Nuprl Lemma : rnexp-convex2
∀a,b:ℝ. ((r0 ≤ a) ⇒ (r0 ≤ b) ⇒ (∀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,
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,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T}
Lemmas referenced :
rnexp-convex,
rmax_wf,
rmin_wf,
rmin_ub,
int-to-real_wf,
rmin-rleq-rmax,
nat_plus_wf,
rleq_wf,
real_wf,
rnexp_wf,
nat_plus_subtype_nat,
rsub_wf,
rabs_wf,
rleq_weakening_equal,
rleq_functionality,
rnexp_functionality,
req_inversion,
rmax-minus-rmin,
req_weakening,
rleq_functionality_wrt_implies,
rsub_functionality,
rmax-rnexp,
rmin-rnexp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
natural_numberEquality,
productElimination,
independent_pairFormation,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}a,b:\mBbbR{}. ((r0 \mleq{} a) {}\mRightarrow{} (r0 \mleq{} b) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. (|a - b|\^{}n \mleq{} |a\^{}n - b\^{}n|)))
Date html generated:
2016_05_18-AM-09_30_09
Last ObjectModification:
2015_12_27-PM-11_20_29
Theory : reals
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