Nuprl Lemma : square-rleq-1-iff
∀x:ℝ. (x^2 ≤ r1 ⇐⇒ |x| ≤ r1)
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,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
nat: ℕ,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T}
Lemmas referenced :
rnexp-rleq-iff,
rabs_wf,
int-to-real_wf,
zero-rleq-rabs,
rleq-int,
false_wf,
less_than_wf,
real_wf,
rnexp_wf,
le_wf,
rleq_wf,
iff_wf,
rleq_functionality,
req_inversion,
rabs-rnexp,
req_weakening,
rmul_wf,
rleq_weakening_equal,
rnexp2-nonneg,
rabs-of-nonneg,
rleq_functionality_wrt_implies,
rnexp2,
rmul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
productElimination,
sqequalRule,
independent_pairFormation,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
because_Cache,
addLevel,
impliesFunctionality,
independent_isectElimination,
promote_hyp,
multiplyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x:\mBbbR{}. (x\^{}2 \mleq{} r1 \mLeftarrow{}{}\mRightarrow{} |x| \mleq{} r1)
Date html generated:
2016_10_26-AM-09_14_30
Last ObjectModification:
2016_10_09-PM-07_11_17
Theory : reals
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