Nuprl Lemma : cantor-to-interval-onto-proper
∀a,b:ℝ. ∀x:ℝ. ((a ≤ x) ⇒ (x ≤ b) ⇒ (∃f:ℕ ⟶ 𝔹. (cantor-to-interval(a;b;f) = x))) supposing a < b
Proof
Definitions occuring in Statement :
cantor-to-interval: cantor-to-interval(a;b;f),
rleq: x ≤ y,
rless: x < y,
req: x = y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
pi1: fst(t),
cand: A c∧ B,
cantor-interval: cantor-interval(a;b;f;n),
primrec: primrec(n;b;c),
pi2: snd(t),
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
less_than: a < b
Lemmas referenced :
cantor-to-interval-onto-lemma,
i-member_wf,
rccint_wf,
cantor-interval_wf,
int_seg_wf,
real_wf,
pi1_wf_top,
equal_wf,
pi2_wf,
bool_wf,
nat_wf,
member_rccint_lemma,
all_wf,
exists_wf,
nat_properties,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
subtype_rel_product,
top_wf,
set_wf,
rleq_wf,
subtype_rel_set,
rless_wf,
primrec-wf,
bfalse_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
sq_stable__i-member,
int_seg_subtype_nat,
ge_wf,
less_than_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
primrec-unroll,
bool_subtype_base,
squash_wf,
true_wf,
eq_int_eq_false,
subtract-add-cancel,
equal-wf-base,
iff_weakening_equal,
sq_stable__req,
cantor-to-interval_wf,
rleq_weakening_rless,
cantor-to-interval-req,
req_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
promote_hyp,
rename,
dependent_pairFormation,
lambdaEquality,
isectElimination,
functionExtensionality,
applyEquality,
natural_numberEquality,
setElimination,
productEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
sqequalRule,
functionEquality,
setEquality,
addEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
applyLambdaEquality,
hyp_replacement,
intWeakElimination,
axiomEquality,
instantiate,
cumulativity,
universeEquality,
baseApply,
closedConclusion
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}x:\mBbbR{}. ((a \mleq{} x) {}\mRightarrow{} (x \mleq{} b) {}\mRightarrow{} (\mexists{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (cantor-to-interval(a;b;f) = x))) supposing a < b
Date html generated:
2017_10_03-AM-09_55_25
Last ObjectModification:
2017_07_28-AM-08_04_22
Theory : reals
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