### Nuprl Lemma : derivative-implies-increasing

`∀I:Interval`
`  (iproper(I)`
`  `` (∀f,f':I ⟶ℝ.`
`        (d(f[x])/dx = λx.f'[x] on I`
`        `` f'[x] continuous for x ∈ I`
`        `` (∀x:{x:ℝ| x ∈ I} . (r0 ≤ f'[x]))`
`        `` f[x] increasing for x ∈ I)))`

Proof

Definitions occuring in Statement :  increasing-on-interval: `f[x] increasing for x ∈ I` derivative: `d(f[x])/dx = λz.g[z] on I` continuous: `f[x] continuous for x ∈ I` rfun: `I ⟶ℝ` i-member: `r ∈ I` iproper: `iproper(I)` interval: `Interval` rleq: `x ≤ y` int-to-real: `r(n)` real: `ℝ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` increasing-on-interval: `f[x] increasing for x ∈ I` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` cand: `A c∧ B` sq_stable: `SqStable(P)` squash: `↓T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rfun: `I ⟶ℝ` label: `...\$L... t` guard: `{T}` uiff: `uiff(P;Q)` uimplies: `b supposing a` exists: `∃x:A. B[x]` continuous: `f[x] continuous for x ∈ I` i-approx: `i-approx(I;n)` rccint: `[l, u]` nat_plus: `ℕ+` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` sq_exists: `∃x:{A| B[x]}` rneq: `x ≠ y` or: `P ∨ Q` rless: `x < y` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` subtype_rel: `A ⊆r B` i-member: `r ∈ I` subinterval: `I ⊆ J ` rsub: `x - y` rev_uimplies: `rev_uimplies(P;Q)` real: `ℝ` rge: `x ≥ y`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality setElimination rename hypothesis productElimination independent_functionElimination sqequalRule imageMemberEquality baseClosed imageElimination independent_pairFormation isectElimination because_Cache lambdaEquality setEquality natural_numberEquality applyEquality dependent_set_memberEquality independent_isectElimination isect_memberEquality functionEquality inrFormation unionElimination dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality computeAll independent_pairEquality minusEquality axiomEquality equalityTransitivity equalitySymmetry promote_hyp addLevel levelHypothesis productEquality multiplyEquality addEquality universeEquality isect_memberFormation

Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{}  (\mforall{}f,f':I  {}\mrightarrow{}\mBbbR{}.
(d(f[x])/dx  =  \mlambda{}x.f'[x]  on  I
{}\mRightarrow{}  f'[x]  continuous  for  x  \mmember{}  I
{}\mRightarrow{}  (\mforall{}x:\{x:\mBbbR{}|  x  \mmember{}  I\}  .  (r0  \mleq{}  f'[x]))
{}\mRightarrow{}  f[x]  increasing  for  x  \mmember{}  I)))

Date html generated: 2017_10_03-PM-00_27_37
Last ObjectModification: 2017_07_28-AM-08_42_09

Theory : reals

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