Nuprl Lemma : rless-int-fractions2
∀a,b:ℤ. ∀d:ℕ+. (r(a) < (r(b)/r(d)) ⇐⇒ a * d < b)
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rless: x < y,
int-to-real: r(n),
nat_plus: ℕ+,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
rless: x < y,
sq_exists: ∃x:{A| B[x]}
Lemmas referenced :
rmul-rdiv-cancel2,
req_weakening,
rmul-int,
rless_functionality,
rmul_wf,
rmul_preserves_rless,
nat_plus_wf,
less_than_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,
rless_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
multiplyEquality,
promote_hyp,
addLevel
Latex:
\mforall{}a,b:\mBbbZ{}. \mforall{}d:\mBbbN{}\msupplus{}. (r(a) < (r(b)/r(d)) \mLeftarrow{}{}\mRightarrow{} a * d < b)
Date html generated:
2016_05_18-AM-07_27_47
Last ObjectModification:
2016_01_17-AM-01_59_13
Theory : reals
Home
Index