Nuprl Lemma : poset-cat-arrow-iff
∀[I:Cname List]. ∀[x,y:cat-ob(poset-cat(I))]. uiff(cat-arrow(poset-cat(I)) x y;{∀i:nameset(I). ((x i) ≤ (y i))})
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
nameset: nameset(L),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
guard: {T},
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a
Definitions unfolded in proof :
guard: {T},
poset-cat: poset-cat(J),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
assert_of_le_int,
nameset_wf,
less_than'_wf,
all_wf,
assert_wf,
le_int_wf,
assert_witness,
le_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
lemma_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
because_Cache,
productElimination,
independent_isectElimination,
lambdaEquality,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
isect_memberEquality,
voidElimination
Latex:
\mforall{}[I:Cname List]. \mforall{}[x,y:cat-ob(poset-cat(I))].
uiff(cat-arrow(poset-cat(I)) x y;\{\mforall{}i:nameset(I). ((x i) \mleq{} (y i))\})
Date html generated:
2016_06_16-PM-06_52_36
Last ObjectModification:
2015_12_28-PM-04_22_50
Theory : cubical!sets
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