Nuprl Lemma : square-rleq-implies
∀x,y:ℝ. ((r0 ≤ y) ⇒ (x^2 ≤ y^2) ⇒ (x ≤ y))
Proof
Definitions occuring in Statement :
rleq: 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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
top: Top,
guard: {T}
Lemmas referenced :
rnexp-rleq-iff,
rabs_wf,
zero-rleq-rabs,
less_than_wf,
rleq_wf,
rnexp_wf,
false_wf,
le_wf,
int-to-real_wf,
real_wf,
rnexp2-nonneg,
rleq_functionality,
req_inversion,
rabs-rnexp,
req_weakening,
rabs-of-nonneg,
rabs-as-rmax,
rleq-rmax,
rminus_wf,
rleq_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productElimination,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}x,y:\mBbbR{}. ((r0 \mleq{} y) {}\mRightarrow{} (x\^{}2 \mleq{} y\^{}2) {}\mRightarrow{} (x \mleq{} y))
Date html generated:
2016_10_26-AM-09_08_50
Last ObjectModification:
2016_09_30-AM-10_54_12
Theory : reals
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