### Nuprl Lemma : poset_functor_extend_same

`∀C:SmallCategory. ∀I:Cname List. ∀L:name-morph(I;[]) ⟶ cat-ob(C). ∀E:i:nameset(I)`
`                                                                      ⟶ c:{c:name-morph(I;[])| (c i) = 0 ∈ ℕ2} `
`                                                                      ⟶ (cat-arrow(C) (L c) (L flip(c;i))).`
`∀x:cat-ob(poset-cat(I)).`
`  ∀[y:cat-ob(poset-cat(I))]`
`    poset_functor_extend(C;I;L;E;x;y) = (cat-id(C) (L x)) ∈ (cat-arrow(C) (L x) (L x)) `
`    supposing x = y ∈ cat-ob(poset-cat(I))`

Proof

Definitions occuring in Statement :  poset_functor_extend: `poset_functor_extend(C;I;L;E;c1;c2)` poset-cat: `poset-cat(J)` name-morph-flip: `flip(f;y)` name-morph: `name-morph(I;J)` nameset: `nameset(L)` coordinate_name: `Cname` cat-id: `cat-id(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` nil: `[]` list: `T List` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` squash: `↓T` prop: `ℙ` subtype_rel: `A ⊆r B` cat-ob: `cat-ob(C)` pi1: `fst(t)` poset-cat: `poset-cat(J)` name-morph: `name-morph(I;J)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` and: `P ∧ Q` implies: `P `` Q` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  equal_wf squash_wf true_wf cat-arrow_wf subtype_rel_self name-morph_wf nil_wf coordinate_name_wf poset_functor_extend_wf all_wf nameset_wf le_wf extd-nameset_subtype_int set_wf equal-wf-T-base int_seg_wf extd-nameset-nil name-morph-flip_wf cat-ob_wf list_wf small-category_wf le_reflexive and_wf poset-cat_wf extd-nameset_wf assert_wf isname_wf subtype_rel_wf cat-id_wf iff_weakening_equal poset_functor_extend_id
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation introduction cut applyEquality thin lambdaEquality sqequalHypSubstitution imageElimination extract_by_obid isectElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry universeEquality functionExtensionality because_Cache sqequalRule independent_isectElimination setElimination rename functionEquality natural_numberEquality baseClosed dependent_functionElimination hyp_replacement dependent_set_memberEquality independent_pairFormation applyLambdaEquality productElimination setEquality imageMemberEquality independent_functionElimination isect_memberEquality axiomEquality

Latex:
\mforall{}C:SmallCategory.  \mforall{}I:Cname  List.  \mforall{}L:name-morph(I;[])  {}\mrightarrow{}  cat-ob(C).  \mforall{}E:i:nameset(I)
{}\mrightarrow{}  c:\{c:name-morph(I;[])|
(c  i)  =  0\}
{}\mrightarrow{}  (cat-arrow(C)  (L  c)
(L  flip(c;i))).
\mforall{}x:cat-ob(poset-cat(I)).
\mforall{}[y:cat-ob(poset-cat(I))].  poset\_functor\_extend(C;I;L;E;x;y)  =  (cat-id(C)  (L  x))  supposing  x  =  y

Date html generated: 2017_10_05-AM-10_30_57
Last ObjectModification: 2017_07_28-AM-11_24_37

Theory : cubical!sets

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