Nuprl Lemma : IVT-from-connected
∀u,v:ℝ. ∀f:[u, v] ⟶ℝ.
((u ≤ v)
⇒ (∀x,y:{x:ℝ| x ∈ [u, v]} . ((x = y) ⇒ (f(x) = f(y))))
⇒ (f(u) < r0)
⇒ (r0 < f(v))
⇒ (∀e:{e:ℝ| r0 < e} . ∃c:{c:ℝ| c ∈ [u, v]} . (|f(c)| < e)))
Proof
Definitions occuring in Statement :
r-ap: f(x),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
or: P ∨ Q,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rgt: x > y,
rge: x ≥ y,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
cand: A c∧ B,
exists: ∃x:A. B[x],
guard: {T},
squash: ↓T,
sq_stable: SqStable(P),
top: Top,
and: P ∧ Q,
rccint: [l, u],
i-member: r ∈ I,
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
connected: Connected(X),
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rabs_wf,
rabs-rless-iff,
rless-cases,
real_term_value_minus_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rless_functionality_wrt_implies,
req-iff-rsub-is-0,
itermMinus_wf,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
rsub_wf,
rless-implies-rless,
rleq_weakening_rless,
rfun_wf,
rleq_wf,
req_wf,
all_wf,
rleq_weakening_equal,
int-to-real_wf,
set_wf,
rless_transitivity1,
req_inversion,
sq_stable__i-member,
rleq_weakening,
rless_transitivity2,
sq_stable__rleq,
member_rccint_lemma,
sq_stable__rless,
rminus_wf,
r-ap_wf,
rless_wf,
real_wf,
rccint_wf,
i-member_wf,
closed-interval-connected
Rules used in proof :
inlFormation,
inrFormation,
unionElimination,
intEquality,
int_eqEquality,
approximateComputation,
productEquality,
dependent_set_memberEquality,
dependent_pairFormation,
functionEquality,
natural_numberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairFormation,
independent_functionElimination,
because_Cache,
setEquality,
independent_isectElimination,
rename,
setElimination,
lambdaEquality,
sqequalRule,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}u,v:\mBbbR{}. \mforall{}f:[u, v] {}\mrightarrow{}\mBbbR{}.
((u \mleq{} v)
{}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [u, v]\} . ((x = y) {}\mRightarrow{} (f(x) = f(y))))
{}\mRightarrow{} (f(u) < r0)
{}\mRightarrow{} (r0 < f(v))
{}\mRightarrow{} (\mforall{}e:\{e:\mBbbR{}| r0 < e\} . \mexists{}c:\{c:\mBbbR{}| c \mmember{} [u, v]\} . (|f(c)| < e)))
Date html generated:
2018_05_22-PM-02_16_48
Last ObjectModification:
2018_05_21-AM-00_31_50
Theory : reals
Home
Index