Nuprl Lemma : strict-increasing-implies-inv-strict-increasing
∀[I:Interval]. ∀[f:I ⟶ℝ].
(∀x,y:{x:ℝ| x ∈ I} . (((f x) < (f y)) ⇒ (x < y))) supposing
((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y)))) and
(∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ ((f x) = (f y)))))
Proof
Definitions occuring in Statement :
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rless: x < y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rfun: I ⟶ℝ,
prop: ℙ,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
false: False
Lemmas referenced :
real-fun-implies-sfun-general,
req_witness,
i-member_wf,
req_wf,
real_wf,
sq_stable__rless,
rless_wf,
set_wf,
all_wf,
rfun_wf,
interval_wf,
rless_transitivity2,
rleq_weakening_rless,
rless_irreflexivity
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
independent_functionElimination,
setEquality,
independent_isectElimination,
lambdaFormation,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality,
inlFormation,
unionElimination,
voidElimination
Latex:
\mforall{}[I:Interval]. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}].
(\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . (((f x) < (f y)) {}\mRightarrow{} (x < y))) supposing
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x < y) {}\mRightarrow{} ((f x) < (f y)))) and
(\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} ((f x) = (f y)))))
Date html generated:
2017_10_03-AM-09_57_15
Last ObjectModification:
2017_08_31-PM-01_21_40
Theory : reals
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