Nuprl Lemma : IVT-strict-decreasing
∀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))))
⇒ (∀a,b:{x:ℝ| x ∈ I} .
((a < b) ⇒ (∀x:ℝ. ((((f b) ≤ x) ∧ (x ≤ (f a))) ⇒ (∃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],
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,
subinterval: I ⊆ J ,
member: t ∈ T,
sq_stable: SqStable(P),
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rfun: I ⟶ℝ,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
guard: {T},
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
true: True,
less_than': less_than'(a;b),
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
rneq: x ≠ y
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{}a,b:\{x:\mBbbR{}| x \mmember{} I\} .
((a < b)
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. ((((f b) \mleq{} x) \mwedge{} (x \mleq{} (f a))) {}\mRightarrow{} (\mexists{}c:\mBbbR{}. (((a \mleq{} c) \mwedge{} (c \mleq{} b)) \mwedge{} ((f c) = x))))))))
Date html generated:
2020_05_20-PM-00_28_12
Last ObjectModification:
2020_01_08-AM-10_39_59
Theory : reals
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