Nuprl Lemma : square-rge-1-iff
∀x:ℝ. (r1 ≤ x^2 ⇐⇒ r1 ≤ |x|)
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
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,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
not: ¬A,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True
Lemmas referenced :
square-rleq-1-iff,
not-rless,
rabs_wf,
int-to-real_wf,
rless_wf,
rleq_wf,
rnexp_wf,
false_wf,
le_wf,
real_wf,
rleq_antisymmetry,
rleq_functionality_wrt_implies,
rleq_weakening_rless,
rleq_weakening_equal,
rleq_weakening,
itermSubtract_wf,
itermConstant_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
rnexp2-nonneg,
req_functionality,
req_inversion,
rabs-rnexp,
req_weakening,
rabs-of-nonneg,
rnexp-rless,
zero-rleq-rabs,
less_than_wf,
rless_transitivity1,
rless_irreflexivity,
rless_functionality,
rnexp-one,
rnexp_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
isectElimination,
natural_numberEquality,
independent_isectElimination,
dependent_set_memberEquality,
sqequalRule,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
lambdaEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}x:\mBbbR{}. (r1 \mleq{} x\^{}2 \mLeftarrow{}{}\mRightarrow{} r1 \mleq{} |x|)
Date html generated:
2017_10_03-AM-08_50_22
Last ObjectModification:
2017_06_23-PM-05_29_21
Theory : reals
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