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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