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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