Nuprl Lemma : rlog_functionality_wrt_rleq
∀x,y:{t:ℝ| r0 < t} . ((x ≤ y) ⇒ (rlog(x) ≤ rlog(y)))
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
top: Top,
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
sq_stable: SqStable(P),
squash: ↓T,
increasing-on-interval: f[x] increasing for x ∈ I,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A
Lemmas referenced :
derivative-implies-increasing,
roiint_wf,
int-to-real_wf,
iproper-roiint,
rlog_wf,
member_roiint_lemma,
real_wf,
i-member_wf,
rdiv_wf,
sq_stable__rless,
rless_wf,
derivative-rlog,
set_wf,
rleq_wf,
function-is-continuous,
req_functionality,
rdiv_functionality,
req_weakening,
req_wf,
rmul_preserves_rleq,
rmul_wf,
rleq-int,
false_wf,
uiff_transitivity,
rleq_functionality,
rmul-rdiv-cancel2,
rmul-zero-both
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesisEquality,
setEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
inrFormation,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaFormation,
productElimination,
independent_pairFormation
Latex:
\mforall{}x,y:\{t:\mBbbR{}| r0 < t\} . ((x \mleq{} y) {}\mRightarrow{} (rlog(x) \mleq{} rlog(y)))
Date html generated:
2016_10_26-PM-00_27_30
Last ObjectModification:
2016_10_05-PM-08_21_33
Theory : reals_2
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