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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