Nuprl Lemma : same-face-edge-arrows-commute1
∀[C:SmallCategory]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(C);I;J;x;i)].
∀f:name-morph(I;[]). ∀a,b:nameset(I).
∀[v:I-face(cubical-nerve(C);I)]
((cat-comp(C) nerve_box_label(box;f) nerve_box_label(box;flip(f;a)) nerve_box_label(box;flip(flip(f;a);b))
nerve_box_edge(box;f;a)
nerve_box_edge(box;flip(f;a);b))
= (cat-comp(C) nerve_box_label(box;f) nerve_box_label(box;flip(f;b)) nerve_box_label(box;flip(flip(f;b);a))
nerve_box_edge(box;f;b)
nerve_box_edge(box;flip(f;b);a))
∈ (cat-arrow(C) nerve_box_label(box;f) nerve_box_label(box;flip(flip(f;a);b)))) supposing
((v ∈ box) and
(¬(dimension(v) = b ∈ Cname)) and
(¬(dimension(v) = a ∈ Cname)) and
(¬(a = b ∈ nameset(I))) and
((f b) = 0 ∈ ℕ2) and
(((f a) = 0 ∈ ℕ2) ∧ (direction(v) = (f dimension(v)) ∈ ℕ2)))
supposing (∃j1∈J. ¬(j1 = a ∈ Cname)) ∧ (∃j2∈J. ¬(j2 = b ∈ Cname))
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),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
small-category: SmallCategory,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
nil: [],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
name-morph: name-morph(I;J),
int_seg: {i..j-},
open_box: open_box(X;I;J;x;i),
nameset: nameset(L),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
same-face-edge-arrows-commute0,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
nameset_wf,
top_wf,
equal_wf,
extd-nameset_subtype_int,
nil_wf,
coordinate_name_wf,
name-morph-flip_wf,
name-morph_wf,
int_seg_wf,
l_member_wf,
I-face_wf,
cubical-nerve_wf,
face-dimension_wf,
equal-wf-T-base,
extd-nameset-nil,
face-direction_wf,
l_exists_wf,
open_box_wf,
list_wf,
small-category_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
independent_isectElimination,
productElimination,
inlFormation,
independent_pairFormation,
productEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
sqequalRule,
intEquality,
setElimination,
rename,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
setEquality
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{}f:name-morph(I;[]). \mforall{}a,b:nameset(I).
\mforall{}[v:I-face(cubical-nerve(C);I)]
((cat-comp(C) nerve\_box\_label(box;f) nerve\_box\_label(box;flip(f;a))
nerve\_box\_label(box;flip(flip(f;a);b))
nerve\_box\_edge(box;f;a)
nerve\_box\_edge(box;flip(f;a);b))
= (cat-comp(C) nerve\_box\_label(box;f) nerve\_box\_label(box;flip(f;b))
nerve\_box\_label(box;flip(flip(f;b);a))
nerve\_box\_edge(box;f;b)
nerve\_box\_edge(box;flip(f;b);a))) supposing
((v \mmember{} box) and
(\mneg{}(dimension(v) = b)) and
(\mneg{}(dimension(v) = a)) and
(\mneg{}(a = b)) and
((f b) = 0) and
(((f a) = 0) \mwedge{} (direction(v) = (f dimension(v)))))
supposing (\mexists{}j1\mmember{}J. \mneg{}(j1 = a)) \mwedge{} (\mexists{}j2\mmember{}J. \mneg{}(j2 = b))
Date html generated:
2017_10_05-PM-03_38_47
Last ObjectModification:
2017_07_28-AM-11_26_02
Theory : cubical!sets
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