Nuprl Lemma : intermediate-value-lemma
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
∀c:{c:ℝ| (f(a) ≤ c) ∧ (c ≤ f(b))}
((∃d:{d:ℝ| (r0 ≤ d) ∧ (d < r1)}
∀a1:{a1:ℝ| (a1 ∈ [a, b]) ∧ (f(a1) ≤ c)} . ∀b1:{b1:ℝ| (b1 ∈ [a, b]) ∧ (c ≤ f(b1)) ∧ (a1 ≤ b1)} .
∃a2:{a2:ℝ| (a2 ∈ [a, b]) ∧ (f(a2) ≤ c)} . (∃b2:{b2:ℝ| (b2 ∈ [a, b]) ∧ (c ≤ f(b2))} [((a1 ≤ a2) ∧ (a2 ≤ b2) ∧ (\000Cb2 ≤ b1) ∧ ((b2 - a2) ≤ ((b1 - a1) * d)))]))
⇒ (∃x:ℝ [((x ∈ [a, b]) ∧ (f(x) = c))]))
supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y]))
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,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_apply: x[s],
rfun: I ⟶ℝ,
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
superlevelset: superlevelset(I;f;c),
sublevelset: sublevelset(I;f;c),
top: Top,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
i-member: r ∈ I,
rccint: [l, u],
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
r-ap: f(x),
i-closed: i-closed(I),
isl: isl(x),
outl: outl(x),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
bor: p ∨bq,
bfalse: ff,
assert: ↑b,
true: True,
closed-rset: closed-rset(A),
member-closure: y ∈ closure(A),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B
Lemmas referenced :
req_witness,
sq_stable__rleq,
rleq_weakening_rless,
closures-meet-sq-ext,
sublevelset_wf,
rccint_wf,
superlevelset_wf,
member_rccint_lemma,
istype-void,
rleq_weakening_equal,
rleq_wf,
r-ap_wf,
subtype_rel_sets_simple,
real_wf,
i-member_wf,
rsub_wf,
rmul_wf,
function-is-continuous,
req_wf,
int-to-real_wf,
rless_wf,
rfun_wf,
sublevelset-closed,
sq_stable__and,
le_witness_for_triv,
istype-nat,
converges-to_wf,
subtype_rel_self,
superlevelset-closed,
rleq_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
independent_functionElimination,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
rename,
setElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productElimination,
dependent_set_memberEquality_alt,
productIsType,
universeIsType,
dependent_set_memberFormation_alt,
setIsType,
productEquality,
functionIsType,
natural_numberEquality,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a < b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
\mforall{}c:\{c:\mBbbR{}| (f(a) \mleq{} c) \mwedge{} (c \mleq{} f(b))\}
((\mexists{}d:\{d:\mBbbR{}| (r0 \mleq{} d) \mwedge{} (d < r1)\}
\mforall{}a1:\{a1:\mBbbR{}| (a1 \mmember{} [a, b]) \mwedge{} (f(a1) \mleq{} c)\} . \mforall{}b1:\{b1:\mBbbR{}| (b1 \mmember{} [a, b]) \mwedge{} (c \mleq{} f(b1)) \mwedge{} (a1 \mleq{} b1)\}\000C .
\mexists{}a2:\{a2:\mBbbR{}| (a2 \mmember{} [a, b]) \mwedge{} (f(a2) \mleq{} c)\}
(\mexists{}b2:\{b2:\mBbbR{}| (b2 \mmember{} [a, b]) \mwedge{} (c \mleq{} f(b2))\} [((a1 \mleq{} a2)
\mwedge{} (a2 \mleq{} b2)
\mwedge{} (b2 \mleq{} b1)
\mwedge{} ((b2 - a2) \mleq{} ((b1 - a1) * d)))]))
{}\mRightarrow{} (\mexists{}x:\mBbbR{} [((x \mmember{} [a, b]) \mwedge{} (f(x) = c))]))
supposing \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} (f[x] = f[y]))
Date html generated:
2019_10_30-AM-07_48_08
Last ObjectModification:
2019_01_08-PM-11_16_45
Theory : reals
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