Nuprl Lemma : rnexp-rleq-iff
∀x,y:ℝ. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (∀n:ℕ+. (x ≤ y ⇐⇒ x^n ≤ y^n)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
not: ¬A,
guard: {T},
false: False
Lemmas referenced :
rnexp-rleq,
rleq_wf,
rnexp_wf,
nat_plus_wf,
int-to-real_wf,
real_wf,
not-rless,
rnexp-rless,
rless_transitivity1,
nat_plus_subtype_nat,
rless_irreflexivity,
rless_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
isectElimination,
natural_numberEquality,
independent_isectElimination,
voidElimination
Latex:
\mforall{}x,y:\mBbbR{}. ((r0 \mleq{} x) {}\mRightarrow{} (r0 \mleq{} y) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. (x \mleq{} y \mLeftarrow{}{}\mRightarrow{} x\^{}n \mleq{} y\^{}n)))
Date html generated:
2016_05_18-AM-07_19_36
Last ObjectModification:
2015_12_28-AM-00_45_58
Theory : reals
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