Nuprl Lemma : rational-fun-zero_wf
∀a,b:ℤ × ℕ+. ∀f:(ℤ × ℕ+) ⟶ (ℤ × ℕ+).
∀[g:{x:ℝ| x ∈ [ratreal(a), ratreal(b)]} ⟶ ℝ]
rational-fun-zero(f;a;b) ∈ {c:ℝ| (c ∈ [ratreal(a), ratreal(b)]) ∧ (g[c] = r0)}
supposing (ratreal(a) ≤ ratreal(b))
∧ (ratreal(f[a]) ≤ r0)
∧ (r0 ≤ ratreal(f[b]))
∧ (∀x,y:{x:ℝ| x ∈ [ratreal(a), ratreal(b)]} . ((x = y) ⇒ (g[x] = g[y])))
∧ (∀r:ℤ × ℕ+. ((ratreal(r) ∈ [ratreal(a), ratreal(b)]) ⇒ (g[ratreal(r)] = ratreal(f[r]))))
Proof
Definitions occuring in Statement :
rational-fun-zero: rational-fun-zero(f;a;b),
ratreal: ratreal(r),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
sq_exists: ∃x:A [B[x]],
rational-fun-zero: rational-fun-zero(f;a;b),
rational-IVT-2,
top: Top,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
rleq_wf,
ratreal_wf,
int-to-real_wf,
req_wf,
i-member_wf,
rccint_wf,
real_wf,
istype-int,
nat_plus_wf,
rational-IVT-2,
subtype_rel_self,
sq_exists_wf,
member_rccint_lemma,
istype-void,
subtype_rel_sets,
sq_stable__rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
productIsType,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
natural_numberEquality,
functionIsType,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality_alt,
setIsType,
inhabitedIsType,
instantiate,
functionEquality,
productEquality,
intEquality,
isectEquality,
setEquality,
lambdaEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
closedConclusion,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype
Latex:
\mforall{}a,b:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. \mforall{}f:(\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}) {}\mrightarrow{} (\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}).
\mforall{}[g:\{x:\mBbbR{}| x \mmember{} [ratreal(a), ratreal(b)]\} {}\mrightarrow{} \mBbbR{}]
rational-fun-zero(f;a;b) \mmember{} \{c:\mBbbR{}| (c \mmember{} [ratreal(a), ratreal(b)]) \mwedge{} (g[c] = r0)\}
supposing (ratreal(a) \mleq{} ratreal(b))
\mwedge{} (ratreal(f[a]) \mleq{} r0)
\mwedge{} (r0 \mleq{} ratreal(f[b]))
\mwedge{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [ratreal(a), ratreal(b)]\} . ((x = y) {}\mRightarrow{} (g[x] = g[y])))
\mwedge{} (\mforall{}r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. ((ratreal(r) \mmember{} [ratreal(a), ratreal(b)]) {}\mRightarrow{} (g[ratreal(r)] = ratreal(f[r]))))
Date html generated:
2019_10_30-AM-10_01_54
Last ObjectModification:
2019_01_11-PM-02_36_23
Theory : reals
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