Nuprl Lemma : real-continuity4-ext
∀a,b:ℝ.
∀f:[a, b] ⟶ℝ
(∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y)
⇒ ((f x) = (f y)))
⇐⇒ ∀k:ℕ+. ∃d:{d:ℝ| r0 < d} . ∀x,y:{x:ℝ| x ∈ [a, b]} . ((|x - y| ≤ d)
⇒ (|(f x) - f y| ≤ (r1/r(k)))))
supposing a < b
Proof
Definitions occuring in Statement :
rfun: I ⟶ℝ
,
rccint: [l, u]
,
i-member: r ∈ I
,
rdiv: (x/y)
,
rleq: x ≤ y
,
rless: x < y
,
rabs: |x|
,
rsub: x - y
,
req: x = y
,
int-to-real: r(n)
,
real: ℝ
,
nat_plus: ℕ+
,
uimplies: b supposing a
,
all: ∀x:A. B[x]
,
exists: ∃x:A. B[x]
,
iff: P
⇐⇒ Q
,
implies: P
⇒ Q
,
set: {x:A| B[x]}
,
apply: f a
,
natural_number: $n
Definitions unfolded in proof :
bfalse: ff
,
it: ⋅
,
btrue: tt
,
ifthenelse: if b then t else f fi
,
subtract: n - m
,
so_apply: x[s]
,
so_lambda: λ2x.t[x]
,
uimplies: b supposing a
,
so_apply: x[s1;s2]
,
top: Top
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2;s3;s4]
,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w])
,
uall: ∀[x:A]. B[x]
,
cantor-to-int-uniform-continuity,
real-continuity1,
real-continuity4,
member: t ∈ T
Lemmas referenced :
cantor-to-int-uniform-continuity,
real-continuity1,
real-continuity4,
lifting-strict-callbyvalue,
strict4-spread
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}
(\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} ((f x) = (f y)))
\mLeftarrow{}{}\mRightarrow{} \mforall{}k:\mBbbN{}\msupplus{}
\mexists{}d:\{d:\mBbbR{}| r0 < d\}
\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((|x - y| \mleq{} d) {}\mRightarrow{} (|(f x) - f y| \mleq{} (r1/r(k)))))
supposing a < b
Date html generated:
2016_05_18-AM-11_13_11
Last ObjectModification:
2016_01_17-AM-00_16_29
Theory : reals
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