Nuprl Lemma : rexp-difference-bound
∀x,y:ℝ. ((x < y) ⇒ ((e^y - e^x) ≤ (e^y * (y - x))))
Proof
Definitions occuring in Statement :
rexp: e^x,
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
rmul: a * b,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
iproper: iproper(I),
top: Top,
rfun: I ⟶ℝ,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subinterval: I ⊆ J ,
squash: ↓T,
sq_stable: SqStable(P),
rccint: [l, u],
i-member: r ∈ I,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
itermConstant: "const",
guard: {T},
rge: x ≥ y,
rgt: x > y,
cand: A c∧ B,
ml-term-to-poly: ml-term-to-poly(t),
nil: [],
it: ⋅,
has-value: (a)↓
Lemmas referenced :
rless_wf,
real_wf,
rccint_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
i-finite_wf,
rexp_wf,
i-member_wf,
req_functionality,
rexp_functionality,
req_weakening,
req_wf,
set_wf,
mean-value-for-bounded-derivative,
derivative-rexp,
rleq_wf,
member_riiint_lemma,
member_rccint_lemma,
derivative_functionality_wrt_subinterval,
riiint_wf,
sq_stable__rleq,
rabs-of-nonneg,
rleq_functionality,
req-iff-rsub-is-0,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_term_value_const_lemma,
int-to-real_wf,
itermVar_wf,
itermSubtract_wf,
real_term_polynomial,
rleq_weakening,
rleq_weakening_equal,
rexp_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
rabs_wf,
rexp-positive,
rleq_weakening_rless,
rsub_wf,
rmul_wf,
rsub_functionality_wrt_rleq,
rexp_functionality_wrt_rless,
real_polynomial_null,
itermConstant_wf,
evalall-sqequal,
rmul_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
productEquality,
natural_numberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
intEquality,
int_eqEquality,
computeAll,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
sqleReflexivity,
mlComputation
Latex:
\mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} ((e\^{}y - e\^{}x) \mleq{} (e\^{}y * (y - x))))
Date html generated:
2017_10_04-PM-10_17_56
Last ObjectModification:
2017_07_28-AM-08_48_04
Theory : reals_2
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