Nuprl Lemma : rexp-positive
∀y:ℝ. (r0 < e^y)
Proof
Definitions occuring in Statement :
rexp: e^x,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rge: x ≥ y,
guard: {T}
Lemmas referenced :
real_wf,
rmax_wf,
int-to-real_wf,
rminus_wf,
rleq-rmax,
rleq_wf,
radd_wf,
radd-preserves-rleq,
uiff_transitivity,
rleq_functionality,
radd_comm,
req_transitivity,
radd-ac,
radd_functionality,
req_weakening,
radd-rminus-both,
radd-zero-both,
rexp-of-nonneg,
rexp_wf,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
req_wf,
req_functionality,
req_inversion,
radd-assoc,
rless_functionality,
rexp_functionality,
rmul_wf,
rexp-radd,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
rmul_comm,
rexp0,
rmul_preserves_rless,
rless_wf,
rmul-assoc,
rmul-one-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
dependent_pairFormation,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
productElimination,
independent_pairFormation,
productEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
minusEquality,
addEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}y:\mBbbR{}. (r0 < e\^{}y)
Date html generated:
2016_10_26-PM-00_12_09
Last ObjectModification:
2016_09_19-PM-10_59_22
Theory : reals_2
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