Nuprl Lemma : r-archimedean-implies2
∀x:ℝ. ∀d:{d:ℝ| r0 < d} . ∃M:ℕ+. ((x/r(M)) ≤ d)
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
small-reciprocal-real,
r-archimedean-implies,
rleq_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
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,
set_wf,
real_wf,
rmul_preserves_req,
rmul_wf,
req_wf,
req_weakening,
rless_transitivity2,
rleq_weakening_rless,
uiff_transitivity,
req_functionality,
req_inversion,
rmul-assoc,
rmul_functionality,
rmul_comm,
rmul-ac,
rmul-rdiv-cancel,
rmul-rdiv-cancel2,
rmul-one-both,
rleq_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
independent_isectElimination,
sqequalRule,
inrFormation,
independent_functionElimination,
natural_numberEquality,
unionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}x:\mBbbR{}. \mforall{}d:\{d:\mBbbR{}| r0 < d\} . \mexists{}M:\mBbbN{}\msupplus{}. ((x/r(M)) \mleq{} d)
Date html generated:
2016_10_26-AM-09_21_54
Last ObjectModification:
2016_08_19-PM-00_50_19
Theory : reals
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