Nuprl Lemma : integer-bound
∀x:ℝ. ∃n:ℕ+. (|x| ≤ r(n))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
real: ℝ,
nat: ℕ,
nat_plus: ℕ+,
int_upper: {i...},
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
and: P ∧ Q,
le: A ≤ B,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
canonical-bound_wf,
rabs_wf,
subtype_rel_set,
int_upper_wf,
nat_plus_wf,
le_wf,
absval_wf,
istype-int_upper,
subtype_rel_sets_simple,
istype-int,
less_than_wf,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
istype-void,
zero-add,
le-add-cancel,
istype-le,
canonical-bound-property,
rleq_wf,
int-to-real_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
dependent_pairFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
natural_numberEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
independent_isectElimination,
intEquality,
independent_pairFormation,
productElimination,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_functionElimination,
isect_memberEquality_alt,
universeIsType
Latex:
\mforall{}x:\mBbbR{}. \mexists{}n:\mBbbN{}\msupplus{}. (|x| \mleq{} r(n))
Date html generated:
2019_10_29-AM-10_09_35
Last ObjectModification:
2019_01_31-AM-09_51_05
Theory : reals
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