Nuprl Lemma : real-subset-connected
∀X:ℝ ⟶ ℙ
((∀x:ℝ. SqStable(X x))
⇒ (∀x:ℝ. ∀y:{y:ℝ| x = y} . ((X y) ⇒ (X x)))
⇒ dense-in-interval((-∞, ∞);X)
⇒ (∀Q:{x:ℝ| X x} ⟶ 𝔹. ∃Q':ℝ ⟶ 𝔹. ∀x:{x:ℝ| X x} . Q' x = Q x)
⇒ Connected({x:ℝ| X x} ))
Proof
Definitions occuring in Statement :
connected: Connected(X),
dense-in-interval: dense-in-interval(I;X),
riiint: (-∞, ∞),
req: x = y,
real: ℝ,
bool: 𝔹,
sq_stable: SqStable(P),
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
istype: istype(T),
or: P ∨ Q,
and: P ∧ Q,
guard: {T},
nat_plus: ℕ+,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
real: ℝ,
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
accelerate: accelerate(k;f),
int_seg: {i..j-},
has-value: (a)↓,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
pi1: fst(t),
sq_exists: ∃x:A [B[x]],
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
connected: Connected(X),
bfalse: ff,
isl: isl(x),
isr: isr(x),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q)
Lemmas referenced :
real_wf,
subtype_rel_self,
bool_wf,
dense-in-interval_wf,
riiint_wf,
subtype_rel_dep_function,
i-member_wf,
member_riiint_lemma,
istype-void,
req_wf,
sq_stable_wf,
istype-assert,
real-subset-connected-lemma,
dense-in-reals-implies,
istype-nat,
converges-to_functionality,
req_weakening,
req_inversion,
connectedness-main-lemma-ext,
nat_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
accelerate_wf,
subtype_rel_sets_simple,
nat_plus_wf,
regular-int-seq_wf,
real-regular,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat_plus,
istype-false,
accelerate-req,
value-type-has-value,
int-value-type,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
subtype_rel_set,
weak-continuity-principle-real-double,
btrue_wf,
iff_imp_equal_bool,
iff_weakening_equal,
istype-universe,
true_wf,
squash_wf,
equal_wf,
bool_subtype_base,
sq_stable__req,
isr_wf,
isl_wf,
outl_wf,
assert_of_bnot,
outr_wf,
bfalse_wf,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
functionIsType,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
universeEquality,
productIsType,
because_Cache,
equalityIstype,
setElimination,
rename,
cumulativity,
lambdaEquality_alt,
setEquality,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
isect_memberFormation_alt,
unionIsType,
inhabitedIsType,
dependent_set_memberEquality_alt,
independent_functionElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
multiplyEquality,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
functionEquality,
intEquality,
callbyvalueReduce,
sqleReflexivity,
divideEquality,
baseClosed,
sqequalBase,
imageElimination,
applyLambdaEquality,
closedConclusion,
promote_hyp,
inrFormation_alt,
inlFormation_alt,
imageMemberEquality,
unionEquality,
inrEquality_alt,
voidEquality
Latex:
\mforall{}X:\mBbbR{} {}\mrightarrow{} \mBbbP{}
((\mforall{}x:\mBbbR{}. SqStable(X x))
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. \mforall{}y:\{y:\mBbbR{}| x = y\} . ((X y) {}\mRightarrow{} (X x)))
{}\mRightarrow{} dense-in-interval((-\minfty{}, \minfty{});X)
{}\mRightarrow{} (\mforall{}Q:\{x:\mBbbR{}| X x\} {}\mrightarrow{} \mBbbB{}. \mexists{}Q':\mBbbR{} {}\mrightarrow{} \mBbbB{}. \mforall{}x:\{x:\mBbbR{}| X x\} . Q' x = Q x)
{}\mRightarrow{} Connected(\{x:\mBbbR{}| X x\} ))
Date html generated:
2019_10_30-AM-07_35_42
Last ObjectModification:
2019_01_31-AM-10_42_28
Theory : reals
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