Nuprl Lemma : inverse-of-strict-monotonic-function
∀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))))
⇒ (∀J:Interval
((∀t:{t:ℝ| t ∈ I} . (f t ∈ J))
⇒ (∀x:{x:ℝ| x ∈ J} . ∃a,b:{t:ℝ| t ∈ I} . (a ≠ b ∧ ((((f a) ≤ x) ∧ (x ≤ (f 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 x) < (g y)))) ∨ (∀x,y:{x:ℝ| x ∈ J} . ((x < y) ⇒ ((g y) < (g x)))\000C))
∧ (∀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,
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,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
cand: A c∧ B,
false: False,
uimplies: b supposing a,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
rneq: x ≠ y,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rfun_wf,
inverse-of-strict-decreasing-function-exists,
interval_wf,
rneq_wf,
or_wf,
req_wf,
all_wf,
set_wf,
rless_irreflexivity,
rless_transitivity1,
real_wf,
exists_wf,
i-member_wf,
rleq_wf,
rless_wf,
inverse-of-strict-increasing-function-exists
Rules used in proof :
inrFormation,
setEquality,
functionExtensionality,
functionEquality,
inlFormation,
voidElimination,
independent_isectElimination,
lambdaEquality,
sqequalRule,
dependent_set_memberEquality,
applyEquality,
rename,
setElimination,
isectElimination,
productEquality,
independent_pairFormation,
because_Cache,
dependent_pairFormation,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
unionElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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{}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 \mneq{} b \mwedge{} ((((f a) \mleq{} x) \mwedge{} (x \mleq{} (f b))) \mvee{} (((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 x) < (g y))))
\mvee{} (\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_34_14
Last ObjectModification:
2017_07_31-PM-01_59_50
Theory : reals
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