Nuprl Lemma : rnexp2-positive
∀x:ℝ. (x ≠ r0 ⇒ (r0 < x^2))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rless: x < y,
rnexp: x^k1,
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
nat: ℕ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
uiff: uiff(P;Q),
and: P ∧ Q,
req_int_terms: t1 ≡ t2,
guard: {T},
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rneq_wf,
int-to-real_wf,
real_wf,
rnexp-positive,
rminus_wf,
rless-implies-rless,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
rless_transitivity1,
rnexp_wf,
rleq_weakening,
req_functionality,
rmul_wf,
rnexp2,
itermMultiply_wf,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
unionElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
setElimination,
rename,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
productElimination,
int_eqEquality
Latex:
\mforall{}x:\mBbbR{}. (x \mneq{} r0 {}\mRightarrow{} (r0 < x\^{}2))
Date html generated:
2019_10_29-AM-09_39_41
Last ObjectModification:
2019_03_20-PM-00_17_13
Theory : reals
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