### Nuprl Lemma : poset-cat-dist-zero

`∀[I:Cname List]. ∀[x,y:cat-ob(poset-cat(I))].`
`  uiff(x = y ∈ cat-ob(poset-cat(I));poset-cat-dist(I;x;y) ≤ 0) supposing cat-arrow(poset-cat(I)) x y`

Proof

Definitions occuring in Statement :  poset-cat-dist: `poset-cat-dist(I;x;y)` poset-cat: `poset-cat(J)` coordinate_name: `Cname` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` list: `T List` uiff: `uiff(P;Q)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` apply: `f a` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` uiff: `uiff(P;Q)` and: `P ∧ Q` poset-cat: `poset-cat(J)` cat-ob: `cat-ob(C)` pi1: `fst(t)` name-morph: `name-morph(I;J)` cat-arrow: `cat-arrow(C)` pi2: `snd(t)` poset-cat-dist: `poset-cat-dist(I;x;y)` le: `A ≤ B` not: `¬A` implies: `P `` Q` false: `False` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` nameset: `nameset(L)` less_than': `less_than'(a;b)` l_all: `(∀x∈L.P[x])` squash: `↓T` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` less_than: `a < b` sq_type: `SQType(T)` eq_int: `(i =z j)` band: `p ∧b q` ifthenelse: `if b then t else f fi ` btrue: `tt` assert: `↑b` bfalse: `ff` bool: `𝔹` unit: `Unit` it: `⋅` cand: `A c∧ B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_stable: `SqStable(P)` l_member: `(x ∈ l)` coordinate_name: `Cname` int_upper: `{i...}` nat: `ℕ` cons: `[a / b]` ge: `i ≥ j ` true: `True`
Lemmas referenced :  extd-nameset-nil less_than'_wf poset-cat-dist_wf equal_wf cat-ob_wf poset-cat_wf nameset_wf all_wf assert_wf isname_wf nil_wf coordinate_name_wf extd-nameset_wf le_wf cat-arrow_wf list_wf list-subtype filter_is_nil band_wf eq_int_wf extd-nameset_subtype_int length_of_nil_lemma false_wf int_seg_wf length_wf subtype_base_sq extd-nameset_subtype_base 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 decidable__equal_int int_subtype_base int_seg_subtype int_seg_cases bool_wf eqtt_to_assert assert_of_eq_int assert_of_le_int equal-wf-T-base decidable__cand member_filter iff_transitivity iff_weakening_uiff assert_of_band sq_stable__l_member decidable__equal-coordinate_name set_subtype_base select_member lelt_wf filter_wf5 l_member_wf list-cases null_nil_lemma btrue_wf member-implies-null-eq-bfalse btrue_neq_bfalse product_subtype_list length_of_cons_lemma cons_wf non_neg_length itermAdd_wf int_term_value_add_lemma intformeq_wf int_formula_prop_eq_lemma not_wf equal-wf-base nsub2_subtype_extd-nameset
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation extract_by_obid sqequalHypSubstitution sqequalRule setElimination thin rename productElimination independent_pairEquality lambdaEquality dependent_functionElimination hypothesisEquality because_Cache isectElimination natural_numberEquality hypothesis applyEquality axiomEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality functionExtensionality functionEquality isect_memberEquality voidElimination independent_isectElimination lambdaFormation applyLambdaEquality imageMemberEquality baseClosed imageElimination instantiate cumulativity unionElimination dependent_pairFormation int_eqEquality intEquality voidEquality computeAll independent_functionElimination hypothesis_subsumption addEquality equalityElimination productEquality setEquality promote_hyp

Latex:
\mforall{}[I:Cname  List].  \mforall{}[x,y:cat-ob(poset-cat(I))].
uiff(x  =  y;poset-cat-dist(I;x;y)  \mleq{}  0)  supposing  cat-arrow(poset-cat(I))  x  y

Date html generated: 2017_10_05-AM-10_28_41
Last ObjectModification: 2017_07_28-AM-11_23_45

Theory : cubical!sets

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