Nuprl Lemma : rexp-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],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
uiff: uiff(P;Q)
Lemmas referenced :
rexp_functionality_wrt_rleq,
rleq_wf,
rlog_functionality_wrt_rleq,
rexp-positive,
rexp_wf,
rless_wf,
int-to-real_wf,
real_wf,
rlog_wf,
rleq_functionality,
rlog-rexp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
because_Cache,
independent_isectElimination,
productElimination
Latex:
\mforall{}x,y:\mBbbR{}. (x \mleq{} y \mLeftarrow{}{}\mRightarrow{} e\^{}x \mleq{} e\^{}y)
Date html generated:
2016_10_26-PM-00_39_12
Last ObjectModification:
2016_10_14-PM-02_50_45
Theory : reals_2
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