Nuprl Lemma : rexp_functionality_wrt_rless
∀x,y:ℝ. ((x < y) ⇒ (e^x < e^y))
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,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rsub: x - y,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rexp-of-positive,
rless_wf,
real_wf,
req_wf,
radd_wf,
rminus_wf,
req_weakening,
rexp_wf,
rsub_wf,
rmul_wf,
uiff_transitivity,
req_functionality,
radd_functionality,
radd_comm,
radd-rminus-assoc,
rless_functionality,
rexp_functionality,
rexp-radd,
radd-preserves-rless,
int-to-real_wf,
radd-zero-both,
rmul_preserves_rless,
rexp-positive,
rmul-one-both,
rmul_functionality,
rmul_comm
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
productElimination,
dependent_functionElimination,
natural_numberEquality,
addLevel,
levelHypothesis,
promote_hyp
Latex:
\mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} (e\^{}x < e\^{}y))
Date html generated:
2016_10_26-PM-00_12_14
Last ObjectModification:
2016_09_19-PM-10_59_01
Theory : reals_2
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