Nuprl Lemma : square-rless-implies
∀x,y:ℝ.  ((r0 ≤ y) 
⇒ (x^2 < y^2) 
⇒ (x < y))
Proof
Definitions occuring in Statement : 
rleq: x ≤ y
, 
rless: 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]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
prop: ℙ
, 
nat: ℕ
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
itermConstant: "const"
, 
req_int_terms: t1 ≡ t2
, 
top: Top
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
or: P ∨ Q
, 
rev_uimplies: rev_uimplies(P;Q)
, 
guard: {T}
, 
rge: x ≥ y
Lemmas referenced : 
radd-preserves-rless, 
rnexp_wf, 
rminus_wf, 
rless_wf, 
false_wf, 
le_wf, 
rleq_wf, 
int-to-real_wf, 
real_wf, 
radd_wf, 
rmul_wf, 
rsub_wf, 
real_term_polynomial, 
itermSubtract_wf, 
itermAdd_wf, 
itermMinus_wf, 
itermMultiply_wf, 
itermVar_wf, 
real_term_value_const_lemma, 
real_term_value_sub_lemma, 
real_term_value_add_lemma, 
real_term_value_minus_lemma, 
real_term_value_mul_lemma, 
real_term_value_var_lemma, 
req-iff-rsub-is-0, 
rmul-is-positive, 
rless-implies-rless, 
itermConstant_wf, 
rless_functionality, 
req_transitivity, 
radd_functionality, 
req_weakening, 
req_functionality, 
rnexp2, 
rminus_functionality, 
rless_transitivity1, 
rless_functionality_wrt_implies, 
radd_functionality_wrt_rleq, 
rleq_weakening_equal
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
isectElimination, 
because_Cache, 
hypothesis, 
productElimination, 
independent_functionElimination, 
dependent_set_memberEquality, 
natural_numberEquality, 
sqequalRule, 
independent_pairFormation, 
hypothesisEquality, 
computeAll, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_isectElimination, 
unionElimination, 
equalityTransitivity, 
equalitySymmetry
Latex:
\mforall{}x,y:\mBbbR{}.    ((r0  \mleq{}  y)  {}\mRightarrow{}  (x\^{}2  <  y\^{}2)  {}\mRightarrow{}  (x  <  y))
Date html generated:
2017_10_03-AM-08_49_51
Last ObjectModification:
2017_07_28-AM-07_33_55
Theory : reals
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