Nuprl Lemma : r-archimedean-implies
∀x:ℝ. ∀m:ℕ+. ∃M:ℕ+. (((r1/r(M)) * x) ≤ (r1/r(m)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
exists: ∃x:A. B[x],
and: P ∧ Q,
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
rneq: x ≠ y,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_plus_wf,
real_wf,
r-archimedean,
rmul_wf,
int-to-real_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
rleq_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rleq-int,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
rleq_transitivity,
equal_wf,
rmul_preserves_rleq,
req_wf,
req_weakening,
uiff_transitivity,
rleq_functionality,
rmul-rdiv-cancel2,
req_inversion,
rmul-assoc,
rmul_functionality,
rmul_comm,
req_functionality,
req_transitivity,
rmul-ac,
rmul-rdiv-cancel,
rmul-one-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
hypothesisEquality,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
inrFormation,
int_eqEquality,
computeAll,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x:\mBbbR{}. \mforall{}m:\mBbbN{}\msupplus{}. \mexists{}M:\mBbbN{}\msupplus{}. (((r1/r(M)) * x) \mleq{} (r1/r(m)))
Date html generated:
2017_10_03-AM-09_22_30
Last ObjectModification:
2017_07_28-AM-07_45_48
Theory : reals
Home
Index