬↑null(J)))
Proof
Definitions occuring in Statement : 
nerve-box-common-face: nerve-box-common-face(box;L;z)
, 
cubical-nerve: cubical-nerve(X)
, 
poset-cat: poset-cat(J)
, 
open_box: open_box(X;I;J;x;i)
, 
face-direction: direction(f)
, 
face-dimension: dimension(f)
, 
I-face: I-face(X;I)
, 
name-morph-flip: flip(f;y)
, 
nameset: nameset(L)
, 
coordinate_name: Cname
, 
cat-ob: cat-ob(C)
, 
small-category: SmallCategory
, 
l_exists: (∃x∈L. P[x])
, 
l_member: (x ∈ l)
, 
null: null(as)
, 
list: T List
, 
int_seg: {i..j-}
, 
assert: ↑b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
not: ¬A
, 
or: P ∨ Q
, 
and: P ∧ Q
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
apply: f a
, 
natural_number: $n
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
poset-cat: poset-cat(J)
, 
cat-ob: cat-ob(C)
, 
pi1: fst(t)
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
name-morph: name-morph(I;J)
, 
so_lambda: λ2x.t[x]
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
nameset: nameset(L)
, 
and: P ∧ Q
, 
top: Top
, 
or: P ∨ Q
, 
iff: P 
⇐⇒ Q
, 
exists: ∃x:A. B[x]
, 
rev_implies: P 
⇐ Q
, 
decidable: Dec(P)
, 
not: ¬A
, 
false: False
, 
guard: {T}
Lemmas referenced : 
nerve-box-common-face_wf2, 
subtype_rel_self, 
nameset_wf, 
extd-nameset_wf, 
nil_wf, 
coordinate_name_wf, 
all_wf, 
assert_wf, 
isname_wf, 
equal_wf, 
or_wf, 
l_exists_wf, 
l_member_wf, 
not_wf, 
int_seg_wf, 
extd-nameset-nil, 
null_wf3, 
subtype_rel_list, 
top_wf, 
name-morph_wf, 
open_box_wf, 
cubical-nerve_wf, 
list_wf, 
small-category_wf, 
l_exists_iff, 
exists_wf, 
decidable__equal-coordinate_name
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
applyEquality, 
setEquality, 
functionEquality, 
because_Cache, 
hypothesis, 
lambdaEquality, 
functionExtensionality, 
independent_isectElimination, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
lambdaFormation, 
setElimination, 
rename, 
dependent_functionElimination, 
productEquality, 
natural_numberEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
unionElimination, 
inlFormation, 
productElimination, 
independent_functionElimination, 
addLevel, 
existsFunctionality, 
independent_pairFormation, 
andLevelFunctionality, 
promote_hyp, 
dependent_pairFormation, 
inrFormation
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:cat-ob(poset-cat(I))].  \mforall{}[z:nameset(I)].
    nerve-box-common-face(box;L;z)  \mmember{}  \{f:I-face(cubical-nerve(C);I)| 
                                                                        (f  \mmember{}  box)
                                                                        \mwedge{}  (direction(f)  =  (L  dimension(f)))
                                                                        \mwedge{}  (direction(f)  =  (flip(L;z)  dimension(f)))\}   
    supposing  (\mexists{}j1\mmember{}J.  (\mexists{}j2\mmember{}J.  \mneg{}(j1  =  j2)))  \mvee{}  (((L  x)  =  i)  \mwedge{}  (\mneg{}\muparrow{}null(J)))
Date html generated:
2017_10_05-PM-03_36_52
Last ObjectModification:
2017_07_28-AM-11_25_24
Theory : cubical!sets
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