Nuprl Lemma : rexp_functionality_wrt_rleq
∀x,y:ℝ. ((x ≤ y) ⇒ (e^x ≤ e^y))
Proof
Definitions occuring in Statement :
rexp: e^x,
rleq: x ≤ y,
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,
uall: ∀[x:A]. B[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
real: ℝ,
guard: {T}
Lemmas referenced :
rexp-of-nonneg,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
int-to-real_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
req-iff-rsub-is-0,
radd_wf,
rsub_wf,
rleq_wf,
real_wf,
rexp_wf,
rmul_wf,
rleq_functionality,
req_weakening,
rexp_functionality,
rexp-radd,
rleq-implies-rleq,
itermConstant_wf,
rmul_preserves_rleq2,
less_than'_wf,
nat_plus_wf,
rminus_wf,
itermMultiply_wf,
real_term_value_mul_lemma,
rmul_functionality,
itermMinus_wf,
real_term_value_minus_lemma,
req_transitivity,
rleq_weakening_rless,
rexp-positive
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
hypothesisEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_isectElimination,
because_Cache,
independent_functionElimination,
isect_memberFormation,
independent_pairEquality,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x,y:\mBbbR{}. ((x \mleq{} y) {}\mRightarrow{} (e\^{}x \mleq{} e\^{}y))
Date html generated:
2017_10_04-PM-10_17_52
Last ObjectModification:
2017_07_28-AM-08_48_02
Theory : reals_2
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