Nuprl Lemma : poset_functor_extend_id

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)).
  (poset_functor_extend(C;I;L;E;x;x) (cat-id(C) (L x)) ∈ (cat-arrow(C) (L x) (L x)))


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: List int_seg: {i..j-} all: x:A. B[x] set: {x:A| B[x]}  apply: a function: x:A ⟶ B[x] natural_number: $n equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T uall: [x:A]. B[x] name-morph: name-morph(I;J) subtype_rel: A ⊆B prop: poset-cat: poset-cat(J) cat-ob: cat-ob(C) pi1: fst(t) poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2) nameset: nameset(L) uimplies: supposing a nil: [] it: ifthenelse: if then else fi  null: null(as) btrue: tt cat-id: cat-id(C) pi2: snd(t) l_all: (∀x∈L.P[x]) guard: {T} int_seg: {i..j-} lelt: i ≤ j < k and: P ∧ Q decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False implies:  Q not: ¬A top: Top less_than: a < b squash: T iff: ⇐⇒ Q uiff: uiff(P;Q) rev_implies:  Q
Lemmas referenced :  cat-ob_wf poset-cat_wf nameset_wf name-morph_wf nil_wf coordinate_name_wf equal-wf-T-base int_seg_wf extd-nameset-nil cat-arrow_wf name-morph-flip_wf list_wf small-category_wf filter_is_nil band_wf eq_int_wf extd-nameset_subtype_int l_member_wf list-subtype cat-id_wf select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma extd-nameset_wf equal_wf length_wf intformeq_wf int_formula_prop_eq_lemma assert_wf not_wf iff_transitivity iff_weakening_uiff assert_of_band assert_of_eq_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality functionEquality setEquality natural_numberEquality applyEquality setElimination rename sqequalRule baseClosed functionExtensionality because_Cache lambdaEquality equalityTransitivity equalitySymmetry independent_isectElimination callbyvalueReduce sqleReflexivity productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination productEquality addLevel impliesFunctionality

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)).
    (poset\_functor\_extend(C;I;L;E;x;x)  =  (cat-id(C)  (L  x)))



Date html generated: 2017_10_05-AM-10_30_51
Last ObjectModification: 2017_07_28-AM-11_24_32

Theory : cubical!sets


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