Nuprl Lemma : rexp-non-decreasing
∀[a,b:ℝ]. e^a ≤ e^b supposing a ≤ b
Proof
Definitions occuring in Statement :
rexp: e^x,
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rgt: x > y,
guard: {T}
Lemmas referenced :
rexp-radd,
rsub_wf,
less_than'_wf,
rexp_wf,
real_wf,
nat_plus_wf,
rleq_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
rmul_wf,
real_polynomial_null,
int-to-real_wf,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
req_functionality,
rexp_functionality,
req_weakening,
rleq_functionality,
rexp-of-nonneg,
rleq-implies-rleq,
itermConstant_wf,
equal_wf,
rmul_preserves_rleq2,
rmul-identity1,
rmul_comm,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_weakening_rless,
rexp-positive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality,
independent_functionElimination,
lambdaFormation
Latex:
\mforall{}[a,b:\mBbbR{}]. e\^{}a \mleq{} e\^{}b supposing a \mleq{} b
Date html generated:
2017_10_04-PM-10_17_45
Last ObjectModification:
2017_06_05-PM-11_55_03
Theory : reals_2
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