Nuprl Lemma : real-subset-connected-lemma
∀X:ℝ ⟶ ℙ
(dense-in-interval((-∞, ∞);X)
⇒ (∀a,b:{x:ℝ| X x} ⟶ 𝔹.
((∃x:{x:ℝ| X x} . (↑(a x)))
⇒ (∃x:{x:ℝ| X x} . (↑(b x)))
⇒ (∀x:{x:ℝ| X x} . ((↑(a x)) ∨ (↑(b x))))
⇒ (∃f,g:ℕ ⟶ {x:ℝ| X x}
∃x:ℝ. ((∀n:ℕ. (↑(a (f n)))) ∧ (∀n:ℕ. (↑(b (g n)))) ∧ lim n→∞.f n = x ∧ lim n→∞.g n = x)))))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
riiint: (-∞, ∞),
converges-to: lim n→∞.x[n] = y,
real: ℝ,
nat: ℕ,
assert: ↑b,
bool: 𝔹,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
dense-in-interval: dense-in-interval(I;X),
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
rgt: x > y,
pi1: fst(t),
pi2: snd(t),
sq_stable: SqStable(P),
nat: ℕ,
le: A ≤ B,
real: ℝ,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
label: ...$L... t,
rbetween: x≤y≤z,
rleq: x ≤ y,
rnonneg: rnonneg(x),
int_upper: {i...}
Lemmas referenced :
dense-in-interval_wf,
riiint_wf,
subtype_rel_dep_function,
real_wf,
i-member_wf,
member_riiint_lemma,
subtype_rel_self,
set_wf,
rless_wf,
assert_wf,
all_wf,
or_wf,
bool_wf,
rdiv_wf,
radd_wf,
rmul_wf,
int-to-real_wf,
rless-int,
true_wf,
rmul_preserves_rless,
rmul_preserves_rleq,
rsub_wf,
rleq_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
equal-wf-base,
nequal_wf,
rless-implies-rless,
req-iff-rsub-is-0,
rless_functionality,
req_transitivity,
radd_functionality,
rmul-rinv3,
int-rinv-cancel,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rleq_functionality,
rless_transitivity2,
rleq_weakening_rless,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rless1,
rleq_weakening_equal,
rleq_weakening,
radd-preserves-rleq,
sq_stable__rless,
pi1_wf_top,
pi2_wf,
equal_wf,
exists_wf,
subtype_rel_product,
top_wf,
primrec_wf,
int_seg_wf,
nat_wf,
primrec0_lemma,
false_wf,
le_wf,
sq_stable__less_than,
nat_properties,
nat_plus_wf,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
sq_stable__assert,
add-subtract-cancel,
bool_subtype_base,
squash_wf,
eq_int_eq_false,
bfalse_wf,
iff_weakening_equal,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
not_functionality_wrt_uiff,
less_than_wf,
intformless_wf,
int_formula_prop_less_lemma,
ifthenelse_wf,
equal_functionality_wrt_subtype_rel2,
rless_transitivity1,
real-regular,
and_wf,
regular-int-seq_wf,
rmul_preserves_rleq2,
rleq-int,
rless_functionality_wrt_implies,
common-limit-squeeze,
rnexp_wf,
rnexp-converges-ext,
rabs_wf,
rleq-int-fractions2,
rless-int-fractions3,
rabs-of-nonneg,
req_weakening,
rmul-limit,
constant-limit,
converges-to_functionality,
rmul_comm,
rmul-zero,
ge_wf,
less_than'_wf,
subtract_wf,
int_term_value_subtract_lemma,
exp0_lemma,
rsub_functionality_wrt_rleq,
rminus_wf,
itermMinus_wf,
rnexp_zero_lemma,
real_term_value_minus_lemma,
exp-nondecreasing,
subtract-add-cancel,
rbetween_wf,
rnexp_step,
rless-int-fractions2,
rleq-implies-rleq,
rmul_functionality,
converges-to_wf,
rless-cases,
rneq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
applyEquality,
instantiate,
cumulativity,
sqequalRule,
lambdaEquality,
universeEquality,
setEquality,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
functionEquality,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
inrFormation,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
productEquality,
intEquality,
unionElimination,
approximateComputation,
equalityTransitivity,
equalitySymmetry,
minusEquality,
int_eqEquality,
imageElimination,
independent_pairEquality,
functionExtensionality,
inlFormation,
promote_hyp,
addEquality,
baseApply,
closedConclusion,
equalityElimination,
applyLambdaEquality,
multiplyEquality,
intWeakElimination,
axiomEquality,
hyp_replacement
Latex:
\mforall{}X:\mBbbR{} {}\mrightarrow{} \mBbbP{}
(dense-in-interval((-\minfty{}, \minfty{});X)
{}\mRightarrow{} (\mforall{}a,b:\{x:\mBbbR{}| X x\} {}\mrightarrow{} \mBbbB{}.
((\mexists{}x:\{x:\mBbbR{}| X x\} . (\muparrow{}(a x)))
{}\mRightarrow{} (\mexists{}x:\{x:\mBbbR{}| X x\} . (\muparrow{}(b x)))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| X x\} . ((\muparrow{}(a x)) \mvee{} (\muparrow{}(b x))))
{}\mRightarrow{} (\mexists{}f,g:\mBbbN{} {}\mrightarrow{} \{x:\mBbbR{}| X x\}
\mexists{}x:\mBbbR{}
((\mforall{}n:\mBbbN{}. (\muparrow{}(a (f n)))) \mwedge{} (\mforall{}n:\mBbbN{}. (\muparrow{}(b (g n)))) \mwedge{} lim n\mrightarrow{}\minfty{}.f n = x \mwedge{} lim n\mrightarrow{}\minfty{}.g n = x)))))
Date html generated:
2019_10_30-AM-07_21_21
Last ObjectModification:
2018_08_31-AM-10_00_27
Theory : reals
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