Nuprl Lemma : rabs-neq-zero
∀x:ℝ. (x ≠ r0 ⇒ (r0 < |x|))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
int-to-real: r(n),
rabs: |x|,
uall: ∀[x:A]. B[x],
real: ℝ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
decidable: Dec(P),
subtype_rel: A ⊆r B
Lemmas referenced :
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__lt,
add-is-int-iff,
intformnot_wf,
itermMinus_wf,
int_formula_prop_not_lemma,
int_term_value_minus_lemma,
false_wf,
int-to-real_wf,
real_wf,
rabs_wf,
rneq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
setElimination,
rename,
introduction,
dependent_set_memberEquality,
hypothesisEquality,
sqequalRule,
cut,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
minusEquality,
natural_numberEquality,
because_Cache,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
computeAll,
promote_hyp,
instantiate,
cumulativity,
addEquality,
multiplyEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}x:\mBbbR{}. (x \mneq{} r0 {}\mRightarrow{} (r0 < |x|))
Date html generated:
2017_10_03-AM-08_28_25
Last ObjectModification:
2017_07_28-AM-07_25_13
Theory : reals
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