Nuprl Lemma : int-rdiv-is-positive
∀x:ℝ. ∀k:ℤ-o. (r0 < (x)/k ⇐⇒ ((r0 < x) ∧ 0 < k) ∨ ((x < r0) ∧ k < 0))
Proof
Definitions occuring in Statement :
rless: x < y,
int-rdiv: (a)/k1,
int-to-real: r(n),
real: ℝ,
int_nzero: ℤ-o,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
int_nzero: ℤ-o,
uimplies: b supposing a,
not: ¬A,
rless: x < y,
sq_exists: ∃x:A [B[x]],
false: False,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
rless_wf,
int-to-real_wf,
int-rdiv_wf,
istype-less_than,
int_nzero_wf,
real_wf,
rdiv_wf,
rneq-int,
nat_plus_properties,
int_nzero_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
set_subtype_base,
nequal_wf,
int_subtype_base,
rless_functionality,
req_weakening,
int-rdiv-req,
rdiv-is-positive,
rless-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
sqequalRule,
unionIsType,
productIsType,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_functionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityIstype,
applyEquality,
intEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
unionElimination,
inlFormation_alt,
promote_hyp,
inrFormation_alt
Latex:
\mforall{}x:\mBbbR{}. \mforall{}k:\mBbbZ{}\msupminus{}\msupzero{}. (r0 < (x)/k \mLeftarrow{}{}\mRightarrow{} ((r0 < x) \mwedge{} 0 < k) \mvee{} ((x < r0) \mwedge{} k < 0))
Date html generated:
2019_10_29-AM-10_05_55
Last ObjectModification:
2019_04_01-PM-11_11_32
Theory : reals
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