Nuprl Lemma : rational-IVT-ext
∀a,b:ℝ. ∀f:(ℤ × ℕ+) ⟶ (ℤ × ℕ+).
∀[g:{x:ℝ| x ∈ [a, b]} ⟶ ℝ]
∃c:{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 :
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],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
member: t ∈ T,
so_apply: x[s],
subtract: n - m,
pi1: fst(t),
ratreal: ratreal(r),
rat-to-real: r(a/b),
int-rdiv: (a)/k1,
int-to-real: r(n),
rlessw: rlessw(x;y),
quick-find: quick-find(p;n),
ifthenelse: if b then t else f fi ,
int-rat-mul: int-rat-mul(n;x),
so_lambda: λ2x.t[x],
rational_fun_zero: rational_fun_zero(f;a;b),
rational-IVT,
sq_stable__and,
sq_stable__rless,
rleq_functionality,
rat-zero-cases,
sq_stable__rleq,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a
Lemmas referenced :
rational-IVT,
lifting-strict-callbyvalue,
istype-void,
strict4-spread,
lifting-strict-decide,
strict4-decide,
lifting-strict-spread,
lifting-strict-less,
sq_stable__and,
sq_stable__rless,
rleq_functionality,
rat-zero-cases,
sq_stable__rleq
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
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{}]
\mexists{}c:\{c:\mBbbR{}| c \mmember{} (a, b)\} [(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_07
Last ObjectModification:
2019_01_14-PM-00_30_53
Theory : reals
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