Nuprl Lemma : partition-choice-member
∀I:Interval
(icompact(I)
⇒ (∀p:partition(I). ∀x:partition-choice(full-partition(I;p)). ∀i:ℕ||full-partition(I;p)|| - 1. (x i ∈ {x:ℝ| x ∈ I} )\000C))
Proof
Definitions occuring in Statement :
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
real: ℝ,
length: ||as||,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
partition-choice: partition-choice(p),
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
int_seg: {i..j-},
guard: {T},
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,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
so_apply: x[s],
sq_stable: SqStable(P),
i-member: r ∈ I,
rccint: [l, u],
le: A ≤ B,
subtract: n - m,
l_all: (∀x∈L.P[x])
Lemmas referenced :
add-member-int_seg2,
lelt_wf,
full-partition-point-member,
i-member-between,
sq_stable__i-member,
interval_wf,
icompact_wf,
partition_wf,
partition-choice_wf,
int_seg_wf,
equal_wf,
int_term_value_add_lemma,
itermAdd_wf,
false_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
intformless_wf,
subtract-is-int-iff,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
subtract_wf,
int_seg_properties,
full-partition_wf,
select_wf,
rccint_wf,
i-member_wf,
real_wf,
set_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
dependent_set_memberEquality,
applyEquality,
hypothesisEquality,
sqequalRule,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
addEquality,
independent_functionElimination,
introduction,
imageMemberEquality
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}p:partition(I). \mforall{}x:partition-choice(full-partition(I;p)). \mforall{}i:\mBbbN{}||full-partition(I;p)|| - 1.
(x i \mmember{} \{x:\mBbbR{}| x \mmember{} I\} )))
Date html generated:
2016_05_18-AM-09_04_16
Last ObjectModification:
2016_01_17-AM-02_34_31
Theory : reals
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