Nuprl Lemma : non-zero-deriv-non-constant

`∀a,b:ℝ.`
`  ((a < b)`
`  `` (∀f,f':[a, b] ⟶ℝ.`
`        (d(f(x))/dx = λx.f'(x) on [a, b]`
`        `` (∃z:{z:ℝ| z ∈ [a, b]} . f'(z) ≠ r0)`
`        `` (∀c:ℝ. ∃z:{z:ℝ| z ∈ [a, b]} . f(z) ≠ c))))`

Proof

Definitions occuring in Statement :  derivative: `d(f[x])/dx = λz.g[z] on I` r-ap: `f(x)` rfun: `I ⟶ℝ` rccint: `[l, u]` i-member: `r ∈ I` rneq: `x ≠ y` rless: `x < y` int-to-real: `r(n)` real: `ℝ` all: `∀x:A. B[x]` exists: `∃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` exists: `∃x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` i-member: `r ∈ I` rccint: `[l, u]` and: `P ∧ Q` top: `Top` sq_stable: `SqStable(P)` squash: `↓T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` label: `...\$L... t` rfun: `I ⟶ℝ` iff: `P `⇐⇒` Q` guard: `{T}` derivative: `d(f[x])/dx = λz.g[z] on I` nat_plus: `ℕ+` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` i-approx: `i-approx(I;n)` iproper: `iproper(I)` i-finite: `i-finite(I)` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` cand: `A c∧ B` sq_exists: `∃x:{A| B[x]}` rev_uimplies: `rev_uimplies(P;Q)` rneq: `x ≠ y` or: `P ∨ Q` rev_implies: `P `` Q` rless: `x < y` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` rge: `x ≥ y` subtype_rel: `A ⊆r B` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` uiff: `uiff(P;Q)` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` rdiv: `(x/y)` rsub: `x - y`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid dependent_functionElimination isectElimination because_Cache hypothesisEquality setElimination rename hypothesis independent_isectElimination sqequalRule independent_pairFormation isect_memberEquality voidElimination voidEquality independent_functionElimination imageMemberEquality baseClosed imageElimination setEquality lambdaEquality natural_numberEquality dependent_set_memberEquality productEquality functionEquality multiplyEquality inrFormation unionElimination dependent_pairFormation int_eqEquality intEquality computeAll applyEquality equalityTransitivity equalitySymmetry minusEquality independent_pairEquality axiomEquality isect_memberFormation inlFormation addEquality universeEquality promote_hyp

Latex:
\mforall{}a,b:\mBbbR{}.
((a  <  b)
{}\mRightarrow{}  (\mforall{}f,f':[a,  b]  {}\mrightarrow{}\mBbbR{}.
(d(f(x))/dx  =  \mlambda{}x.f'(x)  on  [a,  b]
{}\mRightarrow{}  (\mexists{}z:\{z:\mBbbR{}|  z  \mmember{}  [a,  b]\}  .  f'(z)  \mneq{}  r0)
{}\mRightarrow{}  (\mforall{}c:\mBbbR{}.  \mexists{}z:\{z:\mBbbR{}|  z  \mmember{}  [a,  b]\}  .  f(z)  \mneq{}  c))))

Date html generated: 2017_10_03-PM-00_34_57
Last ObjectModification: 2017_07_28-AM-08_43_35

Theory : reals

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