Nuprl Lemma : real-continuity2
∀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],
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),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
or: P ∨ Q,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rfun: I ⟶ℝ,
real-fun: real-fun(f;a;b),
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],
real-sfun: real-sfun(f;a;b),
real-cont: real-cont(f;a;b)
Lemmas referenced :
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformless_wf,
full-omega-unsat,
nat_plus_properties,
req_weakening,
rneq_functionality,
rneq_wf,
not-rneq,
req_wf,
i-member_wf,
rleq_wf,
rccint_wf,
rfun_wf,
real-sfun_wf,
nat_plus_wf,
real_wf,
rsub_wf,
less_than'_wf,
real-continuity-ext
Rules used in proof :
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
imageElimination,
unionElimination,
independent_functionElimination,
because_Cache,
setEquality,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
natural_numberEquality,
minusEquality,
rename,
setElimination,
applyEquality,
isectElimination,
voidElimination,
independent_pairEquality,
productElimination,
lambdaEquality,
isect_memberFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
extract_by_obid,
introduction,
cut,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
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))
{}\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:
2018_05_22-PM-02_11_26
Last ObjectModification:
2018_05_21-AM-00_27_05
Theory : reals
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