Nuprl Lemma : rnexp-convex3
∀a,b:ℝ. ((((r0 ≤ a) ∧ (r0 ≤ b)) ∨ ((a ≤ r0) ∧ (b ≤ r0))) ⇒ (∀n:ℕ+. (|a - b|^n ≤ |a^n - b^n|)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
absval: |i|
Lemmas referenced :
rnexp-convex2,
rminus_wf,
rleq-implies-rleq,
int-to-real_wf,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rsub_wf,
nat_plus_wf,
or_wf,
rleq_wf,
real_wf,
req_wf,
squash_wf,
true_wf,
rabs-rminus,
rabs_wf,
iff_weakening_equal,
radd_wf,
req_weakening,
rnexp_wf,
nat_plus_subtype_nat,
req_functionality,
rabs_functionality,
itermAdd_wf,
real_term_value_add_lemma,
rleq_functionality,
rnexp_functionality,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rmul_wf,
exp_wf2,
absval_wf,
uiff_transitivity,
rsub_functionality,
rminus-as-rmul,
req_transitivity,
rnexp-rmul,
req_inversion,
rmul-rsub-distrib,
rabs-rmul,
rmul_functionality,
rabs-rnexp,
nat_wf,
rabs-int,
rnexp-int,
rleq_weakening,
itermMultiply_wf,
real_term_value_mul_lemma,
exp-one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
productElimination,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
natural_numberEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
minusEquality,
callbyvalueReduce,
sqleReflexivity
Latex:
\mforall{}a,b:\mBbbR{}. ((((r0 \mleq{} a) \mwedge{} (r0 \mleq{} b)) \mvee{} ((a \mleq{} r0) \mwedge{} (b \mleq{} r0))) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. (|a - b|\^{}n \mleq{} |a\^{}n - b\^{}n|)))
Date html generated:
2017_10_03-AM-10_36_39
Last ObjectModification:
2017_07_28-AM-08_14_08
Theory : reals
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