Nuprl Lemma : r-archimedean-rabs
∀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: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
top: Top,
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
cand: A c∧ B,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
true: True,
squash: ↓T,
req_int_terms: t1 ≡ t2
Lemmas referenced :
r-archimedean,
rabs-as-rmax,
rmax_lb,
rminus_wf,
int-to-real_wf,
rleq_wf,
rabs_wf,
real_wf,
rmul_reverses_rleq,
rleq-int,
false_wf,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
itermMinus_wf,
req-iff-rsub-is-0,
rminus-rminus,
rleq_functionality,
req_transitivity,
req_weakening,
squash_wf,
true_wf,
rminus-int,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation,
sqequalRule,
isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
minusEquality,
natural_numberEquality,
independent_functionElimination,
promote_hyp,
because_Cache,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
approximateComputation,
int_eqEquality,
intEquality
Latex:
\mforall{}x:\mBbbR{}. \mexists{}n:\mBbbN{}. (|x| \mleq{} r(n))
Date html generated:
2017_10_03-AM-09_22_41
Last ObjectModification:
2017_07_28-AM-07_45_56
Theory : reals
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