Nuprl Lemma : square-rless-1-iff
∀x:ℝ. (x^2 < r1 ⇐⇒ |x| < r1)
Proof
Definitions occuring in Statement :
rless: 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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
or: P ∨ Q,
rsub: x - y,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
true: True,
guard: {T},
nat_plus: ℕ+
Lemmas referenced :
rless_wf,
rnexp_wf,
false_wf,
le_wf,
int-to-real_wf,
rabs_wf,
real_wf,
rmul_wf,
radd-preserves-rless,
rminus_wf,
rless_functionality,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_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,
rnexp2,
req_weakening,
rsub_wf,
rabs-rless-iff,
rmul-is-positive,
equal_wf,
radd-rminus-both,
radd-ac,
radd-zero-both,
radd_comm,
rmul-zero-both,
radd-int,
rmul_functionality,
rmul-distrib2,
rmul-identity1,
req_inversion,
radd-assoc,
req_transitivity,
rminus-as-rmul,
radd_functionality,
rless-int,
rless_transitivity2,
rleq_weakening_rless,
rless_irreflexivity,
exp-one,
true_wf,
squash_wf,
rnexp-int,
rabs-rnexp,
exp_wf2,
less_than_wf,
zero-rleq-rabs,
rnexp-rless,
rleq-rmax,
rabs-as-rmax
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalitySymmetry,
equalityTransitivity,
productEquality,
promote_hyp,
addEquality,
minusEquality,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality
Latex:
\mforall{}x:\mBbbR{}. (x\^{}2 < r1 \mLeftarrow{}{}\mRightarrow{} |x| < r1)
Date html generated:
2017_10_03-AM-08_50_12
Last ObjectModification:
2017_07_28-AM-07_34_01
Theory : reals
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