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
Home
Index