Nuprl Lemma : rnexp-rless2
∀x,y:ℝ. ((x < y) ⇒ (r0 < y) ⇒ (∀n:ℕ+. ((↑isOdd(n)) ⇒ (x^n < y^n))))
Proof
Definitions occuring in Statement :
rless: x < y,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
isOdd: isOdd(n),
nat_plus: ℕ+,
assert: ↑b,
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,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
prop: ℙ,
nat_plus: ℕ+,
uimplies: b supposing a,
not: ¬A,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rnexp-positive,
nat_plus_subtype_nat,
rless-cases1,
int-to-real_wf,
rnexp_wf,
rnexp-rless,
assert_wf,
isOdd_wf,
nat_plus_wf,
rless_wf,
real_wf,
not-rless,
rmul_reverses_rless,
rless-int,
rmul_wf,
rminus_wf,
rless_functionality,
rmul-int,
req_weakening,
rmul-minus,
rmul_over_rminus,
rminus_functionality,
rmul-one-both,
ifthenelse_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
req_functionality,
rnexp-rminus,
rless_transitivity2,
rleq_weakening_rless,
rless_irreflexivity,
req_transitivity,
rminus-rminus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
applyEquality,
sqequalRule,
isectElimination,
natural_numberEquality,
because_Cache,
unionElimination,
setElimination,
rename,
independent_isectElimination,
minusEquality,
productElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
multiplyEquality,
promote_hyp,
equalityElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
voidElimination
Latex:
\mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} (r0 < y) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. ((\muparrow{}isOdd(n)) {}\mRightarrow{} (x\^{}n < y\^{}n))))
Date html generated:
2017_10_03-AM-08_40_01
Last ObjectModification:
2017_07_28-AM-07_31_05
Theory : reals
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