Nuprl Lemma : inverse-of-strict-decreasing-function-exists
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀J:Interval
((∀t:{t:ℝ| t ∈ I} . (f t ∈ J))
⇒ (∀x:{x:ℝ| x ∈ J} . ∃a,b:{t:ℝ| t ∈ I} . ((a < b) ∧ ((f b) ≤ x) ∧ (x ≤ (f a))))
⇒ (∃g:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I}
((∀x:{x:ℝ| x ∈ J} . ((f (g x)) = x))
∧ (∀x:{x:ℝ| x ∈ I} . ((g (f x)) = x))
∧ (∀x,y:{x:ℝ| x ∈ J} . ((x < y) ⇒ ((g y) < (g x))))
∧ (∀x,y:{t:ℝ| t ∈ J} . ((x = y) ⇒ ((g x) = (g y)))))))))
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],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
prop: ℙ,
uall: ∀[x:A]. B[x],
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
guard: {T},
subtype_rel: A ⊆r B,
pi1: fst(t)
Lemmas referenced :
rfun_wf,
interval_wf,
rleq_wf,
rless_wf,
exists_wf,
real_wf,
all_wf,
req_wf,
i-member_wf,
sq_stable__i-member,
i-member-between,
IVT-strict-decreasing,
inverse-of-strict-decreasing-function,
set_wf,
equal_wf
Rules used in proof :
functionEquality,
productEquality,
lambdaEquality,
setEquality,
applyEquality,
isectElimination,
because_Cache,
dependent_set_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
dependent_pairFormation,
independent_pairFormation,
rename,
setElimination,
productElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
equalitySymmetry,
equalityTransitivity,
functionExtensionality,
promote_hyp
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
((\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{}J:Interval
((\mforall{}t:\{t:\mBbbR{}| t \mmember{} I\} . (f t \mmember{} J))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} J\} . \mexists{}a,b:\{t:\mBbbR{}| t \mmember{} I\} . ((a < b) \mwedge{} ((f b) \mleq{} x) \mwedge{} (x \mleq{} (f a))))
{}\mRightarrow{} (\mexists{}g:\{x:\mBbbR{}| x \mmember{} J\} {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\}
((\mforall{}x:\{x:\mBbbR{}| x \mmember{} J\} . ((f (g x)) = x))
\mwedge{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . ((g (f x)) = x))
\mwedge{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} J\} . ((x < y) {}\mRightarrow{} ((g y) < (g x))))
\mwedge{} (\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} J\} . ((x = y) {}\mRightarrow{} ((g x) = (g y)))))))))
Date html generated:
2017_10_03-AM-10_33_36
Last ObjectModification:
2017_07_30-AM-11_57_37
Theory : reals
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