Nuprl Lemma : rleq-int-fractions2
∀[a,b:ℤ]. ∀[d:ℕ+]. uiff(r(a) ≤ (r(b)/r(d));(a * d) ≤ b)
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat_plus: ℕ+,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rleq: x ≤ y,
rnonneg: rnonneg(x),
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmul_preserves_rleq,
rmul-rdiv-cancel2,
req_weakening,
rmul-int,
rleq_functionality,
uiff_transitivity,
rmul_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
rleq-int,
rmul_preserves_rleq2,
le_wf,
nat_plus_wf,
rsub_wf,
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
int-to-real_wf,
rleq_wf,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
multiplyEquality,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inrFormation,
independent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality,
minusEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[d:\mBbbN{}\msupplus{}]. uiff(r(a) \mleq{} (r(b)/r(d));(a * d) \mleq{} b)
Date html generated:
2016_05_18-AM-07_27_26
Last ObjectModification:
2016_01_17-AM-01_58_44
Theory : reals
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