Nuprl Lemma : inverse-of-strict-decreasing-function
∀I:Interval. ∀f:I ⟶ℝ. ∀J:Interval. ∀g:x:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I} .
((∀t:{t:ℝ| t ∈ I} . (f t ∈ J))
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀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,
rless: x < y,
req: x = y,
real: ℝ,
all: ∀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 :
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T},
false: False,
or: P ∨ Q,
rneq: x ≠ y,
not: ¬A,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
sq_stable: SqStable(P),
subinterval: I ⊆ J ,
top: Top,
real-fun: real-fun(f;a;b),
real-sfun: real-sfun(f;a;b),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q)
Lemmas referenced :
rfun_wf,
interval_wf,
all_wf,
req_wf,
rless_wf,
i-member_wf,
real_wf,
set_wf,
rneq_wf,
req_inversion,
rless_irreflexivity,
rleq_weakening,
rless_transitivity1,
not-rneq,
rleq_weakening_rless,
rless_transitivity2,
rless_functionality,
not-rless,
rleq_wf,
equal_wf,
sq_stable__i-member,
rcc-subinterval,
rccint_wf,
rfun_subtype,
real-fun-implies-sfun,
member_rccint_lemma,
subtype_rel_sets,
rleq_weakening_equal,
req_functionality
Rules used in proof :
functionEquality,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
setEquality,
because_Cache,
rename,
setElimination,
independent_pairFormation,
hypothesisEquality,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_functionElimination,
voidElimination,
independent_functionElimination,
unionElimination,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
voidEquality,
isect_memberEquality,
inlFormation,
productEquality
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}J:Interval. \mforall{}g:x:\{x:\mBbbR{}| x \mmember{} J\} {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\} .
((\mforall{}t:\{t:\mBbbR{}| t \mmember{} I\} . (f t \mmember{} J))
{}\mRightarrow{} (\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{}x:\{x:\mBbbR{}| x \mmember{} J\} . ((f (g x)) = x))
{}\mRightarrow{} ((\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_18
Last ObjectModification:
2017_07_30-AM-11_56_01
Theory : reals
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