Nuprl Lemma : nerve_box_edge'_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)]. ∀[y:nameset(I)]. ∀[c:{c:name-morph(I;[])| (c y) = 0 ∈ ℕ2} ].
nerve_box_edge'(box; c; y) ∈ cat-arrow(C) nerve_box_label(box;c) nerve_box_label(box;flip(c;y))
supposing (∃j∈J. ¬(j = y ∈ Cname)) ∨ (((c x) = i ∈ ℕ2) ∧ (¬↑null(J)))
Proof
Definitions occuring in Statement :
nerve_box_edge': nerve_box_edge'(box; c; y),
nerve_box_label: nerve_box_label(box;L),
cubical-nerve: cubical-nerve(X),
open_box: open_box(X;I;J;x;i),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
small-category: SmallCategory,
l_exists: (∃x∈L. P[x]),
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,
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 :
nerve_box_edge': nerve_box_edge'(box; c; y)
Lemmas referenced :
nerve_box_edge_wf2
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
hypothesis
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{}[y:nameset(I)]. \mforall{}[c:\{c:name-morph(I;[])| (c y) = 0\} ].
nerve\_box\_edge'(box; c; y) \mmember{} cat-arrow(C) nerve\_box\_label(box;c) nerve\_box\_label(box;flip(c;y))
supposing (\mexists{}j\mmember{}J. \mneg{}(j = y)) \mvee{} (((c x) = i) \mwedge{} (\mneg{}\muparrow{}null(J)))
Date html generated:
2016_06_16-PM-07_04_22
Last ObjectModification:
2015_12_28-PM-04_17_13
Theory : cubical!sets
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