Nuprl Lemma : rational_fun_zero_wf
∀a,b:ℝ. ∀f:(ℤ × ℕ+) ⟶ (ℤ × ℕ+).
∀[g:{x:ℝ| x ∈ [a, b]} ⟶ ℝ]
rational_fun_zero(f;a;b) ∈ {c:ℝ| (c ∈ (a, b)) ∧ (g[c] = r0)}
supposing (a < b)
∧ ((g[a] * g[b]) < r0)
∧ (∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (g[x] = g[y])))
∧ (∀r:ℤ × ℕ+. ((ratreal(r) ∈ [a, b]) ⇒ (g[ratreal(r)] = ratreal(f[r]))))
Proof
Definitions occuring in Statement :
rational_fun_zero: rational_fun_zero(f;a;b),
ratreal: ratreal(r),
rooint: (l, u),
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
req: x = y,
rmul: a * b,
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,
rational-IVT-ext,
rational_fun_zero: rational_fun_zero(f;a;b),
mu-ge: mu-ge(f;n),
ifthenelse: if b then t else f fi ,
band: p ∧b q,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
and: P ∧ Q,
top: Top,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
i-member: r ∈ I,
rccint: [l, u],
rooint: (l, u),
so_lambda: λ2x.t[x],
prop: ℙ,
cand: A c∧ B,
guard: {T},
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
rational-IVT-ext,
member_rccint_lemma,
istype-void,
member_rooint_lemma,
real_wf,
subtype_rel_self,
all_wf,
uall_wf,
rleq_wf,
isect_wf,
rless_wf,
rmul_wf,
rleq_weakening_equal,
rleq_weakening_rless,
int-to-real_wf,
req_wf,
nat_plus_wf,
ratreal_wf,
sq_exists_wf,
subtype_rel_sets,
subtype_rel_sets_simple,
i-member_wf,
rccint_wf,
sq_stable__rless,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
applyEquality,
introduction,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
functionExtensionality,
hypothesisEquality,
because_Cache,
isectElimination,
functionEquality,
lambdaEquality_alt,
setEquality,
productEquality,
inhabitedIsType,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
productIsType,
universeIsType,
closedConclusion,
natural_numberEquality,
setIsType,
intEquality,
functionIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:(\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}) {}\mrightarrow{} (\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}).
\mforall{}[g:\{x:\mBbbR{}| x \mmember{} [a, b]\} {}\mrightarrow{} \mBbbR{}]
rational\_fun\_zero(f;a;b) \mmember{} \{c:\mBbbR{}| (c \mmember{} (a, b)) \mwedge{} (g[c] = r0)\}
supposing (a < b)
\mwedge{} ((g[a] * g[b]) < r0)
\mwedge{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} (g[x] = g[y])))
\mwedge{} (\mforall{}r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. ((ratreal(r) \mmember{} [a, b]) {}\mRightarrow{} (g[ratreal(r)] = ratreal(f[r]))))
Date html generated:
2019_10_30-AM-10_05_39
Last ObjectModification:
2019_01_14-PM-00_34_14
Theory : reals
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