Nuprl Lemma : quasilinear-weighted-mean_wf
∀[I,J:Interval]. ∀[f:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I} ]. ∀[g:{x:ℝ| x ∈ I} ⟶ {x:ℝ| x ∈ J} ].
(quasilinear-weighted-mean(f;g) ∈ {x:ℝ| x ∈ I}
⟶ {x:ℝ| x ∈ I}
⟶ r:{r:ℝ| r0 ≤ r}
⟶ {s:ℝ| (r0 ≤ s) ∧ (r0 < (r + s))}
⟶ {x:ℝ| x ∈ I} )
Proof
Definitions occuring in Statement :
quasilinear-weighted-mean: quasilinear-weighted-mean(f;g),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
quasilinear-weighted-mean: quasilinear-weighted-mean(f;g),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
interval_wf,
convex-comb_wf,
radd_wf,
rless_wf,
int-to-real_wf,
rleq_wf,
real_wf,
i-member_wf
Rules used in proof :
isect_memberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_set_memberEquality,
dependent_functionElimination,
because_Cache,
setEquality,
functionExtensionality,
applyEquality,
productEquality,
natural_numberEquality,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation,
lambdaEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[I,J:Interval]. \mforall{}[f:\{x:\mBbbR{}| x \mmember{} J\} {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\} ]. \mforall{}[g:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} J\} ].
(quasilinear-weighted-mean(f;g) \mmember{} \{x:\mBbbR{}| x \mmember{} I\}
{}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\}
{}\mrightarrow{} r:\{r:\mBbbR{}| r0 \mleq{} r\}
{}\mrightarrow{} \{s:\mBbbR{}| (r0 \mleq{} s) \mwedge{} (r0 < (r + s))\}
{}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\} )
Date html generated:
2017_10_04-PM-11_13_26
Last ObjectModification:
2017_07_29-PM-06_27_40
Theory : reals_2
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