Nuprl Lemma : IVT-strictly-increasing-open
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
f[x] strictly-increasing for x ∈ (a, b) ⇒ (∀c:ℝ. ((f(a) < c) ⇒ (c < f(b)) ⇒ (∃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 :
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
r-ap: f(x),
rfun: I ⟶ℝ,
rooint: (l, u),
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_apply: x[s],
rfun: I ⟶ℝ,
subinterval: I ⊆ J ,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
subtype_rel: A ⊆r B,
i-member: r ∈ I,
rccint: [l, u],
rooint: (l, u),
locally-non-constant: locally-non-constant(f;a;b;c),
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
or: P ∨ Q,
exists: ∃x:A. B[x],
rneq: x ≠ y,
r-ap: f(x)
Lemmas referenced :
IVT-locally-non-constant-open,
req_witness,
member_rooint_lemma,
istype-void,
member_rccint_lemma,
rleq_weakening_rless,
rless_wf,
strictly-increasing-on-interval_wf,
rooint_wf,
subtype_rel_sets_simple,
real_wf,
i-member_wf,
rccint_wf,
req_wf,
rfun_wf,
rless_transitivity1,
rless_transitivity2,
rleq_wf,
rless-cases,
rleq_weakening_equal,
r-ap_wf,
rleq_transitivity,
rneq_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation_alt,
sqequalRule,
lambdaEquality_alt,
isectElimination,
applyEquality,
because_Cache,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
productElimination,
independent_pairFormation,
setElimination,
productIsType,
universeIsType,
setIsType,
functionIsType,
dependent_set_memberEquality_alt,
unionElimination,
dependent_pairFormation_alt,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a < b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
f[x] strictly-increasing for x \mmember{} (a, b)
{}\mRightarrow{} (\mforall{}c:\mBbbR{}. ((f(a) < c) {}\mRightarrow{} (c < f(b)) {}\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-09_09_49
Last ObjectModification:
2019_04_02-PM-00_03_33
Theory : reals
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