Nuprl Lemma : square-positive-iff
∀x:ℝ. (r0 < (x * x) ⇐⇒ x ≠ r0)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rless: x < y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False
Lemmas referenced :
rnexp2-positive-iff,
rless_wf,
int-to-real_wf,
rmul_wf,
rneq_wf,
real_wf,
rnexp_wf,
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,
rless_functionality,
req_weakening,
rnexp2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
universeIsType,
isectElimination,
natural_numberEquality,
promote_hyp,
because_Cache,
dependent_set_memberEquality_alt,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule
Latex:
\mforall{}x:\mBbbR{}. (r0 < (x * x) \mLeftarrow{}{}\mRightarrow{} x \mneq{} r0)
Date html generated:
2019_10_29-AM-10_07_42
Last ObjectModification:
2019_03_20-PM-00_50_44
Theory : reals
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