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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