Nuprl Lemma : rational-IVT-2
∀a,b:ℤ × ℕ+. ∀f:(ℤ × ℕ+) ⟶ (ℤ × ℕ+).
∀[g:{x:ℝ| x ∈ [ratreal(a), ratreal(b)]} ⟶ ℝ]
∃c:{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 :
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],
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],
pi1: fst(t),
so_lambda: λ2x.t[x],
accelerate: accelerate(k;f),
rational-fun-zero: rational-fun-zero(f;a;b),
rational-IVT-1,
common-limit-squeeze,
rat-zero-cases,
ravg-weak-between,
ravg-dist-when-rleq,
sq_stable__rleq,
iff_weakening_uiff,
rleq_functionality,
req_functionality,
rleq_weakening_equal,
converges-to_functionality,
const-rmul-limit-with-bound,
rinv-exp-converges-ext,
converges-iff-cauchy,
sq-all-large-and,
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-1,
lifting-strict-spread,
istype-void,
strict4-spread,
lifting-strict-callbyvalue,
lifting-strict-decide,
strict4-decide,
lifting-strict-less,
common-limit-squeeze,
rat-zero-cases,
ravg-weak-between,
ravg-dist-when-rleq,
sq_stable__rleq,
iff_weakening_uiff,
rleq_functionality,
req_functionality,
rleq_weakening_equal,
converges-to_functionality,
const-rmul-limit-with-bound,
rinv-exp-converges-ext,
converges-iff-cauchy,
sq-all-large-and
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:\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{}]
\mexists{}c:\{c:\mBbbR{}| c \mmember{} [ratreal(a), ratreal(b)]\} [(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_19
Last ObjectModification:
2019_04_02-AM-09_43_13
Theory : reals
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