Nuprl Lemma : rpositive-implies-rnonneg
∀[x:ℝ]. (rpositive(x) ⇒ rnonneg(x))
Proof
Definitions occuring in Statement :
rnonneg: rnonneg(x),
rpositive: rpositive(x),
real: ℝ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
real: ℝ,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
false: False,
rpositive2: rpositive2(x),
exists: ∃x:A. B[x],
rnonneg2: rnonneg2(x),
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
int_upper: {i...},
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
nat: ℕ
Lemmas referenced :
rnonneg-iff,
rpositive-iff,
rpositive_wf,
less_than'_wf,
nat_plus_wf,
real_wf,
int_upper_wf,
all_wf,
le_wf,
less_than_transitivity1,
less_than_wf,
int_upper_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
mul_preserves_le,
nat_plus_subtype_nat,
mul_cancel_in_le,
mul-swap,
mul-commutes,
le_functionality,
le_weakening,
mul_bounds_1a,
int_upper_subtype_nat,
nat_wf,
nat_properties,
itermMultiply_wf,
itermConstant_wf,
intformeq_wf,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
because_Cache,
hypothesis,
setElimination,
rename,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_pairEquality,
voidElimination,
applyEquality,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
multiplyEquality,
dependent_set_memberEquality,
independent_isectElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
applyLambdaEquality
Latex:
\mforall{}[x:\mBbbR{}]. (rpositive(x) {}\mRightarrow{} rnonneg(x))
Date html generated:
2017_10_03-AM-08_23_41
Last ObjectModification:
2017_07_28-AM-07_23_00
Theory : reals
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