Nuprl Lemma : rexp-increasing
∀a,b:ℝ. ((a < b) ⇒ (e^a < e^b))
Proof
Definitions occuring in Statement :
rexp: e^x,
rless: x < y,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rexp-radd,
rsub_wf,
rless_wf,
real_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
rexp_wf,
rmul_wf,
rexp-of-positive,
rless-implies-rless,
int-to-real_wf,
equal_wf,
itermConstant_wf,
real_polynomial_null,
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,
rless_functionality,
rmul_preserves_rless,
rexp-positive,
rmul-identity1,
rmul_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
because_Cache,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}a,b:\mBbbR{}. ((a < b) {}\mRightarrow{} (e\^{}a < e\^{}b))
Date html generated:
2017_10_04-PM-10_17_49
Last ObjectModification:
2017_06_05-PM-11_58_08
Theory : reals_2
Home
Index