Nuprl Lemma : qexp-positive-iff
∀n:ℕ. ∀r:ℚ. (0 < r ↑ n ⇐⇒ (n = 0 ∈ ℤ) ∨ 0 < r ∨ (r < 0 ∧ (↑isEven(n))))
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qless: r < s,
rationals: ℚ,
isEven: isEven(n),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
or: P ∨ Q
Lemmas referenced :
nat_wf,
rationals_wf,
equal-wf-T-base,
iff_wf,
qexp_wf,
int-subtype-rationals,
isEven_wf,
assert_wf,
and_wf,
qless_wf,
equal_wf,
or_wf,
qexp-sign
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
isectElimination,
intEquality,
setElimination,
rename,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
addLevel,
impliesFunctionality,
independent_functionElimination,
baseClosed,
productEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}r:\mBbbQ{}. (0 < r \muparrow{} n \mLeftarrow{}{}\mRightarrow{} (n = 0) \mvee{} 0 < r \mvee{} (r < 0 \mwedge{} (\muparrow{}isEven(n))))
Date html generated:
2016_05_15-PM-11_09_10
Last ObjectModification:
2016_01_16-PM-09_25_50
Theory : rationals
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