Nuprl Lemma : strictly-increasing-on-closed-interval2
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y])))
⇒ f[x] strictly-increasing for x ∈ (a, b)
⇒ (∀x:{x:ℝ| x ∈ [a, b]} . ((f[a] ≤ f[x]) ∧ (f[x] ≤ f[b])))
⇒ f[x] strictly-increasing for x ∈ [a, b])
Proof
Definitions occuring in Statement :
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
rfun: I ⟶ℝ,
rooint: (l, u),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
top: Top,
subinterval: I ⊆ J ,
member: t ∈ T,
all: ∀x:A. B[x],
increasing-on-interval: f[x] increasing for x ∈ I,
false: False,
or: P ∨ Q,
not: ¬A,
stable: Stable{P},
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
rfun_wf,
req_wf,
subtype_rel_sets,
rooint_wf,
strictly-increasing-on-interval_wf,
rleq_transitivity,
rleq_weakening_equal,
rleq_wf,
rccint_wf,
i-member_wf,
all_wf,
real_wf,
rless_wf,
rleq_weakening_rless,
member_rccint_lemma,
member_rooint_lemma,
strictly-increasing-on-closed-interval,
set_wf,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
not_wf,
or_wf,
false_wf,
stable__rleq,
rless_transitivity2,
rless_transitivity1,
rleq_weakening,
rleq_antisymmetry,
not-rless,
req_inversion,
sq_stable__rleq
Rules used in proof :
functionEquality,
dependent_set_memberEquality,
because_Cache,
applyEquality,
rename,
setElimination,
lambdaEquality,
setEquality,
independent_functionElimination,
productEquality,
independent_pairFormation,
independent_isectElimination,
isectElimination,
productElimination,
voidEquality,
voidElimination,
isect_memberEquality,
sqequalRule,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
unionElimination,
imageElimination,
baseClosed,
imageMemberEquality
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} (f[x] = f[y])))
{}\mRightarrow{} f[x] strictly-increasing for x \mmember{} (a, b)
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((f[a] \mleq{} f[x]) \mwedge{} (f[x] \mleq{} f[b])))
{}\mRightarrow{} f[x] strictly-increasing for x \mmember{} [a, b])
Date html generated:
2017_10_03-PM-00_29_43
Last ObjectModification:
2017_07_30-PM-08_56_24
Theory : reals
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