Nuprl Lemma : real-continuity3
∀a,b:ℝ.
∀f:[a, b] ⟶ℝ
(∀x,y:{x:ℝ| x ∈ [a, b]} . (f x ≠ f y ⇒ x ≠ 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),
rneq: x ≠ y,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: 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 :
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
sq_exists: ∃x:{A| B[x]},
rless: x < y,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
so_apply: x[s],
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
real: ℝ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
continuous: f[x] continuous for x ∈ I,
rccint: [l, u],
i-approx: i-approx(I;n),
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B
Lemmas referenced :
sq_stable__rless,
set_wf,
icompact_wf,
i-approx_wf,
real-continuity2,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
continuous-rneq,
rccint_wf,
i-member_wf,
all_wf,
exists_wf,
rless_wf,
int-to-real_wf,
rleq_wf,
rabs_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rfun_wf
Rules used in proof :
computeAll,
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
inrFormation,
functionEquality,
independent_functionElimination,
because_Cache,
setEquality,
dependent_set_memberEquality,
independent_pairFormation,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
natural_numberEquality,
minusEquality,
rename,
setElimination,
applyEquality,
isectElimination,
voidElimination,
independent_pairEquality,
productElimination,
lambdaEquality,
sqequalRule,
introduction,
isect_memberFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lemma_by_obid,
cut,
productEquality,
imageElimination,
baseClosed,
imageMemberEquality,
dependent_set_memberFormation
Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}
(\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (f x \mneq{} f y {}\mRightarrow{} x \mneq{} 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 \mleq{} b
Date html generated:
2016_05_18-AM-11_12_27
Last ObjectModification:
2016_01_17-AM-00_18_03
Theory : reals
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