Nuprl Lemma : IVT-strict-monotonic
∀I:Interval. ∀f:I ⟶ℝ.
(((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y)))) ∨ (∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x)))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀a,b:{x:ℝ| x ∈ I} .
(a ≠ b
⇒ (∀x:ℝ
(((((f a) ≤ x) ∧ (x ≤ (f b))) ∨ (((f b) ≤ x) ∧ (x ≤ (f a))))
⇒ (∃c:ℝ. ((((a ≤ c) ∧ (c ≤ b)) ∨ ((b ≤ c) ∧ (c ≤ a))) ∧ ((f c) = x))))))))
Proof
Definitions occuring in Statement :
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rneq: x ≠ y,
rleq: x ≤ y,
rless: x < y,
req: x = y,
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
or: P ∨ Q,
rneq: x ≠ y,
sq_exists: ∃x:A [B[x]],
subinterval: I ⊆ J ,
top: Top,
exists: ∃x:A. B[x],
rfun: I ⟶ℝ,
guard: {T},
uimplies: b supposing a,
false: False
Lemmas referenced :
rcc-subinterval,
sq_stable__i-member,
rleq_wf,
IVT-strict-increasing,
member_rccint_lemma,
istype-void,
sq_stable__rleq,
sq_stable__req,
i-member_wf,
req_wf,
rless_transitivity1,
rless_irreflexivity,
rleq_weakening_rless,
rless_transitivity2,
IVT-strict-decreasing,
real_wf,
rneq_wf,
rless_wf,
rfun_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
productElimination,
independent_functionElimination,
setElimination,
rename,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
universeIsType,
isectElimination,
unionElimination,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
inlFormation_alt,
productIsType,
applyEquality,
dependent_set_memberEquality_alt,
unionIsType,
independent_isectElimination,
inrFormation_alt,
inhabitedIsType,
setIsType,
functionIsType
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x < y) {}\mRightarrow{} ((f x) < (f y))))
\mvee{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x < y) {}\mRightarrow{} ((f y) < (f x)))))
{}\mRightarrow{} (\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} ((f x) = (f y))))
{}\mRightarrow{} (\mforall{}a,b:\{x:\mBbbR{}| x \mmember{} I\} .
(a \mneq{} b
{}\mRightarrow{} (\mforall{}x:\mBbbR{}
(((((f a) \mleq{} x) \mwedge{} (x \mleq{} (f b))) \mvee{} (((f b) \mleq{} x) \mwedge{} (x \mleq{} (f a))))
{}\mRightarrow{} (\mexists{}c:\mBbbR{}. ((((a \mleq{} c) \mwedge{} (c \mleq{} b)) \mvee{} ((b \mleq{} c) \mwedge{} (c \mleq{} a))) \mwedge{} ((f c) = x))))))))
Date html generated:
2019_10_30-AM-07_51_05
Last ObjectModification:
2019_04_03-AM-00_22_40
Theory : reals
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