Nuprl Lemma : rneq-zero
∀x:ℝ. (x ≠ r0 ⇐⇒ rpositive(x) ∨ rpositive(-(x)))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rpositive: rpositive(x),
rminus: -(x),
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
rminus: -(x),
rpositive: rpositive(x),
int-to-real: r(n),
rneq: x ≠ y,
rless: x < y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
guard: {T},
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
real: ℝ,
prop: ℙ,
decidable: Dec(P),
false: False,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
real_wf,
minus-is-int-iff,
or_wf,
nat_plus_wf,
sq_exists_wf,
false_wf,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
add-is-int-iff,
less_than_wf,
decidable__lt,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
hypothesis,
inrFormation,
setElimination,
rename,
introduction,
dependent_set_memberEquality,
hypothesisEquality,
lemma_by_obid,
isectElimination,
dependent_functionElimination,
natural_numberEquality,
minusEquality,
applyEquality,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
productElimination,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
inlFormation,
addEquality,
multiplyEquality
Latex:
\mforall{}x:\mBbbR{}. (x \mneq{} r0 \mLeftarrow{}{}\mRightarrow{} rpositive(x) \mvee{} rpositive(-(x)))
Date html generated:
2016_05_18-AM-07_10_38
Last ObjectModification:
2016_01_17-AM-01_51_20
Theory : reals
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