Nuprl Lemma : IVT-strict-increasing
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y))))
⇒ (∀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))) ⇒ (∃c:ℝ [(((a ≤ c) ∧ (c ≤ b)) ∧ ((f c) = x))]))))))
Proof
Definitions occuring in Statement :
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
req: x = y,
real: ℝ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
implies: 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,
and: P ∧ Q,
member: t ∈ T,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
rfun: I ⟶ℝ,
subinterval: I ⊆ J ,
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
r-ap: f(x),
guard: {T}
Lemmas referenced :
strict-monotonic-is-locally-non-constant,
rless_wf,
member_rccint_lemma,
istype-void,
i-member-between,
sq_stable__i-member,
rleq_wf,
IVT-locally-non-constant,
rfun_subtype,
rccint_wf,
subtype_rel_sets_simple,
real_wf,
i-member_wf,
req_wf,
rfun_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
inlFormation_alt,
hypothesis,
sqequalRule,
functionIsType,
because_Cache,
universeIsType,
isectElimination,
setElimination,
rename,
applyEquality,
isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productIsType,
dependent_set_memberEquality_alt,
independent_isectElimination,
lambdaEquality_alt,
inhabitedIsType,
setIsType
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x < y) {}\mRightarrow{} ((f x) < (f y))))
{}\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 < b)
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. ((((f a) \mleq{} x) \mwedge{} (x \mleq{} (f b))) {}\mRightarrow{} (\mexists{}c:\mBbbR{} [(((a \mleq{} c) \mwedge{} (c \mleq{} b)) \mwedge{} ((f c) = x))]))))))
Date html generated:
2019_10_30-AM-07_50_14
Last ObjectModification:
2019_02_12-AM-11_00_07
Theory : reals
Home
Index